Semi-advanced mathematics

## Linear form

FromWikipedia: Linear form

In linear algebra, a **linear form** (also known as a **linear functional**, a **one-form**, or a **covector**) is a linear map from a vector space to its field of scalars. In **ℝ***n*, if vectors are represented as column vectors, then linear functionals are represented as row vectors, and their action on vectors is given by the matrix product with the row vector on the left and the column vector on the right.

Suppose that vectors in the real coordinate space **R***n* are represented as column vectors

x
→

=

[

x
1

⋮

x
n

]

.

{\displaystyle \vec{x} = \begin{bmatrix}x\_1\\ \vdots\\ x\_n\end{bmatrix}.}

For each row vector [*a*1 … *a**n*] there is a linear functional *f* defined by

f
(

x
→

# )

a
1

x
1

⋯

a
n

x
n

,

{\displaystyle f(\vec{x}) = a\_1x\_1 + \cdots + a\_n x\_n,}

and each linear functional can be expressed in this form.

This can be interpreted as either the matrix product or the dot product of the row vector [*a*1 ... *a**n*] and the column vector

x
→

{\displaystyle \vec{x}}

:

f
(

x
→

# )

[

a
1

…

a
n

]

[

x
1

⋮

x
n

]

.

{\displaystyle f(\vec{x}) = \left[a\_1 \dots a\_n\right] \begin{bmatrix}x\_1\\ \vdots\\ x\_n\end{bmatrix}.}

In finite dimensions, a linear functional can be visualized in terms of its level sets, the sets of vectors which map to a given value.  In three dimensions, the level sets of a linear functional are a family of mutually parallel planes; in higher dimensions, they are parallel hyperplanes.  This method of visualizing linear functionals is sometimes introduced in general relativity texts, such as *Gravitation* by Misner, Thorne & Wheeler (1973).

Every non-degenerate bilinear form on a finite-dimensional vector space *V* induces an isomorphism *V* → *V*∗ : *v* ↦ *v*∗ such that

v
∗

(
w
)
:=
⟨
v
,
w
⟩

∀
w
∈
V
,

{\displaystyle v^\*(w) := \langle v, w\rangle \quad \forall w \in V ,}

where the bilinear form on *V* is denoted ⟨ , ⟩ (for instance, in Euclidean space ⟨*v*, *w*⟩ = *v* ⋅ *w* is the dot product of *v* and *w*).

The inverse isomorphism is *V*∗ → *V* : *v*∗ ↦ *v*, where *v* is the unique element of *V* such that

⟨
v
,
w
# ⟩

v
∗

(
w
)

∀
w
∈
V
.

{\displaystyle \langle v, w\rangle = v^\*(w) \quad \forall w \in V .}

The above defined vector *v*∗ ∈ *V*∗ is said to be the **dual vector** of *v* ∈ *V*.

In an infinite dimensional Hilbert space, analogous results hold by the Riesz representation theorem.  There is a mapping *V* → *V*∗ into the *continuous dual space* *V*∗.  However, this mapping is antilinear rather than linear.

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### Dual space

See also: Wikipedia: Dual system
From Wikipedia: Dual space

In mathematics, any vector space *V* has a corresponding **dual vector space** (or just **dual space** for short) consisting of all linear functionals (covectors) on *V*, together with the vector space structure of pointwise addition and scalar multiplication by constants.

The pairing of a functional *φ* in the dual space *V*∗ and an element *x* of *V* is sometimes denoted by a bracket: *φ*(*x*) = [*x*,*φ*]
or *φ*(*x*) = ⟨*φ*,*x*⟩. This pairing defines a nondegenerate  bilinear mapping ⟨·,·⟩ : *V*∗ × *V* → *F* called the  natural pairing.

If *V* is finite-dimensional, then *V*∗ has the same dimension as *V*. Given a basis {**e**1, ..., **e***n*}  in *V*, it is possible to construct a specific basis in *V*∗, called the **dual basis**. This dual basis is a set {**e**1, ..., **e***n*}  of linear functionals on *V*, defined by the relation

e

i

(

e

j

# )

δ

j

i

where

δ

i
j

=

{

0

if
i
≠
j
,

1

if
# i
j
.

{\displaystyle \mathbf{e}^i(\mathbf{e}\_j) = \delta^{i}\_{j} \quad \text{where} \quad \delta\_{ij} = \begin{cases}
0 &\text{if } i \neq j, \\
1 &\text{if } i=j. \end{cases}}

where

δ
j
i

{\displaystyle \delta^i\_j}

 is the Kronecker delta symbol. This property is referred to as *biorthogonality property*.

For example, if *V* is **R**2, let its basis be chosen as {**e**1 = (1/2, 1/2), **e**2 = (0, 1)}. Note that the basis vectors are not orthogonal to each other. Then, **e**1 and **e**2 are one-forms (functions that map a vector to a scalar) such that **e**1(**e**1) = 1, **e**1(**e**2) = 0, **e**2(**e**1) = 0, and **e**2(**e**2) = 1. (Note: The superscript here is the index, not an exponent.) We can express this system of equations using matrix notation as

[

e

11

e

12

e

21

e

22

]

[

e

11

e

21

e

12

e

22

]

=

[

1

0

0

1

]

.

{\displaystyle
\begin{bmatrix}
e\_{11} & e\_{12} \\
e\_{21} & e\_{22}
\end{bmatrix}
\begin{bmatrix}
e^{11} & e^{21} \\
e^{12} & e^{22}
# \end{bmatrix}
\begin{bmatrix}
1 & 0 \\
0 & 1
\end{bmatrix}.
}

Solving this equation, we find the dual basis to be {**e**1 = (2, 0), **e**2 = (−1, 1)}. Recalling that **e**1 and **e**2 are functionals, we can rewrite them as **e**1(*x*, *y*) = 2*x* and **e**2(*x*, *y*) = −*x* + *y*.

In general, when *V* is **R***n*, if **E** = (**e**1, ..., **e***n*) is a matrix whose columns are the basis vectors and **Ê** = (**e**1, ..., **e***n*) is a matrix whose columns are the dual basis vectors, then

E
T

E
^

=

I
n

,

{\displaystyle E^T \hat{E} = I\_n,}

where *I**n* is an identity matrix of order *n*. The biorthogonality property of these two basis sets allows us to represent any point **x** in *V* as

x

=

∑
i

⟨

x

,

e

i

⟩

e

i

=

∑
i

⟨

x

,

e

i

⟩

e

i

,

{\displaystyle \mathbf{x} = \sum\_i \langle\mathbf{x},\mathbf{e}^i \rangle \mathbf{e}\_i = \sum\_i \langle \mathbf{x}, \mathbf{e}\_i \rangle \mathbf{e}^i,}

even when the basis vectors are not orthogonal to each other.

Strictly speaking, the above statement only makes sense once the inner product

⟨
⋅
,
⋅
⟩

{\displaystyle \langle\cdot , \cdot\rangle}

 and the corresponding duality pairing are introduced.

Back to top

## Einstein notation

From Wikipedia:Einstein notation:

According to this convention, when an index variable appears twice in a single term and is not otherwise defined (see free and bound variables), it implies summation of that term over all the values of the index. So where the indices can range over the set {1, 2, 3},

# i
1

3

c
i

x
i

=

c
1

x
1

c
2

x
2

c
3

x
3

{\displaystyle y = \sum\_{i = 1}^3 c\_i x^i = c\_1 x^1 + c\_2 x^2 + c\_3 x^3}

is simplified by the convention to:

# y

c
i

x
i

{\displaystyle y = c\_i x^i}

.
Any tensor **T** in *V* ⊗ *V* can be written as:

T

=

T

i
j

e

i
j

{\displaystyle \mathbf{T} = T^{ij}\mathbf{e}\_{ij}}

.
The matrix product of two matrices *Aij* and *Bjk* is:

C

i
k

=
(

A

B

)

i
k

=

∑

# j
1

N

A

i
j

B

j
k

{\displaystyle \mathbf{C}\_{ik} = (\mathbf{A} \mathbf{B})\_{ik} =\sum\_{j=1}^N A\_{ij} B\_{jk}}

equivalent to

C
i

k

=

A
i

j

B
j

k

{\displaystyle C^i{}\_k = A^i{}\_j B^j{}\_k }

Back to top

### Metric contraction

Choosing an arbitrary vector and reducing our units down from km to m:

v

=

v
i

e
i

=
1

km

=
1000

m

{\displaystyle
\begin{align}
\mathbf{v} = v^i e\_i &= 1 \, \text{km} \\
&= 1000 \, \text{m}
\end{align}
}

We can see that

v
i

{\displaystyle v^i}

 went up from 1 to 1000 and is therefore contravariant.

But

e
i

{\displaystyle e\_i}

 went down down from km to m and is therefore covariant.

Upper indices represent contravariant components.

Lower indices represent covariant components.

From Wikipedia:Covariance and contravariance of vectors

Covectors can be represented as row vectors.

c

=

c
i

e
i

=

[

c
1

c
2

⋯

c
n

]

[

e
1

e
2

⋮

e
n

]

=

[

c
1

e
1

c
2

e
2

⋯

c
n

e
n

]

=

c
1

e
1

c
2

e
2

⋯

c
n

e
n

{\displaystyle
\mathbf{c} = c\_i e^i =
\begin{bmatrix}c\_1 & c\_2 & \cdots & c\_n\end{bmatrix}
\begin{bmatrix}e^1 \\ e^2 \\ \vdots \\ e^n\end{bmatrix} =
\begin{bmatrix}c\_1 e^1 & c\_2 e^2 & \cdots & c\_n e^n \end{bmatrix} =
c\_1 e^1 + c\_2 e^2 + \cdots + c\_n e^n
}

Vectors can be represented as column vectors.

v

=

v
i

e
i

=

[

v
1

v
2

⋮

v
n

]

[

e
1

e
2

⋯

e
n

]

=

[

v
1

e
1

v
2

e
2

⋮

v
n

e
n

]

=

v
1

e
1

v
2

e
2

⋯

v
n

e
n

{\displaystyle
\mathbf{v} = v^i e\_i =
\begin{bmatrix}v^1 \\ v^2 \\ \vdots \\ v^n\end{bmatrix}
\begin{bmatrix}e\_1 & e\_2 & \cdots & e\_n\end{bmatrix} =
\begin{bmatrix}v^1 e\_1 \\ v^2 e\_2 \\ \vdots \\ v^n e\_n \end{bmatrix} =
v^1 e\_1 + v^2 e\_2 + \cdots + v^n e\_n
\qquad
}

To simplify writing column vectors in-line with other text, sometimes they are written as row vectors with the transpose operation applied to them.

# v

[

v
1

v
2

…

v
n

]

T

{\displaystyle
\boldsymbol{v} =
\begin{bmatrix}
 v^1 \; v^2 \; \dots \; v^n
\end{bmatrix}^{\rm {\color{red} T}}
}

We ordinarily think of the product of a vector and a covector as a scalar. In Einstein notation:

v

⋅

c

=

v

i

c

i

=

v
1

c
1

v
2

c
2

v
3

c
3

{\displaystyle
\mathbf{v} \cdot \mathbf{c} =
v^{\color{red}i} c\_{\color{red}i} =
v^1 c\_1 + v^2 c\_2 + v^3 c\_3
}

This is an example of tensor contraction. The important thing to note is that by the Einstein summation convention when both vectors use the same index *i* then tensor contraction occurs.

But the product of a vector and a covector can also be a (1,1) tensor. In Einstein notation:

A
i

j

=

v

i

c

j

=
(
v
c

)
i

j

{\displaystyle
A^i{}\_j =
v^{\color{red}i} c\_{\color{red}j} =
(v c)^i{}\_j
}

The important thing to note is that by the Einstein summation convention when the vectors use different indices *i* and *j* then there is no tensor contraction.

From Wikipedia:Tensor contraction

Contraction on a pair of indices that are either both contravariant or both covariant is not possible in general. However, in the presence of an inner product (also known as a metric) *g*, such contractions are possible. One uses the metric to raise or lower one of the indices, as needed, and then one uses the usual operation of contraction. The combined operation is known as *metric contraction*.

g

i
j

A
j

=

g

i

j

A

j

=

A
i

{\displaystyle
g^{ij}A\_j =
\cancel{g}^{i \cancel{j}}A\_\cancel{j} =
A^i \,
}

From Wikipedia: Raising and lowering indices

The covariant 4-position is given by

X
μ

=
(
−
c
t
,
x
,
y
,
z
)

{\displaystyle X\_\mu = (-ct, x, y, z)}

with components:

X
0

=
−
c
t
,

X
1

=
x
,

X
2

=
y
,

X
3

=
z

{\displaystyle X\_0 = -ct, \quad X\_1 = x, \quad X\_2 = y, \quad X\_3 = z}

(where x,y,z are the usual Cartesian coordinates).

The Minkowski metric tensor with signature (− + + +) is defined as

η

μ
ν

=

η

μ
ν

=

(

−
1

0

0

0

0

1

0

0

0

0

1

0

0

0

0

1

)

{\displaystyle \eta\_{\mu \nu} = \eta^{\mu \nu} = \begin{pmatrix}
 -1 & 0 & 0 & 0 \\
 0 & 1 & 0 & 0 \\
 0 & 0 & 1 & 0 \\
 0 & 0 & 0 & 1
\end{pmatrix} }

To raise the index, multiply by the tensor and contract:

X
λ

=

η

λ
μ

X
μ

{\displaystyle
X^\lambda =
\eta^{\lambda\mu}X\_\mu
}

So the index-raised contravariant 4-position is:

X
μ

=
(
c
t
,
x
,
y
,
z
)

.

{\displaystyle X^\mu = (ct, x, y, z)\,.}

Metric contraction can be used to find the dot product of 2 vectors

u

⋅

v

=

[

u
1

u
2

u
3

]

[

1

0

0

0

1

0

0

0

1

]

[

v
1

v
2

v
3

]

=

[

u
1

u
2

u
3

]

[

v
1

v
2

v
3

]

=

u
1

v
1

u
2

v
2

u
3

v
3

{\displaystyle
\begin{align}
\mathbf{u} \cdot \mathbf{v} &=
\begin{bmatrix}
 u^1 \\
 u^2 \\
 u^3
\end{bmatrix}
 \begin{bmatrix}
 1 & 0 & 0 \\
 0 & 1 & 0 \\
 0 & 0 & 1 \\
 \end{bmatrix}
\begin{bmatrix}
 v^1 \\
 v^2 \\
 v^3
\end{bmatrix} \\

&=
\begin{bmatrix}
 u^1 \\
 u^2 \\
 u^3
\end{bmatrix}
\begin{bmatrix}
 v\_1 &
 v\_2 &
 v\_3
\end{bmatrix} \\

&=
u^1 v\_1 + u^2 v\_2 + u^3 v\_3
\end{align}
}

which is actually

u

⋅

v

=

[

u
1

e
1

u
2

e
2

u
3

e
3

]

[

1

1

e
1

⊗

1

e
1

0

1

e
1

⊗

1

e
2

0

1

e
1

⊗

1

e
3

0

1

e
2

⊗

1

e
1

1

1

e
2

⊗

1

e
2

0

1

e
2

⊗

1

e
3

0

1

e
3

⊗

1

e
1

0

1

e
3

⊗

1

e
2

1

1

e
3

⊗

1

e
3

]

[

v
1

e
1

v
2

e
2

v
3

e
3

]

=

[

u
1

e
1

u
2

e
2

u
3

e
3

]

[

1

e
1

⊗

e
1

0

e
1

⊗

e
2

0

e
1

⊗

e
3

0

e
2

⊗

e
1

1

e
2

⊗

e
2

0

e
2

⊗

e
3

0

e
3

⊗

e
1

0

e
3

⊗

e
2

1

e
3

⊗

e
3

]

[

v
1

e
1

v
2

e
2

v
3

e
3

]

=

[

u
1

e
1

u
2

e
2

u
3

e
3

]

[

v
1

e
1

v
2

e
2

v
3

e
3

]

=

u
1

v
1

e
1

e
1

u
2

v
2

e
2

e
2

u
3

v
3

e
2

e
2

=

u
1

v
1

u
2

v
2

u
3

v
3

{\displaystyle
\begin{align}
\mathbf{u} \cdot \mathbf{v} &=

\begin{bmatrix}
 u^1 e\_1 \\
 u^2 e\_2 \\
 u^3 e\_3
\end{bmatrix}
 \begin{bmatrix}
 1 \, \frac{1}{e\_1}\otimes\frac{1}{e\_1} && 0 \, \frac{1}{e\_1}\otimes\frac{1}{e\_2} && 0 \, \frac{1}{e\_1}\otimes\frac{1}{e\_3} \\ \\
 0 \, \frac{1}{e\_2}\otimes\frac{1}{e\_1} && 1 \, \frac{1}{e\_2}\otimes\frac{1}{e\_2} && 0 \, \frac{1}{e\_2}\otimes\frac{1}{e\_3} \\ \\
 0 \, \frac{1}{e\_3}\otimes\frac{1}{e\_1} && 0 \, \frac{1}{e\_3}\otimes\frac{1}{e\_2} && 1 \, \frac{1}{e\_3}\otimes\frac{1}{e\_3} \\ \\
 \end{bmatrix}
\begin{bmatrix}
 v^1 e\_1 \\
 v^2 e\_2 \\
 v^3 e\_3
\end{bmatrix} \\

&=
\begin{bmatrix}
 u^1 e\_1 \\
 u^2 e\_2 \\
 u^3 e\_3
\end{bmatrix}
 \begin{bmatrix}
 1 \, e^1 \otimes e^1 & 0 \, e^1 \otimes e^2 & 0 \, e^1 \otimes e^3 \\
 0 \, e^2 \otimes e^1 & 1 \, e^2 \otimes e^2 & 0 \, e^2 \otimes e^3 \\
 0 \, e^3 \otimes e^1 & 0 \, e^3 \otimes e^2 & 1 \, e^3 \otimes e^3 \\
 \end{bmatrix}
\begin{bmatrix}
 v^1 e\_1 \\
 v^2 e\_2 \\
 v^3 e\_3
\end{bmatrix} \\

&=
\begin{bmatrix}
 u^1 e\_1 \\
 u^2 e\_2 \\
 u^3 e\_3
\end{bmatrix}
\begin{bmatrix}
 v\_1 \, e^1 &
 v\_2 \, e^2 &
 v\_3 \, e^3
\end{bmatrix} \\

&=
u^1 v\_1 \, e\_1 e^1 + u^2 v\_2 \, e\_2 e^2 + u^3 v\_3 \, e\_2 e^2 \\
&= u^1 v\_1 + u^2 v\_2 + u^3 v\_3
\end{align}
}

Note that

1

e
1

⊗

1

e
2

e
1

=
0
≠

1

e
2

{\displaystyle
\frac{1}{e\_1} \otimes \frac{1}{e\_2} e\_1 = 0 \neq \frac{1}{e\_2}
}

Not only can the dot product be written using an (0,2) tensor but the dot product is sometimes said to BE an (0,2) tensor which is said to be a map from two vectors to a scalar or a map from a vector to a covector.

u

⋅

v

=
β
(

u

,

v

)

{\displaystyle
\mathbf{u} \cdot \mathbf{v} = \beta(\mathbf{u},\mathbf{v})
}

Back to top

### Musical isomorphism

See also: Wikipedia: Raising and lowering indices
From Wikipedia: Musical isomorphism

The term *musical* refers to the use of the symbols

♭

{\displaystyle \flat}

 (flat) and

♯

{\displaystyle \sharp }

 (sharp).

We may express the pseudo-Riemannian metric (which is a 2-covariant tensor field that is symmetric and nondegenerate) as *g* = *g**ij* **e***i* ⊗ **e***j*

Given a vector field *X* = *X**i* **e***i* , we define its **flat** by

X
♭

:=

g

i
j

X
i

e

j

=

X
j

e

j

.

{\displaystyle X^\flat := g\_{ij} X^i \, \mathbf{e}^j=X\_j \, \mathbf{e}^j.}

This is referred to as "**lowering an index**". Using the traditional diamond bracket notation for the inner product defined by *g*, we obtain the somewhat more transparent relation

X
♭

(
Y
# )
⟨
X
,
Y
⟩

{\displaystyle X^\flat (Y) = \langle X, Y \rangle}

for any vector fields X and Y.

In the same way, given a covector field *ω* = *ω**i* **e***i* , we define its **sharp** by

ω
♯

:=

g

i
j

ω
i

e

j

=

ω
j

e

j

,

{\displaystyle \omega^\sharp := g^{ij} \omega\_i \mathbf{e}\_j = \omega^j \mathbf{e}\_j ,}

where *g**ij* are the components of the inverse metric tensor (given by the entries of the inverse matrix to *g**ij* ). Taking the sharp of a covector field is referred to as "**raising an index**". In inner product notation, this reads

ω
(
Y
# )

⟨

ω
♯

,
Y

⟩

,

{\displaystyle \omega(Y) = \bigl \langle \omega^\sharp, Y \bigr \rangle,}

for any covector field ω and any vector field Y.

In the context of exterior algebra ♭ maps *p*-vectors to *p*-covectors and ♯ maps *p*-covectors to *p*-vectors, all the indices of a totally antisymmetric tensor are simultaneously raised or lowered, and so no index need be indicated:

(
X
∧
…
∧
Z

)
♭

=

X
♭

∧
…
∧

Z
♭

,

(
α
∧
…
∧
γ

)
♯

=

α
♯

∧
…
∧

γ
♯

.

{\displaystyle (X \wedge \ldots \wedge Z)^\flat = X^\flat \wedge \ldots \wedge Z^\flat , \qquad
(\alpha \wedge \ldots \wedge \gamma)^\sharp = \alpha^\sharp \wedge \ldots \wedge \gamma^\sharp .}

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## Interior product

See also: Formula for interior product

The interior product is defined to be the contraction (See Tensor contraction) of a differential form with a vector field.

The interior product is a generalization of the dot product extended to multivectors. It is written as *X*⨼*ω* or *ι**X**ω*.

(

e
x

∧

e
y

)

⌟

e
z

=
0

{\displaystyle
(e\_x \wedge e\_y) \, \lrcorner \, e\_z = 0
}

(

e
1

∧

e
2

∧

e
3

∧

e
4

)

⌟

(

e
1

∧

e
4

# )

e
2

∧

e
3

{\displaystyle
(e\_1 \wedge e\_2 \wedge e\_3 \wedge e\_4) \, \lrcorner \, (e\_1 \wedge e\_4) = e\_2 \wedge e\_3
}

If

u

{\displaystyle \mathbf u}

 is a vector and

α

{\displaystyle \alpha}

 is an r-form and

β

{\displaystyle \beta}

 is an s-form then

u

⌟

(
α
∧
β
# )
(

u

⌟

α
)
∧
β
(
−
1

)
r

α
∧
(

u

⌟

β
)

{\displaystyle
\mathbf{u} \, \lrcorner \, (\alpha \wedge \beta) = (\mathbf{u} \, \lrcorner \, \alpha) \wedge \beta + (-1)^r \alpha \wedge (\mathbf{u} \, \lrcorner \, \beta)
}

If

u

{\displaystyle \mathbf u}

 is a vector and

α

{\displaystyle \alpha}

,

β

{\displaystyle \beta}

, and

γ

{\displaystyle \gamma}

 are one-forms

u

⌟

(
α
)

=

u

⋅
α

{\displaystyle
{\color{red}\mathbf{u} \, \lrcorner \, (\alpha)} = \mathbf{u} \cdot \alpha
}

and

[
l
]
(

u

⌟

(
α
∧
β
)

=

(

u

⌟

α
)

∧
β

−
α
∧
(

u

⌟

β
)

=
(

u

⋅
α
)
⋅
β

−
α
⋅
(

u

⋅
β
)

=
(

u

⋅
α
)
⋅
β

−
(

u

⋅
β
)
⋅
α

{\displaystyle
\begin{matrix}[l]
({\color{blue} \mathbf{u} \, \lrcorner \, (\alpha \wedge \beta)}
&= {\color{red} (\mathbf{u} \, \lrcorner \, \alpha)} \wedge \beta &-
\alpha \wedge (\mathbf{u} \, \lrcorner \, \beta) \\
&= ( \mathbf{u} \cdot \alpha ) \cdot \beta &-
\alpha \cdot ( \mathbf{u} \cdot \beta ) \\
&= ( \mathbf{u} \cdot \alpha ) \cdot \beta &-
( \mathbf{u} \cdot \beta ) \cdot \alpha
\end{matrix}
}

and

[
l
]

u

⌟

(
α
∧
β
∧
γ
)

=

(

u

⌟

(
α
∧
β
)
)

∧
γ

(
α
∧
β
)
∧
(

u

⌟

γ
)

=

(

(

u

⌟

α
)
∧
β

−
α
∧
(

u

⌟

β
)

)

∧
γ

(
α
∧
β
)
∧
(

u

⌟

γ
)

=
(

u

⋅
α
)
⋅
β
∧
γ

−
α
⋅
(

u

⋅
β
)
∧
γ

(
α
∧
β
)
⋅
(

u

⋅
γ
)

=
(

u

⋅
α
)
⋅
β
∧
γ

(

u

⋅
β
)
⋅
γ
∧
α

(

u

⋅
γ
)
⋅
(
α
∧
β
)

{\displaystyle
\begin{matrix}[l]
\mathbf{u} \, \lrcorner \, (\alpha \wedge \beta \wedge \gamma)

&= {\color{blue} (\mathbf{u} \, \lrcorner \, (\alpha \wedge \beta))} \wedge \gamma && +(\alpha \wedge \beta) \wedge (\mathbf{u} \, \lrcorner \, \gamma) \\

&= {\color{blue} ( \, ( \mathbf{u} \, \lrcorner \, \alpha ) \wedge \beta} &
{\color{blue} - \alpha \wedge (\mathbf{u} \, \lrcorner \, \beta) \, ) } \wedge \gamma &+
(\alpha \wedge \beta) \wedge (\mathbf{u} \, \lrcorner \, \gamma) \\

&= ( \mathbf{u} \cdot \alpha ) \cdot \beta \wedge \gamma &-
\alpha \cdot ( \mathbf{u} \cdot \beta ) \wedge \gamma &+
(\alpha \wedge \beta) \cdot (\mathbf{u} \cdot \gamma) \\

&= ( \mathbf{u} \cdot \alpha ) \cdot \beta \wedge \gamma &+
( \mathbf{u} \cdot \beta ) \cdot \gamma \wedge \alpha &+
(\mathbf{u} \cdot \gamma) \cdot (\alpha \wedge \beta)
\end{matrix}
}

If

u

=

u
x

e
x

u
y

e
y

u
z

e
z

{\displaystyle \mathbf{u} = u^x e\_x + u^y e\_y + u^z e\_z }

u

⌟
(

e
x

)

=
(

u
x

e
x

⋅

e
x

)
(

u
y

e
y

⋅

e
x

)
(

u
z

e
z

⋅

e
x

)

=
(

u
x

⋅
1
)
0
0

=

u
x

{\displaystyle
\begin{align}
\mathbf{u} \lrcorner (e^x) &=

(u^x e\_x \cdot e^x) +
(u^y e\_y \cdot e^x) +
(u^z e\_z \cdot e^x) \\ &=

(u^x \cdot 1) +
0 +
0 \\ &=

u^x
\end{align}
}

And

u

⌟
(

e
x

∧

e
y

)

=
(

u
x

e
x

⋅

e
x

∧

e
y

)
(

u
y

e
y

⋅

e
x

∧

e
y

)
(

u
z

e
z

⋅

e
x

∧

e
y

)

=
(

u
x

e
x

⋅

e
x

∧

e
y

)
−
(

u
y

e
y

⋅

e
y

∧

e
x

)
(

u
z

e
z

⋅

e
x

∧

e
y

)

=
(

u
x

e
x

⋅

e
x

)
∧

e
y

−
(

u
y

e
y

⋅

e
y

)
∧

e
x

(

u
z

e
z

⋅

e
x

∧

e
y

)

=
(

u
x

⋅
1
)
⋅

e
y

−
(

u
y

⋅
1
)
⋅

e
x

0

=

u
x

⋅

e
y

−

u
y

⋅

e
x

{\displaystyle
\begin{align}
\mathbf{u} \lrcorner (e^x \wedge e^y ) &=

(u^x e\_x \cdot e^x \wedge e^y) +
(u^y e\_y \cdot e^x \wedge e^y) +
(u^z e\_z \cdot e^x \wedge e^y) \\ &=

(u^x e\_x \cdot e^x \wedge e^y) -
(u^y e\_y \cdot e^y \wedge e^x) +
(u^z e\_z \cdot e^x \wedge e^y) \\ &=

(u^x e\_x \cdot e^x) \wedge e^y -
(u^y e\_y \cdot e^y) \wedge e^x +
(u^z e\_z \cdot e^x \wedge e^y) \\ &=

(u^x \cdot 1) \cdot e^y -
(u^y \cdot 1) \cdot e^x +
0 \\ &=

u^x \cdot e^y -
u^y \cdot e^x
\end{align}
}

And

u

⌟
(

e
x

∧

e
y

∧

e
z

)

=
(

u
x

e
x

⋅

e
x

∧

e
y

∧

e
z

)
(

u
y

e
y

⋅

e
x

∧

e
y

∧

e
z

)
(

u
z

e
z

⋅

e
x

∧

e
y

∧

e
z

)

=
(

u
x

e
x

⋅

e
x

∧

e
y

∧

e
z

)
(

u
y

e
y

⋅

e
y

∧

e
z

∧

e
x

)
(

u
z

e
z

⋅

e
z

∧

e
x

∧

e
y

)

=
(

u
x

e
x

⋅

e
x

)
∧

e
y

∧

e
z

(

u
y

e
y

⋅

e
y

)
∧

e
z

∧

e
x

(

u
z

e
z

⋅

e
z

)
∧

e
x

∧

e
y

=
(

u
x

⋅
1
)
⋅

e
y

∧

e
z

(

u
y

⋅
1
)
⋅

e
z

∧

e
x

(

u
z

⋅
1
)
⋅

e
x

∧

e
y

=

u
x

⋅

e
y

∧

e
z

u
y

⋅

e
z

∧

e
x

u
z

⋅

e
x

∧

e
y

{\displaystyle
\begin{align}
\mathbf{u} \lrcorner (e^x \wedge e^y \wedge e^z) &=

(u^x e\_x \cdot e^x \wedge e^y \wedge e^z) +
(u^y e\_y \cdot e^x \wedge e^y \wedge e^z) +
(u^z e\_z \cdot e^x \wedge e^y \wedge e^z) \\ &=

(u^x e\_x \cdot e^x \wedge e^y \wedge e^z) +
(u^y e\_y \cdot e^y \wedge e^z \wedge e^x) +
(u^z e\_z \cdot e^z \wedge e^x \wedge e^y) \\ &=

(u^x e\_x \cdot e^x) \wedge e^y \wedge e^z +
(u^y e\_y \cdot e^y) \wedge e^z \wedge e^x +
(u^z e\_z \cdot e^z) \wedge e^x \wedge e^y \\ &=

(u^x \cdot 1) \cdot e^y \wedge e^z +
(u^y \cdot 1) \cdot e^z \wedge e^x +
(u^z \cdot 1) \cdot e^x \wedge e^y \\ &=

u^x \cdot e^y \wedge e^z +
u^y \cdot e^z \wedge e^x +
u^z \cdot e^x \wedge e^y
\end{align}
}

The **vector triple product** is defined as the cross product of one vector with the cross product of the other two. The following relationship holds:

a

×
(

b

×

c

# )
(

a

⋅

c

)

b

−
(

a

⋅

b

)

c

{\displaystyle \mathbf{a}\times(\mathbf{b}\times\mathbf{c}) = (\mathbf{a}\cdot\mathbf{c})\mathbf{b} - (\mathbf{a}\cdot\mathbf{b})\mathbf{c}}

.
If geometric algebra is used the cross product **b** × **c** of vectors is expressed as their exterior product **b**∧**c**, a bivector. The second cross product cannot be expressed as an exterior product, otherwise the scalar triple product would result. Instead a left contraction can be used, so the formula becomes

−

a

⌟

(

b

∧

c

)

=

b

∧
(

a

⌟

c

)
−
(

a

⌟

b

)
∧

c

=
(

a

⋅

c

)

b

−
(

a

⋅

b

)

c

{\displaystyle \begin{align}
 -\mathbf{a} \;\big\lrcorner\; (\mathbf{b} \wedge \mathbf{c}) &= \mathbf{b} \wedge (\mathbf{a} \;\big\lrcorner\; \mathbf{c}) - (\mathbf{a} \;\big\lrcorner\; \mathbf{b}) \wedge \mathbf{c} \\
 &= (\mathbf{a} \cdot \mathbf{c}) \mathbf{b} - (\mathbf{a} \cdot \mathbf{b}) \mathbf{c}
\end{align}}

Back to top

## Dual

Not to be confused with  Algebraic dual,  Dual space, or  Hodge dual.
Assuming that

(

e
x

)
2

=
(

e
y

)
2

=
(

e
z

)
2

=
(

e
1

)
2

=
(

e
2

)
2

=
(

e
3

)
2

=
(

e
4

)
2

=
1

{\displaystyle (e\_x)^2= (e\_y)^2= (e\_z)^2= (e\_1)^2= (e\_2)^2= (e\_3)^2= (e\_4)^2= 1}

And

# ω

{\displaystyle \omega = }

 pseudoscalar

The dual of v is defined as

ω

v

{\displaystyle {\omega} v}

In one dimension:
ω
v

=
(

e
x

)
v

ω

1

=

e
x

ω

e
x

=
1

{\displaystyle
\begin{align}
 \text{In one dimension:}
 \omega v &= ( e\_x ) v \\
 {\omega} \, 1 &= e\_x \\
 {\omega} \, e\_x &= 1
\end{align}
}

In two dimensions:
ω
v

=
(

e
x

e
y

)
v

ω

1

=

e
x

e
y

ω

e
x

=
−

e
y

ω

e
y

=

e
x

ω

(

e
x

e
y

)

=
−
1

{\displaystyle
\begin{align}
 \text{In two dimensions:}
 \omega v &= ( e\_x e\_y ) v \\
 {\omega} \, 1 &= e\_x e\_y \\
 {\omega} \, e\_x &= -e\_y \\
 {\omega} \, e\_y &= e\_x \\
 {\omega} ( e\_x e\_y ) &= -1
\end{align}
}

In three dimensions:
ω
v

=
(

e
x

e
y

e
z

)
v

ω

1

=

e
x

e
y

e
z

ω

e
x

=

e
y

e
z

ω

e
y

=
−

e
x

e
z

=

e
z

e
x

ω

e
z

=

e
x

e
y

ω

(

e
x

e
y

)

=
−

e
z

ω

(

e
x

e
z

)

=

e
y

ω

(

e
y

e
z

)

=
−

e
x

ω

(

e
x

e
y

e
z

)

=
−
1

{\displaystyle
\begin{align}
 \text{In three dimensions:}
 \omega v &= ( e\_x e\_y e\_z ) v \\
 {\omega} \, 1 &= e\_x e\_y e\_z \\
 \color{red}{{\omega} \,e\_x} & \color{red}{= e\_y e\_z} \\
 \color{red}{{\omega} \,e\_y} & \color{red}{=-e\_x e\_z \quad = e\_z e\_x} \\
 \color{red}{{\omega} \,e\_z} & \color{red}{= e\_x e\_y} \\
 {\omega} \,( e\_x e\_y ) &= -e\_z \\
 {\omega} \,( e\_x e\_z ) &= e\_y \\
 {\omega} \,( e\_y e\_z ) &= -e\_x \\
 {\omega} \,( e\_x e\_y e\_z ) &= -1
\end{align}
}

In four dimensions:
ω
v

=
(

e
1

e
2

e
3

e
4

)
v

ω

1

=

e
1

e
2

e
3

e
4

ω

e
1

=
−

e
2

e
3

e
4

ω

e
2

=

e
1

e
3

e
4

ω

e
3

=
−

e
1

e
2

e
4

ω

e
4

=

e
1

e
2

e
3

ω

e
1

e
2

=
−

e
3

e
4

ω

e
1

e
3

=

e
2

e
4

ω

e
1

e
4

=
−

e
2

e
3

ω

e
2

e
3

=
−

e
1

e
4

ω

e
2

e
4

=

e
1

e
3

ω

e
3

e
4

=
−

e
1

e
2

ω

e
1

e
2

e
3

=

e
4

ω

e
1

e
2

e
4

=
−

e
3

ω

e
1

e
3

e
4

=

e
2

ω

e
2

e
3

e
4

=
−

e
1

ω

e
1

e
2

e
3

e
4

=
1

{\displaystyle
\begin{align}
 \text{In four dimensions:}
 \omega v &= ( e\_1 e\_2 e\_3 e\_4 ) v \\
 {\omega} \, 1 &= e\_1 e\_2 e\_3 e\_4 \\
 {\omega} \, e\_1 &= - e\_2 e\_3 e\_4 \\
 {\omega} \, e\_2 &= e\_1 e\_3 e\_4 \\
 {\omega} \, e\_3 &= - e\_1 e\_2 e\_4 \\
 {\omega} \, e\_4 &= e\_1 e\_2 e\_3 \\
 {\omega} \, e\_1 e\_2 &= - e\_3 e\_4 \\
 {\omega} \, e\_1 e\_3 &= e\_2 e\_4 \\
 {\omega} \, e\_1 e\_4 &= - e\_2 e\_3 \\
 {\omega} \, e\_2 e\_3 &= - e\_1 e\_4 \\
 {\omega} \, e\_2 e\_4 &= e\_1 e\_3 \\
 {\omega} \, e\_3 e\_4 &= - e\_1 e\_2 \\
 {\omega} \, e\_1 e\_2 e\_3 &= e\_4 \\
 {\omega} \, e\_1 e\_2 e\_4 &= - e\_3 \\
 {\omega} \, e\_1 e\_3 e\_4 &= e\_2 \\
 {\omega} \, e\_2 e\_3 e\_4 &= - e\_1 \\
 {\omega} \, e\_1 e\_2 e\_3 e\_4 &= 1 \\
\end{align}
}

Taking the dual only affects units. It has no effect on coefficients.

In 1 dimension:

ω
# 2
2

e
x

{\displaystyle
\omega 2 = 2 e\_x
}

One can replace

e
x

,

e
y

,

e
z

{\displaystyle e\_x, e\_y, e\_z}

 with

e
x

,

e
y

,

e
z

{\displaystyle e^x, e^y, e^z}

 (dx, dy, dz) of differential forms and it would still be correct because Clifford algebra makes no distinction between them.

Note: The dual of v is sometimes defined as

v
ω

{\displaystyle \frac{v}{\omega}}

# v

e
0

e
1

e
2

e
4

e
6

{\displaystyle v = + e\_0 e\_1 e\_2 e\_4 e\_6}

then

ω
# v
(

e
0

e
1

e
2

e
3

e
4

e
5

e
6

e
7

)
(

e
0

e
1

e
2

e
4

e
6

)

{\displaystyle
\omega v = ( e\_0 e\_1 e\_2 e\_3 e\_4 e\_5 e\_6 e\_7 ) (+ e\_0 e\_1 e\_2 e\_4 e\_6 ) }

It is easy to see that

ω
v

{\displaystyle \omega v }

 is either positive or negative

e
3

e
5

e
7

{\displaystyle e\_3 e\_5 e\_7 }

To determine the sign we need to know 2 things:

1. The number of unit vectors in v that square to -1
2. The number of pairs of unit vectors in

ω
v

{\displaystyle \omega v }

 that switch places (because a unit vector cant square to 1 or -1 until it is next to an identical unit and a unit vector can only move by switching places with its immediate neighbor and each time a pair of unit vectors switch places the sign changes because

e
a

e
b

=
−

e
b

e
a

{\displaystyle e\_a e\_b = -e\_b e\_a}

).

Let us suppose that

(

e
0

)
2

=
−
1

{\displaystyle (e\_0)^2 = -1}

Therefore letting n=7 we can easily see that

e0 will switch places n-0 times before squaring to -1
then
e1 will switch places n-1 times before squaring to 1
then
e2 will switch places n-2 times before squaring to 1
then
e4 will switch places n-4 times before squaring to 1
then
e6 will switch places n-6 times before squaring to 1
The sign is therefore:

(
−
1

)

1

(
n
−
0
)
(
n
−
1
)
(
n
−
2
)
(
n
−
4
)
(
n
−
6
)

=
(
−
1

)

1
5
n
−
(
1
2
4
6
)

=
(
−
1

)

23

=
−
1

{\displaystyle (-1)^{{\color{red} 1}+(n-0)+(n-1)+(n-2)+(n-4)+(n-6)} = (-1)^{1+5n-(1+2+4+6)} = (-1)^{23} = -1}

Therefore

ω
# v
−

e
3

e
5

e
7

{\displaystyle \omega v = -e\_3 e\_5 e\_7 }

Note that even numbers in the exponent can be ignored since they have no effect

(
−
1

)

1
5
n
−
(
1
2
4
6
)

=
(
−
1

)

1
5
n
−
(
1
)

=
(
−
1

)

35

=
−
1

{\displaystyle (-1)^{1+5n-(1+2+4+6)} = (-1)^{1+5n-(1)} = (-1)^{35} = -1 }

Back to top

### Hodge dual of the curl

From Wikipedia: Pauli matrices

σ
x
2

=

σ
y
2

=

σ
z
2

=

I
3

{\displaystyle \sigma\_x^2 = \sigma\_y^2 = \sigma\_z^2 = I\_3}

σ
x

σ
y

σ
z

=
i

{\displaystyle \sigma\_x \sigma\_y \sigma\_z = i}

The real linear span of {*I*, *iσ*1, *iσ*2, *iσ*3} is isomorphic to the real algebra of quaternions ℍ. The isomorphism from ℍ to this set is given by the following map

1

↦
I
,

i

↦

σ
x

σ
y

j

↦

σ
z

σ
x

k

↦

σ
y

σ
z

{\displaystyle
\begin{align}
 1 &\mapsto I, \\
 \mathbf{i} &\mapsto \sigma\_x \sigma\_y \\
 \mathbf{j} &\mapsto \sigma\_z \sigma\_x \\
 \mathbf{k} &\mapsto \sigma\_y \sigma\_z
\end{align}
}

where

i

2

=

j

2

=

k

2

=

i
j
k

=
−
1

{\displaystyle \mathbf{i}^2 = \mathbf{j}^2 = \mathbf{k}^2 = \mathbf{ijk} = -1}

The curl of a vector field is:

∇
×

F

=

e
x

e
y

e
z

∂

∂
x

∂

∂
y

∂

∂
z

F
x

F
y

F
z

=

(

∂

F
z

∂
y

−

∂

F
y

∂
z

)

e
x

(

∂

F
x

∂
z

−

∂

F
z

∂
x

)

e
y

(

∂

F
y

∂
x

−

∂

F
x

∂
y

)

e
z

{\displaystyle
\begin{align}
{\color{blue} \nabla \times \mathbf{F} }

&=
\begin{vmatrix}
\mathbf{e\_x} & \mathbf{e\_y} & \mathbf{e\_z} \\
{\frac{\partial}{\partial x}} & {\frac{\partial}{\partial y}} & {\frac{\partial}{\partial z}} \\
 F\_x & F\_y & F\_z
\end{vmatrix} \\ \\

&=
\left(
 \frac{\partial F\_z}{\partial y} - \frac{\partial F\_y}{\partial z}
\right) \mathbf{e\_x} +
\left(
 \frac{\partial F\_x}{\partial z} - \frac{\partial F\_z}{\partial x}
\right) \mathbf{e\_y} +
\left(
 \frac{\partial F\_y}{\partial x} - \frac{\partial F\_x}{\partial y}
\right) \mathbf{e\_z}
\end{align}
}

The dual of the curl is:

∇

F

=

k
^

ȷ
^

ı
^

∂

∂
x

∂

∂
y

∂

∂
z

F
x

F
y

F
z

=

e

y
z

e

z
x

e

x
y

∂

∂
x

∂

∂
y

∂

∂
z

F
x

F
y

F
z

{\displaystyle
\nabla \mathbf{F} =

\begin{vmatrix}
\boldsymbol{\hat k} &
\boldsymbol{\hat\jmath} &
\boldsymbol{\hat\imath} \\[5pt]

{\dfrac{\partial}{\partial x}} &
{\dfrac{\partial}{\partial y}} &
{\dfrac{\partial}{\partial z}} \\[10pt]

F\_x & F\_y & F\_z
\end{vmatrix} =

\begin{vmatrix} \mathbf{e\_{yz}} & \mathbf{e\_{zx}} & \mathbf{e\_{xy}} \\[5pt]
{\frac{\partial}{\partial x}} & {\frac{\partial}{\partial y}} & {\frac{\partial}{\partial z}} \\[10pt]
 F\_x & F\_y & F\_z
\end{vmatrix}
}

where **k**, **j**, and **i** are the unit "vectors" for the *x*-, *y*-, and *z*-axes, respectively. This expands as follows:

∇

F

=

(

∂

F
z

∂
y

−

∂

F
y

∂
z

)

k
^

(

∂

F
x

∂
z

−

∂

F
z

∂
x

)

ȷ
^

(

∂

F
y

∂
x

−

∂

F
x

∂
y

)

ı
^

{\displaystyle
\nabla \mathbf{F} =
\left(
 \frac{\partial F\_z}{\partial y} - \frac{\partial F\_y}{\partial z}
\right) \boldsymbol{\hat k} +
\left(
 \frac{\partial F\_x}{\partial z} - \frac{\partial F\_z}{\partial x}
\right) \boldsymbol{\hat\jmath} +
\left(
 \frac{\partial F\_y}{\partial x} - \frac{\partial F\_x}{\partial y}
\right) \boldsymbol{\hat\imath}
}

∇

F

=

(

∂

F
z

∂
y

−

∂

F
y

∂
z

)

e

y
z

(

∂

F
x

∂
z

−

∂

F
z

∂
x

)

e

z
x

(

∂

F
y

∂
x

−

∂

F
x

∂
y

)

e

x
y

{\displaystyle
\nabla \mathbf{F} =
{\color{green}
\left(
 \frac{\partial F\_z}{\partial y} -
 \frac{\partial F\_y}{\partial z}
\right)
\mathbf{e\_{yz}}} +
{\color{green}
\left(
 \frac{\partial F\_x}{\partial z} -
 \frac{\partial F\_z}{\partial x}
\right)
\mathbf{e\_{zx}}} +
{\color{green}
\left(
 \frac{\partial F\_y}{\partial x}
 -\frac{\partial F\_x}{\partial y}
\right) \mathbf{e\_{xy}}}
}

Back to top

## Position, velocity, acceleration

See also: Fourth, fifth, and sixth derivatives of position
Position

p

(
t
# )

p
x

(
t
)

e
x

p
y

(
t
)

e
y

p
z

(
t
)

e
z

{\displaystyle
\mathbf{p}(t) =
p\_x(t) e\_x + p\_y(t) e\_y + p\_z(t) e\_z
}

Velocity

v

(
t
# )

d

p

d

t

=

d

p
x

d

t

e
x

d

p
y

d

t

e
y

d

p
z

d

t

e
z

{\displaystyle
\mathbf{v}(t) =
\frac{\mathrm{d}\mathbf{p}}{\mathrm{d}t} = \quad \quad
\frac{\mathrm{d}p\_x} {\mathrm{d}t} e\_x +
\frac{\mathrm{d}p\_y} {\mathrm{d}t} e\_y +
\frac{\mathrm{d}p\_z} {\mathrm{d}t} e\_z
}

Acceleration

a

(
t
# )

d

2

p

d

t
2

=

d

v

d

t

=

d
2

p
x

d
2

t

e
x

d
2

p
y

d
2

t

e
y

d
2

p
z

d
2

t

e
z

{\displaystyle
\mathbf{a}(t) =
\frac{\mathrm{d}^2\mathbf{p}}{\mathrm{d}t^2} =
\frac{\mathrm{d}\mathbf{v}}{\mathrm{d}t} = \quad \quad
\frac{\mathrm{d^2}p\_x} {\mathrm{d^2}t} e\_x +
\frac{\mathrm{d^2}p\_y} {\mathrm{d^2}t} e\_y +
\frac{\mathrm{d^2}p\_z} {\mathrm{d^2}t} e\_z
}

Jerk

j

(
t
# )

d

3

p

d

t
3

=

d

2

v

d

t
2

=

d

a

d

t

=

d
3

p
x

d
3

t

e
x

d
3

p
y

d
3

t

e
y

d
3

p
z

d
3

t

e
z

{\displaystyle
\mathbf{j}(t) =
\frac{\mathrm{d}^3\mathbf{p}}{\mathrm{d}t^3} =
\frac{\mathrm{d}^2\mathbf{v}}{\mathrm{d}t^2} =
\frac{\mathrm{d}\mathbf{a}}{\mathrm{d}t} = \quad \quad
\frac{\mathrm{d^3}p\_x} {\mathrm{d^3}t} e\_x +
\frac{\mathrm{d^3}p\_y} {\mathrm{d^3}t} e\_y +
\frac{\mathrm{d^3}p\_z} {\mathrm{d^3}t} e\_z
}

Back to top

## Derivatives

### Multivector derivatives

From Wikipedia:Gradient

We consider general coordinates, which we write as *x*1, ..., *x**i*, ..., *x**n*, where n is the number of dimensions of the domain. Here, the upper index refers to the position in the list of the coordinate or component, so *x*2 refers to the second component—not the quantity *x* squared. The index variable *i* refers to an arbitrary element *x**i*. Using Einstein notation, the gradient can then be written as:

∇
# f

∂
f

∂

x

i

g

i
j

e

j

{\displaystyle \nabla f = \frac{\partial f}{\partial x^{i}}g^{ij} \mathbf{e}\_j}

 ( Note that its dual is

d

# f

∂
f

∂

x

i

e

i

{\displaystyle \mathrm{d}f= \frac{\partial f}{\partial x^{i}}\mathbf{e}^i}

 ),
where

e

i

=
∂

x

/

∂

x
i

{\displaystyle \mathbf{e}\_i = \partial \mathbf{x}/\partial x^i}

 and

e

i

=

d

x
i

{\displaystyle \mathbf{e}^i = \mathrm{d}x^i}

 refer to the unnormalized local covariant and contravariant bases respectively,

g

i
j

{\displaystyle g^{ij}}

 is the inverse metric tensor, and the Einstein summation convention implies summation over *i* and *j*.

The derivative of a scalar function:

∇

f

=

(

∂

∂
x

e
x

,

∂

∂
y

e
y

,

∂

∂
z

e
z

)

(

f
(
x
,
y
,
z
)

)

=

(

∂
f

∂
x

e
x

∂
f

∂
y

e
y

∂
f

∂
z

e
z

)

=
0
∇
∧

F

{\displaystyle
\begin{align}
\nabla \mathbf{f} &=

\left(
\frac{\partial}{\partial x} e\_x,
\frac{\partial}{\partial y} e\_y,
\frac{\partial}{\partial z} e\_z
\right)
\left( f(x,y,z) \right) \\ \\ \\ &=
\left(
 \frac{\partial f}{\partial x} e\_x
+ \frac{\partial f}{\partial y} e\_y
+ \frac{\partial f}{\partial z} e\_z
\right) \\ \\ \\ &=

0 + \nabla \wedge \mathbf{F}

\end{align}
}

The derivative of a vector function:

∇

F

=

(

∂

∂
x

e
x

,

∂

∂
y

e
y

,

∂

∂
z

e
z

)

(

F
x

(
x
,
y
,
z
)

e
x

F
y

(
x
,
y
,
z
)

e
y

F
z

(
x
,
y
,
z
)

e
z

)

=

(

∂

F
x

∂
x

e

x
x

∂

F
y

∂
x

e

x
y

∂

F
z

∂
x

e

x
z

)

(

∂

F
x

∂
y

e

y
x

∂

F
y

∂
y

e

y
y

∂

F
z

∂
y

e

y
z

)

(

∂

F
x

∂
z

e

z
x

∂

F
y

∂
z

e

z
y

∂

F
z

∂
z

e

z
z

)

=

(

∂

F
x

∂
x

(
1
)

∂

F
y

∂
x

e

x
y

∂

F
z

∂
x

e

x
z

)

(

∂

F
x

∂
y

e

y
x

∂

F
y

∂
y

(
1
)

∂

F
z

∂
y

e

y
z

)

(

∂

F
x

∂
z

e

z
x

∂

F
y

∂
z

e

z
y

∂

F
z

∂
z

(
1
)

)

=

(

∂

F
x

∂
x

∂

F
y

∂
y

∂

F
z

∂
z

)

(

∂

F
y

∂
x

e

x
y

∂

F
x

∂
y

e

y
x

)

(

∂

F
z

∂
x

e

x
z

∂

F
x

∂
z

e

z
x

)

(

∂

F
z

∂
y

e

y
z

∂

F
y

∂
z

e

z
y

)

=

(

∂

F
x

∂
x

∂

F
y

∂
y

∂

F
z

∂
z

)

(

∂

F
y

∂
x

−

∂

F
x

∂
y

)

e

x
y

(

∂

F
x

∂
z

−

∂

F
z

∂
x

)

e

z
x

(

∂

F
z

∂
y

−

∂

F
y

∂
z

)

e

y
z

=
∇
⋅

F

∇
∧

F

{\displaystyle
\begin{align}
\nabla \mathbf{F} &=

\left(
\frac{\partial}{\partial x} e\_x,
\frac{\partial}{\partial y} e\_y,
\frac{\partial}{\partial z} e\_z
\right)
\left( F\_x(x,y,z) e\_x + F\_y(x,y,z) e\_y + F\_z(x,y,z) e\_z \right) \\ &=

\left(
 {\color{red} \frac{\partial F\_x}{\partial x} e\_{xx} }
+ {\color{green} \frac{\partial F\_y}{\partial x} \mathbf{e\_{xy}} }
+ {\color{green} \frac{\partial F\_z}{\partial x} \mathbf{e\_{xz}} }
\right) +
\left(
 {\color{green} \frac{\partial F\_x}{\partial y} \mathbf{e\_{yx}} }
+ {\color{red} \frac{\partial F\_y}{\partial y} e\_{yy} }
+ {\color{green} \frac{\partial F\_z}{\partial y} \mathbf{e\_{yz}} }
\right) +
\left(
 {\color{green} \frac{\partial F\_x}{\partial z} \mathbf{e\_{zx}} }
+ {\color{green} \frac{\partial F\_y}{\partial z} \mathbf{e\_{zy}} }
+ {\color{red} \frac{\partial F\_z}{\partial z} e\_{zz} }
\right) \\ \\ &=

\left(
 {\color{red} \frac{\partial F\_x}{\partial x} (1) }
+ {\color{green} \frac{\partial F\_y}{\partial x} \mathbf{e\_{xy}}}
+ {\color{green} \frac{\partial F\_z}{\partial x} \mathbf{e\_{xz}}}
\right) +
\left(
 {\color{green} \frac{\partial F\_x}{\partial y} \mathbf{e\_{yx}}}
+ {\color{red} \frac{\partial F\_y}{\partial y} (1) }
+ {\color{green} \frac{\partial F\_z}{\partial y} \mathbf{e\_{yz}}}
\right) +
\left(
 {\color{green} \frac{\partial F\_x}{\partial z} \mathbf{e\_{zx}}}
+ {\color{green} \frac{\partial F\_y}{\partial z} \mathbf{e\_{zy}}}
+ {\color{red} \frac{\partial F\_z}{\partial z} (1) }
\right) \\ \\ &=

{\color{red}
\left(
 \frac{\partial F\_x}{\partial x} +
 \frac{\partial F\_y}{\partial y} +
 \frac{\partial F\_z}{\partial z}
\right)} +
{\color{green}
\left(
 \frac{\partial F\_y}{\partial x} \mathbf{e\_{xy}} +
 \frac{\partial F\_x}{\partial y} \mathbf{e\_{yx}}
\right)} +
{\color{green}
\left(
 \frac{\partial F\_z}{\partial x} \mathbf{e\_{xz}} +
 \frac{\partial F\_x}{\partial z} \mathbf{e\_{zx}}
\right)} +
{\color{green}
\left(
 \frac{\partial F\_z}{\partial y} \mathbf{e\_{yz}} +
 \frac{\partial F\_y}{\partial z} \mathbf{e\_{zy}}
\right)} \\ \\ &=

{\color{red}
\left(
 \frac{\partial F\_x}{\partial x} +
 \frac{\partial F\_y}{\partial y} +
 \frac{\partial F\_z}{\partial z}
\right)} +
{\color{green}
\left(
 \frac{\partial F\_y}{\partial x}
- \frac{\partial F\_x}{\partial y}
\right) \mathbf{e\_{xy}}} +
{\color{green}
\left(
 \frac{\partial F\_x}{\partial z} -
 \frac{\partial F\_z}{\partial x}
\right)
\mathbf{e\_{zx}}} +
{\color{green}
\left(
 \frac{\partial F\_z}{\partial y} -
 \frac{\partial F\_y}{\partial z}
\right)
\mathbf{e\_{yz}}} \\ \\ \\ &=

\nabla\cdot\mathbf{F} + \nabla \wedge \mathbf{F}

\end{align}
}

The derivative of a bivector function:

∇

F

=

(

∂

∂
x

e
x

,

∂

∂
y

e
y

,

∂

∂
z

e
z

)

(

F

x
y

(
x
,
y
,
z
)

e

x
y

F

z
x

(
x
,
y
,
z
)

e

z
x

F

y
z

(
x
,
y
,
z
)

e

y
z

)

=

(

∂

F

x
y

∂
x

e

x
x
y

∂

F

z
x

∂
x

e

x
z
x

∂

F

y
z

∂
x

e

x
y
z

)

(

∂

F

x
y

∂
y

e

y
x
y

∂

F

z
x

∂
y

e

y
z
x

∂

F

y
z

∂
y

e

y
y
z

)

(

∂

F

x
y

∂
z

e

z
x
y

∂

F

z
x

∂
z

e

z
z
x

∂

F

y
z

∂
z

e

z
y
z

)

=

(

∂

F

x
y

∂
x

e

y

−

∂

F

z
x

∂
x

e

z

∂

F

y
z

∂
x

e

x
y
z

)

(

−

∂

F

x
y

∂
y

e

x

∂

F

z
x

∂
y

e

x
y
z

∂

F

y
z

∂
y

e

z

)

(

∂

F

x
y

∂
z

e

x
y
z

∂

F

z
x

∂
z

e

x

−

∂

F

y
z

∂
z

e

y

)

=

(

−

∂

F

x
y

∂
y

e

x

∂

F

z
x

∂
z

e

x

)

(

∂

F

x
y

∂
x

e

y

−

∂

F

y
z

∂
z

e

y

)

(

−

∂

F

z
x

∂
x

e

z

∂

F

y
z

∂
y

e

z

)

(

∂

F

x
y

∂
z

e

x
y
z

∂

F

z
x

∂
y

e

x
z
y

∂

F

y
z

∂
x

e

y
z
x

)

=

(

∂

F

z
x

∂
z

−

∂

F

x
y

∂
y

)

e

x

(

∂

F

x
y

∂
x

−

∂

F

y
z

∂
z

)

e

y

(

∂

F

y
z

∂
y

−

∂

F

z
x

∂
x

)

e

z

(

∂

F

x
y

∂
z

∂

F

z
x

∂
y

∂

F

y
z

∂
x

)

e

x
y
z

=
∇
⋅

F

∇
∧

F

{\displaystyle
\begin{align}

\nabla \mathbf{F} &=

\left(
 \frac{\partial}{\partial x} e\_x,
 \frac{\partial}{\partial y} e\_y,
 \frac{\partial}{\partial z} e\_z
\right)

\left( F\_{xy}(x,y,z) e\_{xy} + F\_{zx}(x,y,z) e\_{zx} + F\_{yz}(x,y,z) e\_{yz} \right) \\ &=

\left(
 \frac{\partial F\_{xy}}{\partial x} \mathbf{e\_{xxy}}
 + \frac{\partial F\_{zx}}{\partial x} \mathbf{e\_{xzx}}
 + \frac{\partial F\_{yz}}{\partial x} e\_{xyz}
\right) +
\left(
 \frac{\partial F\_{xy}}{\partial y} \mathbf{e\_{yxy}}
 + \frac{\partial F\_{zx}}{\partial y} e\_{yzx}
 + \frac{\partial F\_{yz}}{\partial y} \mathbf{e\_{yyz}}
\right) +
\left(
 \frac{\partial F\_{xy}}{\partial z} e\_{zxy}
 + \frac{\partial F\_{zx}}{\partial z} \mathbf{e\_{zzx}}
 + \frac{\partial F\_{yz}}{\partial z} \mathbf{e\_{zyz}}
\right) \\ \\ &=

\left(
 {\color{blue} \frac{\partial F\_{xy}}{\partial x} \mathbf{e\_{y}}}
 {\color{blue} \, - \frac{\partial F\_{zx}}{\partial x} \mathbf{e\_{z}}}
 {\color{orange} \, + \frac{\partial F\_{yz}}{\partial x} e\_{xyz}}
\right) +
\left(
 {\color{blue} - \frac{\partial F\_{xy}}{\partial y} \mathbf{e\_{x}}}
 {\color{orange} \, + \frac{\partial F\_{zx}}{\partial y} e\_{xyz}}
 {\color{blue} \, + \frac{\partial F\_{yz}}{\partial y} \mathbf{e\_{z}}}
\right) +
\left(
 {\color{orange} \frac{\partial F\_{xy}}{\partial z} e\_{xyz}}
 {\color{blue} \, + \frac{\partial F\_{zx}}{\partial z} \mathbf{e\_{x}}}
 {\color{blue} \, - \frac{\partial F\_{yz}}{\partial z} \mathbf{e\_{y}}}
\right) \\ \\ &=

{\color{blue}
\left(
 -\frac{\partial F\_{xy}}{\partial y} \mathbf{e\_{x}} +
 \frac{\partial F\_{zx}}{\partial z} \mathbf{e\_{x}}
\right)} +
{\color{blue}
\left(
 \frac{\partial F\_{xy}}{\partial x} \mathbf{e\_{y}} -
 \frac{\partial F\_{yz}}{\partial z} \mathbf{e\_{y}}
\right)} +
{\color{blue}
 \left(
 -\frac{\partial F\_{zx}}{\partial x} \mathbf{e\_{z}} +
\frac{\partial F\_{yz}}{\partial y} \mathbf{e\_{z}}
\right)} +
{\color{orange}
\left(
 \frac{\partial F\_{xy}}{\partial z} e\_{xyz} +
 \frac{\partial F\_{zx}}{\partial y} e\_{xzy} +
 \frac{\partial F\_{yz}}{\partial x} e\_{yzx} \right)} \\ \\ &=

{\color{blue}
\left(
 \frac{\partial F\_{zx}}{\partial z} -
 \frac{\partial F\_{xy}}{\partial y}
 \right) \mathbf{e\_{x}}} +
{\color{blue}
\left(
 \frac{\partial F\_{xy}}{\partial x} -
 \frac{\partial F\_{yz}}{\partial z}
 \right) \mathbf{e\_{y}}} +
{\color{blue}
\left(
 \frac{\partial F\_{yz}}{\partial y} -
 \frac{\partial F\_{zx}}{\partial x}
\right) \mathbf{e\_{z}}} +
{\color{orange}
\left(
 \frac{\partial F\_{xy}}{\partial z} +
 \frac{\partial F\_{zx}}{\partial y} +
 \frac{\partial F\_{yz}}{\partial x}
\right) e\_{xyz}} \\ \\ \\ &=

\nabla\cdot\mathbf{F} + \nabla \wedge \mathbf{F}

\end{align}
}

The derivative of a pseudoscalar function:

∇

F

=

(

∂

∂
x

e
x

,

∂

∂
y

e
y

,

∂

∂
z

e
z

)

(

F
(
x
,
y
,
z
)

e

x
y
z

)

=

(

∂
F

∂
x

e

x
x
y
z

∂
F

∂
y

e

y
x
y
z

∂
F

∂
z

e

z
x
y
z

)

=

(

∂
F

∂
x

e

y
z

∂
F

∂
y

e

z
x

∂
F

∂
z

e

x
y

)

=
∇
⋅

F

0

{\displaystyle
\begin{align}
\nabla \mathbf{F} &=

\left(
 \frac{\partial}{\partial x} e\_x,
 \frac{\partial}{\partial y} e\_y,
 \frac{\partial}{\partial z} e\_z
\right)

\left( F(x,y,z) e\_{xyz} \right) \\ \\ \\ &=
\left(
 \frac{\partial F}{\partial x} e\_{xxyz}
+ \frac{\partial F}{\partial y} e\_{yxyz}
+ \frac{\partial F}{\partial z} e\_{zxyz}
\right) \\ &=

\left(
 {\color{green}\frac{\partial F}{\partial x} e\_{yz}}
+ {\color{green}\frac{\partial F}{\partial y} e\_{zx}}
+ {\color{green}\frac{\partial F}{\partial z} e\_{xy}}
\right) \\ \\ \\ &=

\nabla \cdot \mathbf{F} + 0

\end{align}
}

Back to top

#### Jacobian matrix

From Wikipedia:Jacobian matrix and determinant

The Jacobian matrix

J

f

{\displaystyle \mathbf{J}\_{\mathbf{f}}}

 represents the total derivative of **f** at every point where **f** is differentiable.

The Jacobian matrix of a vector-valued function in several variables is the matrix of all its first-order partial derivatives.

J

=
d

f

=

[

∂

f

∂

x
1

⋯

∂

f

∂

x
n

]

=

[

∂

f
1

∂

x
1

⋯

∂

f
1

∂

x
n

⋮

⋱

⋮

∂

f
m

∂

x
1

⋯

∂

f
m

∂

x
n

]

.

{\displaystyle \mathbf J = d \mathbf{f} = \begin{bmatrix}
 \dfrac{\partial \mathbf{f}}{\partial x\_1} & \cdots & \dfrac{\partial \mathbf{f}}{\partial x\_n} \end{bmatrix}
= \begin{bmatrix}
 \dfrac{\partial f\_1}{\partial x\_1} & \cdots & \dfrac{\partial f\_1}{\partial x\_n}\\
 \vdots & \ddots & \vdots\\
 \dfrac{\partial f\_m}{\partial x\_1} & \cdots & \dfrac{\partial f\_m}{\partial x\_n} \end{bmatrix}.}

where

J

i
j

=

∂

f
i

∂

x
j

{\displaystyle \mathbf J\_{ij} = \frac{\partial f\_i}{\partial x\_j}}

This matrix, whose entries are functions of **x**, is denoted in various ways; common notations include

D

f

{\displaystyle D \mathbf{f}}

,

J

f

{\displaystyle \mathbf{J}\_{\mathbf{f}}}

,

∇

f

{\displaystyle \nabla \mathbf{f}}

, and

∂
(

f
1

,
.
.
,

f
m

)

∂
(

x
1

,
.
.
,

x
n

)

{\displaystyle \frac{\partial(f\_1,..,f\_m)}{\partial(x\_1, ..,x\_n)}}

 . Note also that some authors define the Jacobian as the transpose of the form given above.

Back to top

### Exterior derivative of differential forms

The inverse of a unit vector (wrt to the wedge product) is a unit covector:

1

e
x

=

e

x

e
x

e

x

=

e

x

1

=

e

x

=

e

x

{\displaystyle \frac{1}{e\_x} = \frac{e^{x}}{e\_x e^{x}} = \frac{e^{x}}{1} = e^{x} = e^{x}}

but in Clifford algebra

1

e
x

=

e

x

e
x

e

x

=

e

x

1

=

e

x

=

e

x

{\displaystyle \frac{1}{e\_x} = \frac{e\_{x}}{e\_x e\_{x}} = \frac{e\_{x}}{1} = e\_{x} = e\_{x} }

(Note that both of these equations only work in *L*2 space).
Vectors and covectors both have magnitude and direction and Clifford algebra treats them the same.

Thats why the inverse of a vector in Clifford algebra is a vector.

(There should probably be a word for something that is either a vector or a covector. Omnivector perhaps?)

In Clifford algebra the difference between them lies not in their representation but in what you do with them.

The geometric product seems to be for multiplying vectors.

v

u

=

v

⋅

u

v

∧

u

{\displaystyle
\mathbf{v} \mathbf{u} =
\mathbf{v} \cdot \mathbf{u} + \mathbf{v} \wedge \mathbf{u}
}

The wedge product seems to be for multiplying covectors.

c

∧

b

=
−

b

∧

c

=
(

c

b

−

b

c

)

/

2

=

c
x

∧

b
y

c
x

∧

b
z

c
y

∧

b
x

c
y

∧

b
z

c
z

∧

b
x

c
z

∧

b
y

{\displaystyle
\begin{align}
\mathbf{c} \wedge \mathbf{b} &=
-\mathbf{b} \wedge \mathbf{c} \\ &=
(\mathbf{c} \mathbf{b} - \mathbf{b} \mathbf{c}) / 2 \\ &=
\mathbf{c\_x} \wedge \mathbf{b\_y} +
\mathbf{c\_x} \wedge \mathbf{b\_z} +
\mathbf{c\_y} \wedge \mathbf{b\_x} +
\mathbf{c\_y} \wedge \mathbf{b\_z} +
\mathbf{c\_z} \wedge \mathbf{b\_x} +
\mathbf{c\_z} \wedge \mathbf{b\_y}
\end{align}
}

The dot product seems to be for multiplying vectors with covectors.

v

⋅

c

=

c

⋅

v

=
(

v

c

c

v

)

/

2

=

v
x

c
x

v
y

c
y

v
z

c
z

{\displaystyle
\begin{align}
\mathbf{v} \cdot \mathbf{c} &=
\mathbf{c} \cdot \mathbf{v} \\ &=
(\mathbf{v} \mathbf{c} + \mathbf{c} \mathbf{v}) / 2 \\ &=
v\_x c\_x + v\_y c\_y + v\_z c\_z
\end{align}
}

The gradient is the algebraic dual to the exterior derivative

Δ
f

{\displaystyle \Delta f}

:

* The value of the gradient at a point is a tangent vector – a vector at each point;

∇

f

=

(

∂
f

∂
x

e
x

∂
f

∂
y

e
y

∂
f

∂
z

e
z

)

{\displaystyle
\begin{align}
\nabla \mathbf{f} &=
\left(
 \frac{\partial f}{\partial x} e\_x
+ \frac{\partial f}{\partial y} e\_y
+ \frac{\partial f}{\partial z} e\_z
\right)
\end{align}
}

* The value of the exterior derivative at a point is a *co*tangent vector – a linear function on vectors.

Δ
∧

f

=

(

∂
f

∂
x

e

x

∂
f

∂
y

e

y

∂
f

∂
z

e

z

)

{\displaystyle
\begin{align}
\Delta \wedge \mathbf{f} &=
\left(
 \frac{\partial f}{\partial x} e^{x}
+ \frac{\partial f}{\partial y} e^{y}
+ \frac{\partial f}{\partial z} e^{z}
\right)
\end{align}
}

Clifford algebra makes no distinction between

∇

{\displaystyle \nabla }

 and

Δ

{\displaystyle \Delta}

. I am using two different symbols merely to help the reader know exactly whats being calculated.

# A zero-form
f
(
x
,
y
,
z
)

{\displaystyle
\text{A zero-form} = f(x,y,z)
}

exterior derivative of a zero-form

Δ
∧

f

=

(

∂

∂
x

e

x

,

∂

∂
y

e

y

,

∂

∂
z

e

z

)

∧

(

f
(
x
,
y
,
z
)

)

=

(

∂
f

∂
x

e

x

∂
f

∂
y

e

y

∂
f

∂
z

e

z

)

=

(

∂
f

∂
x

d
x

∂
f

∂
y

d
y

∂
f

∂
z

d
z

)

{\displaystyle
\begin{align}
\Delta \wedge \mathbf{f} &=

\left(
\frac{\partial}{\partial x} e^{x},
\frac{\partial}{\partial y} e^{y},
\frac{\partial}{\partial z} e^{z}
\right) \wedge
\left( f(x,y,z) \right) \\ \\ \\ &=
\left(
 \frac{\partial f}{\partial x} e^{x}
+ \frac{\partial f}{\partial y} e^{y}
+ \frac{\partial f}{\partial z} e^{z}
\right) =
\left(
 \frac{\partial f}{\partial x} dx
+ \frac{\partial f}{\partial y} dy
+ \frac{\partial f}{\partial z} dz
\right)

\end{align}
}

exterior derivative of a one-form

Δ
∧

F

=

(

∂

∂
x

e

x

,

∂

∂
y

e

y

,

∂

∂
z

e

z

)

∧

(

F
x

(
x
,
y
,
z
)

e

x

F
y

(
x
,
y
,
z
)

e

y

F
z

(
x
,
y
,
z
)

e

z

)

=

(

∂

F
x

∂
x

e

x
x

∂

F
y

∂
x

e

x
y

∂

F
z

∂
x

e

x
z

)

(

∂

F
x

∂
y

e

y
x

∂

F
y

∂
y

e

y
y

∂

F
z

∂
y

e

y
z

)

(

∂

F
x

∂
z

e

z
x

∂

F
y

∂
z

e

z
y

∂

F
z

∂
z

e

z
z

)

=

(

∂

F
x

∂
x

(
0
)

∂

F
y

∂
x

e

x
y

∂

F
z

∂
x

e

x
z

)

(

∂

F
x

∂
y

e

y
x

∂

F
y

∂
y

(
0
)

∂

F
z

∂
y

e

y
z

)

(

∂

F
x

∂
z

e

z
x

∂

F
y

∂
z

e

z
y

∂

F
z

∂
z

(
0
)

)

=

(

∂

F
y

∂
x

e

x
y

∂

F
x

∂
y

e

y
x

)

(

∂

F
z

∂
x

e

x
z

∂

F
x

∂
z

e

z
x

)

(

∂

F
z

∂
y

e

y
z

∂

F
y

∂
z

e

z
y

)

=

(

∂

F
y

∂
x

−

∂

F
x

∂
y

)

e

x
y

(

∂

F
x

∂
z

−

∂

F
z

∂
x

)

e

z
x

(

∂

F
z

∂
y

−

∂

F
y

∂
z

)

e

y
z

{\displaystyle
\begin{align}
\Delta \wedge \mathbf{F} &=

\left(
\frac{\partial}{\partial x} e^{x},
\frac{\partial}{\partial y} e^{y},
\frac{\partial}{\partial z} e^{z}
\right) \wedge
\left( F\_x(x,y,z) e^{x} + F\_y(x,y,z) e^{y} + F\_z(x,y,z) e^{z} \right) \\ &=

\left(
 {\color{red} \frac{\partial F\_x}{\partial x} e^{xx}}
+ {\color{green} \frac{\partial F\_y}{\partial x} \mathbf{e^{xy}} }
+ {\color{green} \frac{\partial F\_z}{\partial x} \mathbf{e^{xz}} }
\right) +
\left(
 {\color{green} \frac{\partial F\_x}{\partial y} \mathbf{e^{yx}} }
+ {\color{red} \frac{\partial F\_y}{\partial y} e^{yy} }
+ {\color{green} \frac{\partial F\_z}{\partial y} \mathbf{e^{yz}} }
\right) +
\left(
 {\color{green} \frac{\partial F\_x}{\partial z} \mathbf{e^{zx}} }
+ {\color{green} \frac{\partial F\_y}{\partial z} \mathbf{e^{zy}} }
+ {\color{red} \frac{\partial F\_z}{\partial z} e^{zz} }
\right) \\ \\ &=

\left(
 {\color{red} \frac{\partial F\_x}{\partial x} (0) }
+ {\color{green} \frac{\partial F\_y}{\partial x} \mathbf{e^{xy}}}
+ {\color{green} \frac{\partial F\_z}{\partial x} \mathbf{e^{xz}}}
\right) +
\left(
{\color{green} \frac{\partial F\_x}{\partial y} \mathbf{e^{yx}}}
+ {\color{red} \frac{\partial F\_y}{\partial y} (0) }
+ {\color{green} \frac{\partial F\_z}{\partial y} \mathbf{e^{yz}}}
\right) +
\left(
 {\color{green} \frac{\partial F\_x}{\partial z} \mathbf{e^{zx}}}
+ {\color{green} \frac{\partial F\_y}{\partial z} \mathbf{e^{zy}}}
+ {\color{red} \frac{\partial F\_z}{\partial z} (0) }
\right) \\ \\ &=

{\color{green}
\left(
 \frac{\partial F\_y}{\partial x} \mathbf{e^{xy}} +
 \frac{\partial F\_x}{\partial y} \mathbf{e^{yx}}
\right)} +
{\color{green}
\left(
 \frac{\partial F\_z}{\partial x} \mathbf{e^{xz}} +
 \frac{\partial F\_x}{\partial z} \mathbf{e^{zx}}
\right)} +
{\color{green}
\left(
 \frac{\partial F\_z}{\partial y} \mathbf{e^{yz}} +
 \frac{\partial F\_y}{\partial z} \mathbf{e^{zy}}
\right)} \\ \\ &=

{\color{green}
\left(
 \frac{\partial F\_y}{\partial x}
- \frac{\partial F\_x}{\partial y}
\right) \mathbf{e^{xy}}} +
{\color{green}
\left(
 \frac{\partial F\_x}{\partial z} -
 \frac{\partial F\_z}{\partial x}
\right)
\mathbf{e^{zx}}} +
{\color{green}
\left(
 \frac{\partial F\_z}{\partial y} -
 \frac{\partial F\_y}{\partial z}
\right)
\mathbf{e^{yz}}}

\end{align}
}

exterior derivative of a two-form

Δ
∧

F

=

(

∂

∂
x

e

x

,

∂

∂
y

e

y

,

∂

∂
z

e

z

)

∧

(

F

x
y

(
x
,
y
,
z
)

e

x
y

F

z
x

(
x
,
y
,
z
)

e

z
x

F

y
z

(
x
,
y
,
z
)

e

y
z

)

=

(

∂

F

x
y

∂
x

e

x
x
y

∂

F

z
x

∂
x

e

x
z
x

∂

F

y
z

∂
x

e

x
y
z

)

(

∂

F

x
y

∂
y

e

y
x
y

∂

F

z
x

∂
y

e

y
z
x

∂

F

y
z

∂
y

e

y
y
z

)

(

∂

F

x
y

∂
z

e

z
x
y

∂

F

z
x

∂
z

e

z
z
x

∂

F

y
z

∂
z

e

z
y
z

)

=

(

∂

F

x
y

∂
x

(
0
)

−

∂

F

z
x

∂
x

(
0
)

∂

F

y
z

∂
x

e

x
y
z

)

(

−

∂

F

x
y

∂
y

(
0
)

∂

F

z
x

∂
y

e

y
z
x

∂

F

y
z

∂
y

(
0
)

)

(

∂

F

x
y

∂
z

e

z
x
y

∂

F

z
x

∂
z

(
0
)

−

∂

F

y
z

∂
z

(
0
)

)

=

(

∂

F

x
y

∂
z

e

z
x
y

∂

F

z
x

∂
y

e

y
z
x

∂

F

y
z

∂
x

e

x
y
z

)

=

(

∂

F

x
y

∂
z

∂

F

z
x

∂
y

∂

F

y
z

∂
x

)

e

x
y
z

{\displaystyle
\begin{align}

\Delta \wedge \mathbf{F} &=

\left(
\frac{\partial}{\partial x} e^{x},
\frac{\partial}{\partial y} e^{y},
\frac{\partial}{\partial z} e^{z}
\right) \wedge

\left( F\_{xy}(x,y,z) e^{xy} + F\_{zx}(x,y,z) e^{zx} + F\_{yz}(x,y,z) e^{yz} \right) \\ &=

\left(
 \frac{\partial F\_{xy}}{\partial x} e^{xxy}
 + \frac{\partial F\_{zx}}{\partial x} e^{xzx}
 + \frac{\partial F\_{yz}}{\partial x} e^{xyz}
\right) +
\left(
 \frac{\partial F\_{xy}}{\partial y} e^{yxy}
 + \frac{\partial F\_{zx}}{\partial y} e^{yzx}
 + \frac{\partial F\_{yz}}{\partial y} e^{yyz}
\right) +
\left(
 \frac{\partial F\_{xy}}{\partial z} e^{zxy}
 + \frac{\partial F\_{zx}}{\partial z} e^{zzx}
 + \frac{\partial F\_{yz}}{\partial z} e^{zyz}
\right) \\ \\ &=

\left(
 {\color{blue} \frac{\partial F\_{xy}}{\partial x} (0)}
- {\color{blue} \, \frac{\partial F\_{zx}}{\partial x} (0)}
+ {\color{orange} \, \frac{\partial F\_{yz}}{\partial x} e^{xyz}}
\right) +
\left(
- {\color{blue} \frac{\partial F\_{xy}}{\partial y} (0)}
+ {\color{orange} \, \frac{\partial F\_{zx}}{\partial y} e^{yzx}}
+ {\color{blue} \, \frac{\partial F\_{yz}}{\partial y} (0)}
\right) +
\left(
 {\color{orange} \frac{\partial F\_{xy}}{\partial z} e^{zxy}}
+ {\color{blue} \, \frac{\partial F\_{zx}}{\partial z} (0)}
- {\color{blue} \, \frac{\partial F\_{yz}}{\partial z} (0)}
\right) \\ \\ &=

{\color{orange}
\left(
\frac{\partial F\_{xy}}{\partial z} e^{zxy} +
\frac{\partial F\_{zx}}{\partial y} e^{yzx} +
\frac{\partial F\_{yz}}{\partial x} e^{xyz} \right)} \\ \\ &=

{\color{orange}
\left(
\frac{\partial F\_{xy}}{\partial z} +
\frac{\partial F\_{zx}}{\partial y} +
\frac{\partial F\_{yz}}{\partial x}
\right) e^{xyz}}

\end{align}
}

In three dimensions the exterior derivative of a three-form is zero

Δ
∧

F

=

(

∂

∂
x

e

x

,

∂

∂
y

e

y

,

∂

∂
z

e

z

)

∧

(

F
(
x
,
y
,
z
)

e

x
y
z

)

=

(

∂
F

∂
x

e

x
x
y
z

∂
F

∂
y

e

y
x
y
z

∂
F

∂
z

e

z
x
y
z

)

=

(

∂
F

∂
x

(
0
)

∂
F

∂
y

(
0
)

∂
F

∂
z

(
0
)

)

=
0

{\displaystyle
\begin{align}
\Delta \wedge \mathbf{F} &=

\left(
\frac{\partial}{\partial x} e^{x},
\frac{\partial}{\partial y} e^{y},
\frac{\partial}{\partial z} e^{z}
\right) \wedge

\left( F(x,y,z) e^{xyz} \right) \\ \\ \\ &=
\left(
 \frac{\partial F}{\partial x} e^{xxyz}
+ \frac{\partial F}{\partial y} e^{yxyz}
+ \frac{\partial F}{\partial z} e^{zxyz}
\right) \\ &=

\left(
 {\color{green}\frac{\partial F}{\partial x} (0) }
+ {\color{green}\frac{\partial F}{\partial y} (0) }
+ {\color{green}\frac{\partial F}{\partial z} (0) }
\right) \\ \\ &= 0
\end{align}
}

The differential form formulas take the same form in arbitrary coordinate systems. This is not true for the formulas for the gradient, curl, and divergence (in other words they use the wrong formulas and you should use the formulas for exterior derivative and differential forms instead).

A scalar function

# f
f
(
x
,
y
,
z
)

{\displaystyle f=f(x,y,z)}

 is a 0-form

A vector field

F

=
(

F
x

,

F
y

,

F
z

)

{\displaystyle \mathbf F = (\mathbf{F\_x},\mathbf{F\_y},\mathbf{F\_z})}

 corresponds to the 1-form

# φ

F
x

e

x

F
y

e

y

F
z

e

z

{\displaystyle \varphi=\mathbf{F\_x}\,e^{x}+\mathbf{F\_y}\,e^{y}+\mathbf{F\_z}\,e^{z}
}

∇

f

⇒

Δ
∧
# f

∂
f

∂
x

e

x

∂
f

∂
y

e

y

∂
f

∂
z

e

z

∇

F

⇒

Δ
∧
# φ

(

∂

F
z

∂
y

−

∂

F
y

∂
z

)

e

y
z

(

∂

F
x

∂
z

−

∂

F
z

∂
x

)

e

z
x

(

∂

F
y

∂
x

−

∂

F
x

∂
y

)

e

x
y

∇
×

F

⇒

⋆
−
Δ
∧
# φ

(

∂

F
z

∂
y

−

∂

F
y

∂
z

)

e

x

(

∂

F
x

∂
z

−

∂

F
z

∂
x

)

e

y

(

∂

F
y

∂
x

−

∂

F
x

∂
y

)

e

z

{\displaystyle
\begin{align}

\nabla \, f &\Rightarrow \quad

\Delta \wedge f =
\frac{\partial f}{\partial x} \, e^{x} +
\frac{\partial f}{\partial y} \, e^{y} +
\frac{\partial f}{\partial z} \, e^{z} \\

\nabla \mathbf F &\Rightarrow \quad

\Delta \wedge \varphi =
\left(
\frac{\partial F\_z}{\partial y} -
\frac{\partial F\_y}{\partial z}
\right)
\mathbf{e^{yz}} +

\left(
\frac{\partial F\_x}{\partial z} -
\frac{\partial F\_z}{\partial x}
\right)
\mathbf{e^{zx}} +

\left(
\frac{\partial F\_y}{\partial x}
- \frac{\partial F\_x}{\partial y}
\right)
\mathbf{e^{xy}} \\

\nabla \times \mathbf F &\Rightarrow \quad

\star -\Delta \wedge \varphi =
\left(
 {\partial \mathbf{F\_z} \over \partial y} -
 {\partial \mathbf{F\_y} \over \partial z}
\right) \, e^{x} +
\left(
 {\partial \mathbf{F\_x} \over \partial z} -
 {\partial \mathbf{F\_z} \over \partial x}
\right) \, e^{y} +
\left(
 {\partial \mathbf{F\_y} \over \partial x} -
 {\partial \mathbf{F\_x} \over \partial y}
\right) \, e^{z} \\

\end{align}
}

And since

Δ
2

=
0

{\displaystyle \Delta^2=0}

 it follows that

∇
×
(
∇
(
f
)
)
⇒
⋆
−
Δ
∧
(
Δ
∧
f
# )
0

{\displaystyle \nabla \times (\nabla (f)) \Rightarrow \star -\Delta \wedge (\Delta \wedge f) = 0}

Back to top

#### Laplace operator

The Laplace operator is a second-order differential operator in the *n*-dimensional Euclidean space, defined as the divergence (∇·) of the gradient (∇*f* ). Thus if *f* is a twice-differentiable real-valued function, then the Laplacian of *f* is defined by:

∇
2

# f
∇
⋅
∇
f

{\displaystyle \nabla^2 f = \nabla \cdot \nabla f }

∇
2

f
⇒

Δ
⋅
∇
f

=

(

∂

∂
x

e
x

∂

∂
y

e
y

∂

∂
z

e
z

)

⋅

(

∂

∂
x

e
x

∂

∂
y

e
y

∂

∂
z

e
z

)

f
(
x
,
y
,
z
)

=

(

∂

∂
x

e
x

∂

∂
y

e
y

∂

∂
z

e
z

)

⋅

(

∂
f

∂
x

e
x

∂
f

∂
y

e
y

∂
f

∂
z

e
z

)

=

(

∂
2

f

∂

x
2

∂
2

f

∂

y
2

∂
2

f

∂

z
2

)

{\displaystyle
\begin{align}
\nabla^2 f \Rightarrow \quad
\Delta \cdot \nabla f

&=
\left(
 \frac{\partial }{\partial x} e^x
+ \frac{\partial }{\partial y} e^y
+ \frac{\partial }{\partial z} e^z
\right) \cdot
\left(
 \frac{\partial }{\partial x} e\_x
+ \frac{\partial }{\partial y} e\_y
+ \frac{\partial }{\partial z} e\_z
\right)
f(x,y,z) \\ \\ \\

&=
\left(
 \frac{\partial}{\partial x} e^x
+ \frac{\partial}{\partial y} e^y
+ \frac{\partial}{\partial z} e^z
\right) \cdot
\left(
 \frac{\partial f}{\partial x} e\_x
+ \frac{\partial f}{\partial y} e\_y
+ \frac{\partial f}{\partial z} e\_z
\right)\\ \\ \\

&=
\left(
 \frac{\partial^2 f}{\partial x^2}
+ \frac{\partial^2 f}{\partial y^2}
+ \frac{\partial^2 f}{\partial z^2}
\right)
\end{align}
}

Back to top

# A zero-form
f
(
x
,
y
,
z
)

{\displaystyle
\text{A zero-form} = f(x,y,z)
}

interior derivative of a zero-form

∇
⋅
(

f

)

=

(

∂

∂
x

e

x

,

∂

∂
y

e

y

,

∂

∂
z

e

z

)

⋅

(

f
(
x
,
y
,
z
)

)

=
0

The dot product of a vector and a scalar is undefined

{\displaystyle
\begin{align}
\nabla \cdot ( \mathbf{f} ) &=

\left(
\frac{\partial}{\partial x} e\_{x},
\frac{\partial}{\partial y} e\_{y},
\frac{\partial}{\partial z} e\_{z}
\right) \cdot
\left( f(x,y,z) \right) \\ \\ \\ &=

0 \quad \text{The dot product of a vector and a scalar is undefined}

\end{align}
}

interior derivative of a one-form

∇
⋅
(

F

)

=

(

∂

∂
x

e

x

,

∂

∂
y

e

y

,

∂

∂
z

e

z

)

⋅

(

F
x

(
x
,
y
,
z
)

e

x

F
y

(
x
,
y
,
z
)

e

y

F
z

(
x
,
y
,
z
)

e

z

)

=

(

∂

F
x

∂
x

e

x

e

x

∂

F
y

∂
x

e

x

e

y

∂

F
z

∂
x

e

x

e

z

)

(

∂

F
x

∂
y

e

y

e

x

∂

F
y

∂
y

e

y

e

y

∂

F
z

∂
y

e

y

e

z

)

(

∂

F
x

∂
z

e

z

e

x

∂

F
y

∂
z

e

z

e

y

∂

F
z

∂
z

e

z

e

z

)

=

(

∂

F
x

∂
x

(
1
)

∂

F
y

∂
x

(
0
)

∂

F
z

∂
x

(
0
)

)

(

∂

F
x

∂
y

(
0
)

∂

F
y

∂
y

(
1
)

∂

F
z

∂
y

(
0
)

)

(

∂

F
x

∂
z

(
0
)

∂

F
y

∂
z

(
0
)

∂

F
z

∂
z

(
1
)

)

=

∂

F
x

∂
x

(
1
)

∂

F
y

∂
y

(
1
)

∂

F
z

∂
z

(
1
)

{\displaystyle
\begin{align}
\nabla \cdot ( \mathbf{F} ) &=

\left(
\frac{\partial}{\partial x} e\_{x},
\frac{\partial}{\partial y} e\_{y},
\frac{\partial}{\partial z} e\_{z}
\right) \cdot
\left( F\_x(x,y,z) e^{x} + F\_y(x,y,z) e^{y} + F\_z(x,y,z) e^{z} \right) \\ &=

\left(
 {\color{red} \frac{\partial F\_x}{\partial x} e\_{x} e^{x} }
+ {\color{green} \frac{\partial F\_y}{\partial x} \mathbf{e\_{x} e^{y}} }
+ {\color{green} \frac{\partial F\_z}{\partial x} \mathbf{e\_{x} e^{z}} }
\right) +
\left(
 {\color{green} \frac{\partial F\_x}{\partial y} \mathbf{e\_{y} e^{x}} }
+ {\color{red} \frac{\partial F\_y}{\partial y} e\_{y} e^{y} }
+ {\color{green} \frac{\partial F\_z}{\partial y} \mathbf{e\_{y} e^{z}} }
\right) +
\left(
 {\color{green} \frac{\partial F\_x}{\partial z} \mathbf{e\_{z} e^{x}} }
+ {\color{green} \frac{\partial F\_y}{\partial z} \mathbf{e\_{z} e^{y}} }
+ {\color{red} \frac{\partial F\_z}{\partial z} e\_{z} e^{z} }
\right) \\ \\ &=

\left(
 {\color{red} \frac{\partial F\_x}{\partial x} (1) }
+ {\color{green} \frac{\partial F\_y}{\partial x} (0)}
+ {\color{green} \frac{\partial F\_z}{\partial x} (0)}
\right) +
\left(
{\color{green} \frac{\partial F\_x}{\partial y} (0)}
+ {\color{red} \frac{\partial F\_y}{\partial y} (1) }
+ {\color{green} \frac{\partial F\_z}{\partial y} (0)}
\right) +
\left(
 {\color{green} \frac{\partial F\_x}{\partial z} (0)}
+ {\color{green} \frac{\partial F\_y}{\partial z} (0)}
+ {\color{red} \frac{\partial F\_z}{\partial z} (1) }
\right) \\ \\ &=

 {\color{red} \frac{\partial F\_x}{\partial x} (1) } +
 {\color{red} \frac{\partial F\_y}{\partial y} (1) } +
 {\color{red} \frac{\partial F\_z}{\partial z} (1) }
\end{align}
}

interior derivative of a two-form

∇
⋅
(

F

)

=

(

∂

∂
x

e

x

,

∂

∂
y

e

y

,

∂

∂
z

e

z

)

⋅

(

F

x
y

(
x
,
y
,
z
)

e

x
y

F

z
x

(
x
,
y
,
z
)

e

z
x

F

y
z

(
x
,
y
,
z
)

e

y
z

)

=

(

∂

F

x
y

∂
x

e

x

e

x
y

∂

F

z
x

∂
x

e

x

e

z
x

∂

F

y
z

∂
x

e

x

e

y
z

)

(

∂

F

x
y

∂
y

e

y

e

x
y

∂

F

z
x

∂
y

e

y

e

z
x

∂

F

y
z

∂
y

e

y

e

y
z

)

(

∂

F

x
y

∂
z

e

z

e

x
y

∂

F

z
x

∂
z

e

z

e

z
x

∂

F

y
z

∂
z

e

z

e

y
z

)

=

(

∂

F

x
y

∂
x

e

y

−

∂

F

z
x

∂
x

e

z

∂

F

y
z

∂
x

(
0
)

)

(

−

∂

F

x
y

∂
y

e

x

∂

F

z
x

∂
y

(
0
)

∂

F

y
z

∂
y

e

z

)

(

∂

F

x
y

∂
z

(
0
)

∂

F

z
x

∂
z

e

x

−

∂

F

y
z

∂
z

e

y

)

=

∂

F

x
y

∂
x

e

y

−

∂

F

z
x

∂
x

e

z

−

∂

F

x
y

∂
y

e

x

∂

F

y
z

∂
y

e

z

∂

F

z
x

∂
z

e

x

−

∂

F

y
z

∂
z

e

y

=

(

∂

F

z
x

∂
z

e

x

−

∂

F

x
y

∂
y

e

x

)

(

∂

F

x
y

∂
x

e

y

−

∂

F

y
z

∂
z

e

y

)

(

∂

F

y
z

∂
y

e

z

−

∂

F

z
x

∂
x

e

z

)

=

(

∂

F

z
x

∂
z

−

∂

F

x
y

∂
y

)

e

x

(

∂

F

x
y

∂
x

−

∂

F

y
z

∂
z

)

e

y

(

∂

F

y
z

∂
y

−

∂

F

z
x

∂
x

)

e

z

{\displaystyle
\begin{align}

\nabla \cdot ( \mathbf{F} ) &=

\left(
\frac{\partial}{\partial x} e\_{x},
\frac{\partial}{\partial y} e\_{y},
\frac{\partial}{\partial z} e\_{z}
\right) \cdot

\left(
F\_{xy}(x,y,z) e^{xy} +
F\_{zx}(x,y,z) e^{zx} +
F\_{yz}(x,y,z) e^{yz}
\right) \\ &=

\left(
 \frac{\partial F\_{xy}}{\partial x} e\_{x} e^{xy}
 + \frac{\partial F\_{zx}}{\partial x} e\_{x} e^{zx}
 + \frac{\partial F\_{yz}}{\partial x} e\_{x} e^{yz}
\right) +
\left(
 \frac{\partial F\_{xy}}{\partial y} e\_{y} e^{xy}
 + \frac{\partial F\_{zx}}{\partial y} e\_{y} e^{zx}
 + \frac{\partial F\_{yz}}{\partial y} e\_{y} e^{yz}
\right) +
\left(
 \frac{\partial F\_{xy}}{\partial z} e\_{z} e^{xy}
 + \frac{\partial F\_{zx}}{\partial z} e\_{z} e^{zx}
 + \frac{\partial F\_{yz}}{\partial z} e\_{z} e^{yz}
\right) \\ \\ &=

\left(
 {\color{blue} \frac{\partial F\_{xy}}{\partial x} e^{y}}
- {\color{blue} \, \frac{\partial F\_{zx}}{\partial x} e^{z}}
+ {\color{orange} \, \frac{\partial F\_{yz}}{\partial x} (0)}
\right) +
\left(
- {\color{blue} \frac{\partial F\_{xy}}{\partial y} e^{x}}
+ {\color{orange} \, \frac{\partial F\_{zx}}{\partial y} (0)}
+ {\color{blue} \, \frac{\partial F\_{yz}}{\partial y} e^{z}}
\right) +
\left(
 {\color{orange} \frac{\partial F\_{xy}}{\partial z} (0)}
+ {\color{blue} \, \frac{\partial F\_{zx}}{\partial z} e^{x}}
- {\color{blue} \, \frac{\partial F\_{yz}}{\partial z} e^{y}}
\right) \\ \\ &=

 {\color{blue} \frac{\partial F\_{xy}}{\partial x} e^{y}}
- {\color{blue} \, \frac{\partial F\_{zx}}{\partial x} e^{z}}

- {\color{blue} \frac{\partial F\_{xy}}{\partial y} e^{x}}
+ {\color{blue} \, \frac{\partial F\_{yz}}{\partial y} e^{z}}

+ {\color{blue} \, \frac{\partial F\_{zx}}{\partial z} e^{x}}
- {\color{blue} \, \frac{\partial F\_{yz}}{\partial z} e^{y}} \\ \\ &=

\left(
 {\color{blue} \, \frac{\partial F\_{zx}}{\partial z} e^{x}}
- {\color{blue} \frac{\partial F\_{xy}}{\partial y} e^{x}}
\right) +
\left(
 {\color{blue} \frac{\partial F\_{xy}}{\partial x} e^{y}}
- {\color{blue} \, \frac{\partial F\_{yz}}{\partial z} e^{y}}
\right) +
\left(
 {\color{blue} \, \frac{\partial F\_{yz}}{\partial y} e^{z}}
- {\color{blue} \, \frac{\partial F\_{zx}}{\partial x} e^{z}}
\right) \\ \\ &=

\left(
 {\color{blue} \, \frac{\partial F\_{zx}}{\partial z} }
- {\color{blue} \frac{\partial F\_{xy}}{\partial y} }
\right) e^{x}+
\left(
 {\color{blue} \frac{\partial F\_{xy}}{\partial x} }
- {\color{blue} \, \frac{\partial F\_{yz}}{\partial z} }
\right) e^{y} +
\left(
 {\color{blue} \, \frac{\partial F\_{yz}}{\partial y} }
- {\color{blue} \, \frac{\partial F\_{zx}}{\partial x} }
\right) e^{z}

\end{align}
}

In three dimensions the interior derivative of a three-form is

∇
⋅
(

F

)

=

(

∂

∂
x

e

x

,

∂

∂
y

e

y

,

∂

∂
z

e

z

)

⋅

(

F
(
x
,
y
,
z
)

e

x
y
z

)

=

(

∂
F

∂
x

e

x

e

x
y
z

∂
F

∂
y

e

y

e

x
y
z

∂
F

∂
z

e

z

e

x
y
z

)

=

(

∂
F

∂
x

e

y
z

∂
F

∂
y

e

z
x

∂
F

∂
z

e

x
y

)

{\displaystyle
\begin{align}
\nabla \cdot ( \mathbf{F} ) &=

\left(
\frac{\partial}{\partial x} e\_{x},
\frac{\partial}{\partial y} e\_{y},
\frac{\partial}{\partial z} e\_{z}
\right) \cdot

\left(
F(x,y,z) e^{xyz}
\right) \\ \\ \\ &=

\left(
 \frac{\partial F}{\partial x} e\_{x} e^{xyz}
+ \frac{\partial F}{\partial y} e\_{y} e^{xyz}
+ \frac{\partial F}{\partial z} e\_{z} e^{xyz}
\right) \\ &=

\left(
 {\color{green}\frac{\partial F}{\partial x} e^{yz}}
+ {\color{green}\frac{\partial F}{\partial y} e^{zx}}
+ {\color{green}\frac{\partial F}{\partial z} e^{xy}}
\right)

\end{align}
}

Back to top

### Directional derivative

From Wikipedia: Directional derivative

The *directional derivative* of a scalar function

f
(

x

# )
f
(

x
1

,

x
2

,
…
,

x
n

)

{\displaystyle f(\mathbf{x}) = f(x\_1, x\_2, \ldots, x\_n)}

along a vector

v

=
(

v
1

,
…
,

v
n

)

{\displaystyle \mathbf{v} = (v\_1, \ldots, v\_n)}

is the function

∇

v

f

{\displaystyle \nabla\_{\mathbf{v}}{f}}

 defined by the limit

∇

v

f

(

x

)

=

lim

h
→
0

f
(

x

h

v

)
−
f
(

x

)

h

{\displaystyle
\begin{align}
\nabla\_{\mathbf{v}}{f}(\mathbf{x}) &= \lim\_{h \rightarrow 0}{\frac{f(\mathbf{x} + h\mathbf{v}) - f(\mathbf{x})}{h}}
\end{align}
}

∇

v

f

(

x

)

=
∇
f
(

x

)
⋅

v

=

(

∂
f

∂
x

e
x

∂
f

∂
y

e
y

∂
f

∂
z

e
z

)

⋅

v

=

(

∂
f

∂
x

v
x

∂
f

∂
y

v
y

∂
f

∂
z

v
z

)

{\displaystyle
\begin{align}
\nabla\_{\mathbf{v}}{f}(\mathbf{x}) &= \nabla f(\mathbf{x}) \cdot \mathbf{v} \\

&=
\left(
 \frac{\partial f}{\partial x} e\_x
+ \frac{\partial f}{\partial y} e\_y
+ \frac{\partial f}{\partial z} e\_z
\right) \cdot \mathbf{v} \\

&=
\left(
 \frac{\partial f}{\partial x} v\_x
+ \frac{\partial f}{\partial y} v\_y
+ \frac{\partial f}{\partial z} v\_z
\right)
\end{align}
}

∇

v

f

(

x

)

=

ι
v

Δ
∧
f

=

(

v
x

e
x

v
y

e
y

v
z

e
z

)

(

∂
f

∂
x

d
x

∂
f

∂
y

d
y

∂
f

∂
z

d
z

)

=

(

v
x

e
x

v
y

e
y

v
z

e
z

)

(

∂
f

∂
x

e
x

∂
f

∂
y

e
y

∂
f

∂
z

e
z

)

{\displaystyle
\begin{align}
\nabla\_{\mathbf{v}}{f}(\mathbf{x}) &= \iota\_v \, \Delta \wedge f \\

& =
\left(
 v\_x e\_x +
 v\_y e\_y +
 v\_z e\_z
\right)
\left(
 \frac{\partial f}{\partial x} dx
+ \frac{\partial f}{\partial y} dy
+ \frac{\partial f}{\partial z} dz
\right) \\ \\

& =
\left(
 v\_x e\_x +
 v\_y e\_y +
 v\_z e\_z
\right)
\left(
 \frac{\partial f}{\partial x} e^x
+ \frac{\partial f}{\partial y} e^y
+ \frac{\partial f}{\partial z} e^z
\right)
\end{align}
}

∇

v

f

(

x

)

=
ω
(

v

# ω
Δ
∧
f

=
scalar

{\displaystyle
\begin{align}
\nabla\_{\mathbf{v}}{f}(\mathbf{x}) &= \omega (\mathbf{v}) = \quad \quad \text{where } \omega = \Delta \wedge f \\

&= \text{scalar}
\end{align}
}

The *directional derivative* of a vector function

∇

v

f

(

x

)

=
∇

f

(

x

)
⋅

v

=

J

f

⋅

v

=
vector

{\displaystyle

\begin{align}
\nabla\_{\mathbf{v}}{\mathbf{f}}(\mathbf{x}) &= \nabla \mathbf{f}(\mathbf{x}) \cdot \mathbf{v} \\
&= \mathbf{J}\_{\mathbf{f}} \cdot \mathbf{v} \\
&= \text{vector}
\end{align}
}

The *directional derivative* of a multivector function is

∇
a

F
(
b
# )

lim

ϵ
→
0

F
(
b
ϵ
a
)
−
F
(
b
)

ϵ

,

{\displaystyle \nabla\_a F(b) = \lim\_{\epsilon \rightarrow 0}{\frac{F(b + \epsilon a) - F(b)}{\epsilon}},}

Next, choose a set of basis vectors

{

e
i

}

{\displaystyle \{e\_i\}}

 and consider the operators, denoted

∂

i

{\displaystyle \partial\_{i}}

, that perform directional derivatives in the directions of

e
i

{\displaystyle e\_i}

:

∂
i

:
F
↦
(
x
↦

∇

e
i

F
(
x
)
)
.

{\displaystyle \partial\_i : F \mapsto (x\mapsto \nabla\_{e\_i} F(x)).}

Then, using the Einstein summation notation, consider the operator:

e
i

∂
i

,

{\displaystyle e^i\partial\_i,}

It can be shown that this operator is independent of the choice of frame, and can thus be used to define the *geometric derivative*:

# ∇

e
i

∂
i

.

{\displaystyle \nabla = e^i\partial\_i.}

The direction

a

{\displaystyle a}

 can be written

# a
(
a
⋅

e
i

)

e
i

{\displaystyle a = (a\cdot e^i) e\_i}

, so that:

∇
a

=

∇

(
a
⋅

e
i

)

e
i

=
(
a
⋅

e
i

)

∇

e
i

=
a
⋅
(

e
i

∇

e
i

# )
a
⋅
∇
.

{\displaystyle \nabla\_a = \nabla\_{(a\cdot e^i)e\_i} = (a\cdot e^i)\nabla\_{e\_i} = a\cdot(e^i\nabla\_{e^i}) = a\cdot \nabla.}

For this reason,

∇
a

F
(
x
)

{\displaystyle \nabla\_a F(x)}

 is often noted

a
⋅
∇
F
(
x
)

{\displaystyle a\cdot \nabla F(x)}

.

See also: Geometric gradient

Back to top

#### Covariant derivative

Also called **Intrinsic derivative**.

From Wikipedia: Covariant derivative

In mathematics, the covariant derivative is a way of specifying a derivative along tangent vectors of a manifold. In the special case of a manifold isometrically embedded into a higher-dimensional Euclidean space, the covariant derivative can be viewed as the orthogonal projection of the Euclidean directional derivative onto the manifold's tangent space.

The **covariant derivative** is a generalization of the directional derivative from vector calculus. As with the directional derivative, the covariant derivative is a rule,

∇

u

v

{\displaystyle \nabla\_{\mathbf u}{\mathbf v}}

, which takes as its inputs: (1) a vector, **u**, defined at a point *P*, and (2) a vector field, **v**, defined in a neighborhood of *P*.

* The covariant derivative is also denoted variously by **∂

{\displaystyle \partial}

vu**, **Dvu**, or other notations.

The output is the vector

∇

u

v

(
P
)

{\displaystyle \nabla\_{\mathbf u}{\mathbf v}(P)}

, also at the point *P*. The primary difference from the usual directional derivative is that

∇

u

v

{\displaystyle \nabla\_{\mathbf u}{\mathbf v}}

 must, in a certain precise sense, be *independent* of the manner in which it is expressed in a coordinate system.

In the case of Euclidean space, one tends to define the derivative of a vector field in terms of the difference between two vectors at two nearby points. In such a system one translates one of the vectors to the origin of the other, keeping it parallel.

* With a Cartesian (fixed orthonormal) coordinate system "keeping it parallel" amounts to keeping the components constant.
* In the general case, however, one must take into account the change of the coordinate system. For example, if the same covariant derivative is written in polar coordinates in a two dimensional Euclidean plane, then it contains extra terms that describe how the coordinate grid itself "rotates". In other cases the extra terms describe how the coordinate grid expands, contracts, twists, interweaves, etc. In this case "keeping it parallel" does not amount to keeping components constant under translation.

∇

v

u

=

∇

v
j

e

j

u
i

e

i

=

v
j

∇

e

j

u
i

e

i

=

v
j

u
i

∇

e

j

e

i

v
j

e

i

∇

e

j

u
i

=

v
j

u
i

Γ
k

i
j

e

k

v
j

∂

u
i

∂

x
j

e

i

=

(

v
j

u
i

Γ
k

i
j

v
j

∂

u
k

∂

x
j

)

e

k

{\displaystyle
\begin{align}
 \nabla\_\mathbf{v} \mathbf{u}
 &= \nabla\_{v^j \mathbf{e}\_j} u^i \mathbf{e}\_i \\
 &= v^j \nabla\_{\mathbf{e}\_j} u^i \mathbf{e}\_i \\
 &= v^j u^i \nabla\_{\mathbf{e}\_j} \mathbf{e}\_i + v^j \mathbf{e}\_i \nabla\_{\mathbf{e}\_j} u^i \\
 &= v^j u^i {\Gamma^k}\_{i j}\mathbf{e}\_k + v^j{\partial u^i \over \partial x^j} \mathbf{e}\_i \\
 &= \left(v^j u^i {\Gamma^k}\_{i j} + v^j{\partial u^k\over\partial x^j}\right)\mathbf{e}\_k
\end{align}
}

The first term in this formula is responsible for "twisting" the coordinate system with respect to the covariant derivative and the second for changes of components of the vector field *u*. See Christoffel symbols.

The covariant derivative is the usual derivative along the coordinates with correction terms which tell how the coordinates change

A covariant derivative introduces an extra geometric structure on a manifold that allows vectors in neighboring tangent spaces to be compared: there is no canonical way to compare vectors from different tangent spaces because there is no canonical coordinate system. (When a Riemannian metric is given, there is a canonical choice of connection: the Levi-Civita connection.)

From Wikipedia:Christoffel symbols

In mathematics and physics, the Christoffel symbols are an array of numbers describing a metric connection. The metric connection is a specialization of the affine connection to surfaces or other manifolds endowed with a metric, allowing distances to be measured on that surface. In differential geometry, an affine connection can be defined without reference to a metric, and many additional concepts follow: parallel transport, covariant derivatives, geodesics, etc. also do not require the concept of a metric. However, when a metric is available, these concepts can be directly tied to the "shape" of the manifold itself; that shape is determined by how the tangent space is attached to the cotangent space by the metric tensor. Abstractly, one would say that the manifold has an associated (orthonormal) frame bundle, with each "frame" being a possible choice of a coordinate frame. An invariant metric implies that the structure group of the frame bundle is the orthogonal group O(p, q). As a result, such a manifold is necessarily a (pseudo-)Riemannian manifold. The Christoffel symbols provide a concrete representation of the connection of (pseudo-)Riemannian geometry in terms of coordinates on the manifold.

The Christoffel symbols of the second kind are the connection coefficients—in a coordinate basis—of the Levi-Civita connection.
In other words, the Christoffel symbols of the second kind Γ*k**ij* (sometimes Γ*k*
*ij* or {*k*
*ij*}) are defined as the unique coefficients such that

∇
i

e

j

=

Γ
k

i
j

e

k

{\displaystyle \nabla\_i \mathrm{e}\_j = {\Gamma^k}\_{ij}\mathrm{e}\_k}

,
where ∇*i* is the Levi-Civita connection on *M* taken in the coordinate direction e*i* (i.e., ∇*i* ≡ ∇e*i*) and where e*i* = ∂*i* is a local coordinate (holonomic) basis. Since this connection has zero torsion, and holonomic vector fields commute (i.e.

[

e
i

,

e
j

# ]
[

∂
i

,

∂
j

# ]
0

{\displaystyle [e\_i, e\_j] = [\partial\_i, \partial\_j] = 0}

) we have

∇
i

e

j

=

∇
j

e

i

{\displaystyle \nabla\_i \mathrm{e}\_j = \nabla\_j \mathrm{e}\_i}

.
Hence in this basis the connection coefficients are symmetric:

Γ*k**ij* = Γ*k**ji*.
For this reason, a torsion-free connection is often called *symmetric*.

There is however another generalization of directional derivatives which *is* canonical: the Lie derivative, which evaluates the change of one vector field along the flow of another vector field. Thus, one must know both vector fields in an open neighborhood, not merely at a single point. Note that the antisymmetrized covariant derivative ∇*u**v* − ∇*v**u*, and the Lie derivative *L**u**v* differ by the torsion of the connection, so that if a connection is torsion free, then its antisymmetrization *is* the Lie derivative.

From Wikipedia: Torsion tensor

Torsion gives an intrinsic characterization of how tangent spaces twist about a curve when they are parallel transported; whereas curvature describes how the tangent spaces roll along the curve. Torsion may be described concretely as a two-form on the manifold.

If ∇ is an affine connection on a differential manifold, then the torsion tensor is defined, in terms of vector fields *X* and *Y*, by

T
(
X
,
Y
# )

∇
X

Y
−

∇
Y

X
−
[
X
,
Y
]

{\displaystyle T(X,Y) = \nabla\_XY-\nabla\_YX - [X,Y]}

where [*X*,*Y*] is the Lie bracket of vector fields.

The Lie bracket [*X*, *Y*] is the derivative of *Y* along the flow generated by *X*, and is sometimes denoted *L

X

Y

{\displaystyle \mathcal{L}\_X Y}* ("Lie derivative of Y along X"). This generalizes to the Lie derivative of any tensor field along the flow generated by *X*.

Back to top

#### Lie derivative

From Wikipedia: Lie derivative

The Lie derivative evaluates the change of a tensor field (including scalar functions, vector fields and one-forms), along the flow defined by another vector field. This change is coordinate invariant and therefore the Lie derivative is defined on any differentiable manifold.

A 'naïve' attempt to define the derivative of a tensor field with respect to a vector field would be to take the components of the tensor field and take the directional derivative with respect to the vector field of each component. However, this definition is undesirable because it is not invariant under changes of coordinate system, e.g. the naive derivative expressed in polar or spherical coordinates differs from the naive derivative of the components in Cartesian coordinates.

In differential geometry, there are three main coordinate independent notions of differentiation of tensor fields:

* Exterior derivative of completely anti symmetric (covariant) tensors or differential forms.
* Derivatives with respect to connections
	+ The notion of a connection makes precise the idea of transporting data along a curve or family of curves in a parallel and consistent manner. In general, there are an infinite number of metric connections for a given metric tensor; however, there is a unique connection that is free of torsion, the Levi-Civita connection.
* Lie derivatives

The Lie derivative of a function (a zero-form) with respect to a vector field *X* at *p* is identical to the directional derivative *X*(*f*):

L

X

f

=
X
(
f
)

=
∇
f
(
x
)
⋅
X

=

(

∂
f

∂
x

e
x

∂
f

∂
y

e
y

∂
f

∂
z

e
z

)

⋅

X

=

(

∂
f

∂
x

X

x

∂
f

∂
y

X

y

∂
f

∂
z

X

z

)

=
X
(
f
# )
s
c
a
l
a
r

{\displaystyle
\begin{align}
\mathcal{L}\_Xf &= X(f) \\
&= \nabla f(x) \cdot X \\
&=
\left(
 \frac{\partial f}{\partial x} e\_x
+ \frac{\partial f}{\partial y} e\_y
+ \frac{\partial f}{\partial z} e\_z
\right) \cdot \mathbf{X} \\
&=
\left(
 \frac{\partial f}{\partial x} \mathbf{X}\_x
+ \frac{\partial f}{\partial y} \mathbf{X}\_y
+ \frac{\partial f}{\partial z} \mathbf{X}\_z
\right) \\
&= X(f) = scalar
\end{align}
}

which is the same as the contraction of the exterior derivative of *f* with *X*:

L

X

f

=

ι
X

Δ
∧
f

=

ι
X

(

∂
f

∂
x

d
x

∂
f

∂
y

d
y

∂
f

∂
z

d
z

)

{\displaystyle
\begin{align}
\mathcal{L}\_Xf
&= \iota\_X \, \Delta \wedge f \\
&= \iota\_X
\left(
 \frac{\partial f}{\partial x} dx
+ \frac{\partial f}{\partial y} dy
+ \frac{\partial f}{\partial z} dz
\right)
\end{align}
}

The Lie derivative of a differential form is the anticommutator of the interior product with the exterior derivative:

L

X

# ω

ι
X

Δ
∧
ω
Δ
∧
(

ι

X

ω
)

{\displaystyle
\mathcal{L}\_X\omega = \iota\_X \Delta \wedge \omega + \Delta \wedge (\iota\_{X} \omega)
}

This identity is known variously as **Cartan formula**, **Cartan homotopy formula** or **Cartan's magic formula**. See interior product for details.

The Lie derivative of a vector field *Y* with respect to a vector field *X* is also known as the Lie bracket of *X* and *Y*, and is sometimes denoted

[
X
,
Y
]

{\displaystyle [X,Y]}

. The Lie bracket of *X* and *Y* at *p* is given in local coordinates by the formula

L

X

Y
(
p
# )
[
X
,
Y
]
(
p
)

=

∂
X

Y
(
p
)
−

∂
Y

X
(
p
)
−
T
(
X
,
Y
)

=

∂
X

Y
(
p
)
−

∂
Y

X
(
p
)

(Because its torsion free. See above)

=
∇

Y

⋅

X

−
∇

X

⋅

Y

(The corrective terms cancel)

=
v
e
c
t
o
r

{\displaystyle
\begin{align}
\mathcal{L}\_X Y (p) = [X,Y](p)
&= \partial\_X Y(p) - \partial\_Y X(p) -T(X,Y) \\
&= \partial\_X Y(p) - \partial\_Y X(p)
 \quad \text{(Because its torsion free. See above)} \\
&= \nabla \mathbf{Y} \cdot \mathbf{X} - \nabla \mathbf{X} \cdot \mathbf{Y}
 \quad \text{(The corrective terms cancel)} \\
&= vector
\end{align}
}

where

∂
X

{\displaystyle \partial\_X}

 and

∂
Y

{\displaystyle \partial\_Y}

 denote the operations of taking the Covariant derivatives with respect to *X* and *Y*, respectively.
If *X* is constant then

∇

X

=
0

{\displaystyle \nabla \mathbf{X} = 0}

 and the Lie derivative reduces to the ordinary directional derivative.

From Wikipedia: Levi-Civita\_connection#Christoffel\_symbols

Let ∇ be the connection of the Riemannian metric. Choose local coordinates *x*1 … *xn* and let Γ*ljk* be the Christoffel symbols with respect to these coordinates. The torsion freeness condition is then equivalent to the symmetry

Γ

j
k

l

=

Γ

k
j

l

{\displaystyle \Gamma^l\_{jk} = \Gamma^l\_{kj}}

That is why the corrective terms cancel

Back to top

## Physics

### Classical mechanics

See also: Wikipedia:Phasor
The derivative of potential is force:

# F
∇

E

=

(

∂

∂
x

,

∂

∂
y

,

∂

∂
z

)

(

E
(
x
,
y
,
z
)

)

=

(

∂
E

∂
x

e
x

∂
E

∂
y

e
y

∂
E

∂
z

e
z

)

{\displaystyle
\begin{align}
F =
\nabla \mathbf{E} &=

\left( \frac{\partial}{\partial x}, \frac{\partial}{\partial y}, \frac{\partial}{\partial z} \right)
\left( E(x,y,z) \right) \\ \\ \\ &=
\left(
 \frac{\partial E}{\partial x} e\_x
+ \frac{\partial E}{\partial y} e\_y
+ \frac{\partial E}{\partial z} e\_z
\right)

\end{align}
}

Assuming time is a scalar:

A
c
c
e
l
e
r
a
t
i
o
# n

(

d

d
t

)

V

=

(

d

d
t

)

(

v
x

(
t
)

e
x

v
y

(
t
)

e
y

v
z

(
t
)

e
z

)

=

(

d

v
x

d
t

e

x

d

v
y

d
t

e

y

d

v
z

d
t

e

z

)

{\displaystyle
\begin{align}
Acceleration =
\left( \frac{d}{d t} \right) \mathbf{V} &=

\left( \frac{d}{d t} \right)
\left( v\_x(t) e\_x + v\_y(t) e\_y + v\_z(t) e\_z \right) \\ &=

\left(
 \frac{d v\_x}{d t} \mathbf{e\_{x}}
+ \frac{d v\_y}{d t} \mathbf{e\_{y}}
+ \frac{d v\_z}{d t} \mathbf{e\_{z}}
\right)

\end{align}
}

From Wikipedia:Lorentz force:

F

=
q
(

E

v

×

B

)

{\displaystyle \mathbf{F} = q(\mathbf{E} + \mathbf{v} \times \mathbf{B})}

F
x

=
q
(

E
x

v
y

B
z

−

v
z

B
y

)

e
x

{\displaystyle F\_x = q (E\_x + v\_y B\_z - v\_z B\_y) e\_x}

F
y

=
q
(

E
y

v
z

B
x

−

v
x

B
z

)

e
y

{\displaystyle F\_y = q (E\_y + v\_z B\_x - v\_x B\_z) e\_y}

F
z

=
q
(

E
z

v
x

B
y

−

v
y

B
x

)

e
z

{\displaystyle F\_z = q (E\_z + v\_x B\_y - v\_y B\_x) e\_z}

And:

F

(

r

,

r
˙

,
t
,
q
# )
q
[

E

(

r

,
t
)

r
˙

×

B

(

r

,
t
)
]

{\displaystyle \mathbf{F}(\mathbf{r},\mathbf{\dot{r}},t,q) = q[\mathbf{E}(\mathbf{r},t) + \mathbf{\dot{r}} \times \mathbf{B}(\mathbf{r},t)]}

The rate at which energy is transferred from the electromagnetic field to the particle is:

v

⋅

F

=
q

v

⋅

E

{\displaystyle \mathbf{v} \cdot \mathbf{F} = q \, \mathbf{v} \cdot \mathbf{E}}

Back to top

### Maxwell's equations

From Wikipedia:Mathematical descriptions of the electromagnetic field

**E**(*x*, *y*, *z*, *t*) and **B**(*x*, *y*, *z*, *t*) are vector fields called the electric field and the magnetic field.

The behaviour of electric and magnetic fields, whether in cases of electrostatics, magnetostatics, or electrodynamics (electromagnetic fields), is governed by Maxwell's equations:

∇
⋅

E

=

ρ

ε
0

{\displaystyle \nabla \cdot \mathbf{E} = \frac{\rho}{\varepsilon\_0}}

∇
⋅

B

=
0

{\displaystyle \nabla \cdot \mathbf{B} = 0}

∇
×

B

=

μ
0

J

μ
0

ε
0

∂

E

∂
t

{\displaystyle \nabla \times \mathbf{B} = \mu\_0 \mathbf{J} + \mu\_0\varepsilon\_0 \frac{\partial \mathbf{E}}{\partial t}}

∇
×

E

=
−

∂

B

∂
t

{\displaystyle \nabla \times \mathbf{E} = -\frac {\partial \mathbf{B}}{\partial t}}

where *ρ* is the charge density, *ε*0 is the electric constantw, *μ*0 is the magnetic constantw, and **J** is the current per unit area.

Back to top

#### Algebra of physical space

In geometric algebraw two multivectorsw are introduced. One for the field and one for the current.

The field multivector, known as the Riemann–Silberstein vector, is

F

=

E

I
c

B

=

E
k

σ
k

I
c

B
k

σ
k

=
vector + pseudovector

{\displaystyle \bold{F} = \bold{E} + Ic\bold{B} = E^k\sigma\_k + IcB^k\sigma\_k = \text{vector + pseudovector}}

The current multivector is

c
ρ
−

J

=
c
ρ
−

J
k

σ
k

=
scalar + vector

{\displaystyle c \rho - \bold{J} = c \rho - J^k\sigma\_k = \text{scalar + vector}}

where, in the algebra of physical space (APS)

C

ℓ

3
,
0

(

R

)

{\displaystyle C\ell\_{3,0}(\mathbb{R})}

 with the vector basis

{

σ
k

}

{\displaystyle \{\sigma\_k\}}

. The unit pseudoscalar is

# I

σ
1

σ
2

σ
3

{\displaystyle I=\sigma\_1\sigma\_2\sigma\_3}

 (assuming an orthonormal basis). Orthonormal basis vectors share the algebra of the Pauli matrices, but are usually not equated with them.
After defining the derivative

# ∇

σ
k

∂
k

{\displaystyle \boldsymbol{\nabla} = \sigma^k \partial\_k}

Maxwell's equations are reduced to the single equation

(

1
c

∂

∂
t

∇

)

F

=

μ
0

c
(
c
ρ
−

J

)
.

{\displaystyle \left(\frac{1}{c}\dfrac{\partial }{\partial t} + \boldsymbol{\nabla}\right)\bold{F} = \mu\_0 c (c \rho - \bold{J}). }

In three dimensions, the derivative has a special structure allowing the introduction of a cross product:

∇

F

=
∇
⋅

F

∇
∧

F

=
∇
⋅

F

I
∇
×

F

{\displaystyle \boldsymbol{\nabla}\bold{F} = \boldsymbol{\nabla} \cdot \bold{F} + \boldsymbol{\nabla} \wedge \bold{F} = \boldsymbol{\nabla} \cdot \bold{F} + I \boldsymbol{\nabla} \times \bold{F}}

from which it is easily seen that Gauss's law is the scalar part, the Ampère–Maxwell law is the vector part, Faraday's law is the pseudovector part, and Gauss's law for magnetism is the pseudoscalar part of the equation. After expanding and rearranging, this can be written as

(

∇
⋅

E

−

ρ

ϵ
0

)

−
c

(

∇
×

B

−

μ
0

ϵ
0

∂

E

∂

t

−

μ
0

J

)

I

(

∇
×

E

∂

B

∂

t

)

I
c

(

∇
⋅

B

)

=
0

{\displaystyle
\left( \boldsymbol{\nabla} \cdot \mathbf{E} - \frac{\rho}{\epsilon\_0} \right)- c \left( \boldsymbol{\nabla} \times \mathbf{B} - \mu\_0 \epsilon\_0 \frac{\partial {\mathbf{E}}}{\partial {t}} - \mu\_0 \mathbf{J} \right)+ I \left( \boldsymbol{\nabla} \times \mathbf{E} + \frac{\partial {\mathbf{B}}}{\partial {t}} \right)+ I c \left( \boldsymbol{\nabla} \cdot \mathbf{B} \right)= 0
}

Back to top

#### Spacetime algebra

We can identify APS as a subalgebra of the spacetime algebra (STA)

C

ℓ

1
,
3

(

R

)

{\displaystyle C\ell\_{1,3}(\mathbb{R})}

, defining

σ
k

=

γ
k

γ
0

{\displaystyle \sigma\_k=\gamma\_k\gamma\_0}

 and

# I

γ
0

γ
1

γ
2

γ
3

{\displaystyle I=\gamma\_0\gamma\_1\gamma\_2\gamma\_3}

. The

γ
μ

{\displaystyle \gamma\_\mu}

s have the same algebraic properties of the gamma matrices but their matrix representation is not needed. The derivative is now

# ∇

γ
μ

∂
μ

.

{\displaystyle \nabla = \gamma^\mu \partial\_\mu.}

The Riemann–Silberstein becomes a bivector

# F

E

I
c

B

=

E
1

γ
1

γ
0

E
2

γ
2

γ
0

E
3

γ
3

γ
0

−
c
(

B
1

γ
2

γ
3

B
2

γ
3

γ
1

B
3

γ
1

γ
2

)
,

{\displaystyle F = \bold{E} + Ic\bold{B} = E^1\gamma\_1\gamma\_0 + E^2\gamma\_2\gamma\_0 + E^3\gamma\_3\gamma\_0 -c(B^1\gamma\_2\gamma\_3 + B^2\gamma\_3\gamma\_1 + B^3\gamma\_1\gamma\_2),}

and the charge and current density become a vector

# J

J
μ

γ
μ

=
c
ρ

γ
0

J
k

γ
k

=

γ
0

(
c
ρ
−

J
k

σ
k

)
.

{\displaystyle J = J^\mu \gamma\_\mu = c \rho \gamma\_0 + J^k \gamma\_k = \gamma\_0(c \rho - J^k \sigma\_k).}

Owing to the identity

γ
0

# ∇

γ
0

γ
0

∂
0

γ
0

γ
k

∂
k

=

∂
0

σ
k

∂
k

=

1
c

∂

∂
t

∇
,

{\displaystyle \gamma\_0 \nabla = \gamma\_0\gamma^0 \partial\_0 + \gamma\_0\gamma^k\partial\_k = \partial\_0 + \sigma^k\partial\_k = \frac{1}{c}\dfrac{\partial }{\partial t} + \boldsymbol{\nabla},}

Maxwell's equations reduce to the single equation

∇
# F

μ
0

c
J
.

{\displaystyle \nabla F = \mu\_0 c J. }

Back to top

### Electromagnetic tensor

From Wikipedia: Electromagnetic tensor

The electromagnetic tensor or electromagnetic field tensor (sometimes called the field strength tensor, Faraday tensor or Maxwell bivector) is a mathematical object that describes the electromagnetic field in spacetime.

In contravariant matrix form,

F

μ
ν

=

[

0

−

E
x

/

c

−

E
y

/

c

−

E
z

/

c

E
x

/

c

0

−

B
z

B
y

E
y

/

c

B
z

0

−

B
x

E
z

/

c

−

B
y

B
x

0

]

.

{\displaystyle
 F^{\mu\nu} = \begin{bmatrix}
 0 & -E\_x/c & -E\_y/c & -E\_z/c \\
 E\_x/c & 0 & -B\_z & B\_y \\
 E\_y/c & B\_z & 0 & -B\_x \\
 E\_z/c & -B\_y & B\_x & 0
 \end{bmatrix}.
}

This tensor simplifies and reduces Maxwell's equations as four vector calculus equations into two tensor field equations. In electrostatics and electrodynamics, Gauss's law and Ampère's circuital law are respectively:

∇
⋅

E

=

ρ

ϵ
0

,

∇
×

B

−

1

c
2

∂

E

∂
t

=

μ
0

J

{\displaystyle \nabla \cdot \mathbf{E} = \frac{\rho}{\epsilon\_0},\quad \nabla \times \mathbf{B} - \frac{1}{c^2} \frac{ \partial \mathbf{E}}{\partial t} = \mu\_0 \mathbf{J} }

and reduce to the inhomogeneous Maxwell equation:

∂

α

F

α
β

=

μ
0

J

β

{\displaystyle \partial\_{\alpha} F^{\alpha\beta} = \mu\_0 J^{\beta}}

, where

J

α

=
(
c
ρ
,

J

)

{\displaystyle J^{\alpha} = ( c\rho, \mathbf{J} ) }

 is the four-current.
In magnetostatics and magnetodynamics, Gauss's law for magnetism and Maxwell–Faraday equation are respectively:

∇
⋅

B

=
0
,

∂

B

∂
t

∇
×

E

=
0

{\displaystyle \nabla \cdot \mathbf{B} = 0,\quad \frac{ \partial \mathbf{B}}{ \partial t } + \nabla \times \mathbf{E} = 0 }

which reduce to Bianchi identity:

∂
γ

F

α
β

∂
α

F

β
γ

∂
β

F

γ
α

=
0

{\displaystyle \partial\_\gamma F\_{ \alpha \beta } + \partial\_\alpha F\_{ \beta \gamma } + \partial\_\beta F\_{ \gamma \alpha } = 0}

or using the index notation with square brackets for the antisymmetric part of the tensor:

∂

[
α

F

β
γ
]

=
0

{\displaystyle \partial\_{ [ \alpha } F\_{ \beta \gamma ] } = 0 }

From Wikipedia: Classical electromagnetism and special relativity

The field tensor transforms according to the Lorentz transformation according to

F

′

α
β

=

Λ
μ
α

Λ
ν
β

F

μ
ν

{\displaystyle F'^{\alpha \beta} = \Lambda^\alpha\_\mu \Lambda^\beta\_\nu F^{\mu \nu}}

,
where Λαν is the Lorentz transformation tensor for a change from one reference frame to another. The same tensor is used twice in the summation.

The charge and current density, the sources of the fields, also combine into the four-vector

J
α

=

(

c
ρ
,

J
x

,

J
y

,

J
z

)

{\displaystyle J^\alpha = \left(c \rho, J\_x, J\_y, J\_z \right)}

called the four-current.

The EM field tensor can also be written

F

α
β

=

∂

A
β

∂

x
α

−

∂

A
α

∂

x
β

,

{\displaystyle F^{\alpha \beta} = \frac {\partial A^\beta}{\partial x\_\alpha} - \frac {\partial A^\alpha}{\partial x\_\beta} \, ,}

where

A
α

=

(

φ
c

,

A
x

,

A
y

,

A
z

)

,

{\displaystyle A^\alpha = \left(\frac{\varphi}{c}, A\_x, A\_y, A\_z\right)\,, }

is the four-potential and

x
α

=
(
c
t
,
−
x
,
−
y
,
−
z
)

{\displaystyle x\_\alpha = (ct, -x, -y, -z ) }

is the four-position.

Back to top

### Gauge theory

From Wikipedia: Gauge theory

Historically, the first example of gauge symmetry discovered was classical electromagnetism.

In electrostatics, one can either discuss the electric field, **E**, or its corresponding  electric potential, *V*. Knowledge of one makes it possible to find the other, except that potentials differing by a constant,

V
↦
V
C

{\displaystyle V \mapsto V+C}

, correspond to the same electric field. In terms of vector calculus, the electric field is the gradient of the potential:

E

=
−
∇
V

{\displaystyle \mathbf{E} = -\nabla V}

.

Generalizing from static electricity to electromagnetism, we have a second potential, the vector potential **A**, with

E

=
−
∇
V
−

∂

A

∂
t

{\displaystyle
\begin{align}
 \mathbf{E} &=
 -\nabla V - \frac{\partial \mathbf{A}}{\partial t}
\end{align}
}

Magnetic vector potential, A, is the vector quantity in classical electromagnetism defined so that its curl is equal to the magnetic field:

B

=
∇
×

A

{\displaystyle \mathbf{B} = \nabla \times \mathbf{A} }

The above definition does not define the magnetic vector potential uniquely because, by definition, we can arbitrarily add curl-free components to the magnetic potential without changing the observed magnetic field. Thus, there is a degree of freedom available when choosing A. This condition is known as gauge invariance.

The general gauge transformations now become not just

V
↦
V
C

{\displaystyle V \mapsto V+C}

 but

A

↦

A

∇
f

V

↦
V
−

∂
f

∂
t

{\displaystyle \begin{align}
 \mathbf{A} &\mapsto \mathbf{A} + \nabla f\\
 V &\mapsto V - \frac{\partial f}{\partial t}
\end{align}}

where *f* is any twice differentiable function that depends on position and time. The fields remain the same under the gauge transformation, and therefore Maxwell's equations are still satisfied. That is, Maxwell's equations have a gauge symmetry.

Back to top

## Rotation

### Rotation in 2D

In two dimensions a vector can be represented with a complex number. Multiplying that vector by *iθ* would rotate the vector by 90° and enlarge it by a factor of θ.

To perform a general rotation without enlarging it requires a unit complex number

# z

e

i
θ

=
cos
⁡
θ
i
sin
⁡
# θ
cis
⁡
(
θ
)

{\displaystyle z = e^{i \theta} = \cos \theta + i \sin \theta = \operatorname{cis}(\theta)}

which when multiplied by any vector rotates it through an angle *θ* about the origin:

(

x
′

e

1

y
′

e

2

# )
(
x

e

1

y

e

2

)

e

i
θ

.

{\displaystyle (x'\mathbf{e}\_1 + y'\mathbf{e}\_2) = (x\mathbf{e}\_1 + y\mathbf{e}\_2)e^{i\theta}.}

From Wikipedia:Bivector

With the properties of negative square and unit magnitude, the unit bivector can be identified with the imaginary unit from complex numbers.

A general bivector can be written

θ

e

12

=
i
θ
,

{\displaystyle \theta\mathbf{e}\_{12} = i\theta,}

where *θ* is a real number.

Putting this into the Taylor series for the exponential map and using the property (**e12**)2 = −1 results in a bivector version of Euler's formula,

e

i
θ

=

e

θ

e

12

=
cos
⁡

θ

e

12

sin
⁡

θ

,

{\displaystyle e^{i\theta} = e^{\theta\mathbf{e}\_{12}} = \cos{\theta} + \mathbf{e}\_{12}\sin{\theta},}

The following products all generate the same rotation

v

′

=

v

e

e

12

θ

=

e

−

e

12

θ

v

=

e

−

e

12

θ

2

v

e

e

12

θ

2

.

{\displaystyle \mathbf{v}' = \mathbf{v}e^{\mathbf{e}\_{12}\theta} = e^{-\mathbf{e}\_{12}\theta}\mathbf{v} = e^{\frac{-\mathbf{e}\_{12}\theta}{2}} \mathbf{v}e^{\frac{\mathbf{e}\_{12}\theta}{2}}.}

Of these the last product is the one that generalises into higher dimensions. The quantity needed is called a rotor and is given the symbol *R*, so in two dimensions a rotor that rotates through angle *θ* can be written

# R

e

−
i
θ

2

=

e

−
θ

e

12

2

,

{\displaystyle R = e^{\frac{-i\theta}{2}} = e^{\frac{-\theta\mathbf{e}\_{12}}{2}}, }

and the rotation it generates is

v

′

=
R

v

R

−
1

.

{\displaystyle \mathbf{v}' = R\mathbf{v}R^{-1}.\,}

Let the unit vector *ω* be the axis of rotation and
let *θ* be the angle of rotation.

The rotation vector is given by *ωθ*
The quaternion associated with the rotation is

# q

(

cos
⁡

(

θ
2

)

,
ω
sin
⁡

(

θ
2

)

)

{\displaystyle q = \left(\cos\left(\frac{\theta}{2}\right), \omega \sin\left(\frac{\theta}{2}\right)\right)}

Let *Ω* be a unit bivector in the plane of rotation, and let *θ* be the angle of rotation.

Then the rotation bivector is *Ωθ*.
The quaternion closely corresponds to the exponential of half of the bivector *Ωθ*. That is, the components of the quaternion correspond to the scalar and bivector parts of the following expression:

e

Ω
θ

2

=
cos
⁡

(

θ
2

)

Ω
sin
⁡

(

θ
2

)

{\displaystyle e^{\frac{\Omega\theta}{2}} = \cos\left(\frac{\theta}{2}\right) + \Omega\sin\left(\frac{\theta}{2}\right) }

The exponential can be defined in terms of its power series, and easily evaluated using the fact that *Ω* squared is -1.
Back to top

### Rotation in 3-D

From Wikipedia:Bivector

The bivector **Ω***θ* generates a rotation through the exponential map. The even elements generated rotate a general vector in three dimensions in the same way as quaternions:

v

′

=

e

Ω
θ

2

v

e

−

Ω
θ

2

.

{\displaystyle \mathbf{v}' = e^{\frac{\boldsymbol{\Omega}\theta}{2}}\mathbf{v}e^{-\frac{\boldsymbol{\Omega}\theta}{2}}.}

As in two dimensions the quantity *e***Ω***θ* is called a rotor and written *R*. The quantity *e*−**Ω***θ* is then *R*−1, and they generate rotations as follows

v

′

=
R

v

R

−
1

.

{\displaystyle \mathbf{v}' = R\mathbf{v}R^{-1}.\,}

This is identical to two dimensions, except here rotors are four-dimensional objects isomorphic to the quaternions (instead of complex numbers). This can be generalised to all dimensions, with rotors, elements of the even subalgebra with unit magnitude, being generated by the exponential map from bivectors. They form a double cover over the rotation group, so the rotors *R* and −*R* represent the same rotation.

Back to top

### Rotors

From Wikipedia:Rotor

A **rotor** is an object in Clifford algebra (a bivector) that rotates any blade or general multivector about the origin. They are normally motivated by considering an even number of reflections, which generate rotations (see also the Cartan–Dieudonné theorem).

The term originated with William Kingdon Clifford, in showing that the quaternion algebra is just a special case of Hermann Grassmann's "theory of extension" (Ausdehnungslehre). Hestenes defined a rotor to be any element

R

{\displaystyle R}

 of a geometric algebra that can be written as the geometric product of an even number of unit vectors and satisfies

R
~

# R
1

{\displaystyle \tilde{R} R = 1}

, where

R
~

{\displaystyle \tilde{R}}

 is the "reverse" of

R

{\displaystyle R}

—that is, the product of the same vectors, but in reverse order.

The geometric product of two arbitrary vectors is given by

a
# b
a
⋅
b
a
∧
b

{\displaystyle ab=a \cdot b + a \wedge b }

 = scalar + bivector
The algebra consisting of scalars and bivectors, is isomorphic to the quaternions.
This suggests that the usual split of a quaternion into scalar and vector parts would be better represented as a split into scalar and bivector parts; if this is done the quaternion product is merely the geometric product.

Back to top

#### Using inverse

Reflections along a vector in geometric algebra may be represented as (minus) sandwiching a multivector *M* between a non-null vector *v* perpendicular to the hyperplane of reflection and that vector's inverse *v*−1:

−

v
1

M

v
1

−
1

{\displaystyle -v\_1Mv\_1^{-1}}

and are of even grade. Two reflections is a rotation.

v
2

v
1

M

v
1

−
1

v
2

−
1

{\displaystyle v\_2 v\_1Mv\_1^{-1} v\_2^{-1}}

Multiplying a vector times a vector (

v
2

v
1

{\displaystyle v\_2 v\_1}

) results in a scalar plus a bivector called a rotor.

Under a rotation generated by the rotor *R*, a general multivector *M* will transform double-sidedly as

R
M

R

−
1

.

{\displaystyle RMR^{-1}.}

The formulation above (using inverse) is self-normalizing.

Back to top

#### Using reverse

For a Euclidean space, it may be convenient to consider an alternative formulation, and some authors define the operation of reflection as (minus) the sandwiching of a *unit* (i.e. normalized) multivector:

−
v
M
v
,

v
2

=
1
,

{\displaystyle -vMv, \quad v^2=1 ,}

forming rotors that are automatically normalised:

R

R
†

=

R
†

# R
1
.

{\displaystyle RR^\dagger = R^\dagger R=1 .}

The derived rotor action is then expressed as a sandwich product with the reverse:

R
M

R

†

{\displaystyle RMR^{\dagger}}

where

R
~

{\displaystyle \tilde{R}}

 is the "reverse" of

R

{\displaystyle R}

—that is, the product of the same vectors, but in reverse order.

For a reflection for which the associated vector squares to a negative scalar, as may be the case with a pseudo-Euclidean space, such a vector can only be normalized up to the sign of its square, and additional bookkeeping of the sign of the application the rotor becomes necessary. The formulation in terms of the sandwich product with the inverse as above suffers no such shortcoming.

Two consecutive rotations is given by

R
2

R
1

M

R
1

†

R
2

†

=

R
2

R
1

M
(

R
1

R
2

)

†

{\displaystyle R\_2R\_1MR\_1^{\dagger}R\_2^{\dagger} = R\_2R\_1M(R\_1R\_2)^{\dagger}}

The alternative formulation above (using reverse) is not self-normalizing and motivates the definition of spinor in geometric algebra as an object that transforms single-sidedly – i.e. spinors may be regarded as non-normalised rotors in which the reverse rather than the inverse is used in the sandwich product.

Back to top

### Complex vector space

The 3 Pauli matrices are the tensor representations of orthogonal unit basis vectors

σ
x

=

(

0

1

1

0

)

σ
y

=

(

0

−
i

i

0

)

σ
z

=

(

1

0

0

−
1

)

.

{\displaystyle \begin{align}
 \sigma\_x &=
 \begin{pmatrix}
 0&1\\
 1&0
 \end{pmatrix} \\
 \sigma\_y &=
 \begin{pmatrix}
 0&-i\\
 i&0
 \end{pmatrix} \\
 \sigma\_z &=
 \begin{pmatrix}
 1&0\\
 0&-1
 \end{pmatrix} \,.
\end{align}}

Squaring any Pauli matrix results in the tensor representation of the scalar 1.

σ
x
2

=

σ
y
2

=

σ
z
2

=

(

1

0

0

1

)

{\displaystyle
\sigma\_x^2 = \sigma\_y^2 = \sigma\_z^2 =
 \begin{pmatrix}
 1&0 \\
 0&1
 \end{pmatrix}
}

Multiplying any 2 Pauli matrices results in the tensor representation of a bivector called a quaternion.

σ
y

σ
z

=

(

0

i

i

0

)

=

k

σ
z

σ
x

=

(

0

1

−
1

0

)

=

j

σ
x

σ
y

=

(

i

0

0

−
i

)

=

i

{\displaystyle \begin{align}
 \sigma\_y \sigma\_z &=
 \begin{pmatrix}
 0&i \\
 i&0
 \end{pmatrix} = \mathbf{k}\\
 \sigma\_z \sigma\_x &=
 \begin{pmatrix}
 0&1 \\
 -1&0
 \end{pmatrix} = \mathbf{j}\\
 \sigma\_x \sigma\_y &=
 \begin{pmatrix}
 i&0 \\
 0&-i
 \end{pmatrix} = \mathbf{i}
\end{align}}

Multiplying all three Pauli matrices results in the the tensor representation of the pseudoscalar i.

σ

x

σ

y

σ

z

=

(

i

0

0

i

)

{\displaystyle
 \sigma\_{x} \sigma\_{y} \sigma\_{z} =
 \begin{pmatrix}
 i&0\\
 0&i
 \end{pmatrix}
}

Any 2 by 2 complex matrix can be constructed from (or decomposed into) a sum of the 8 matrices above multiplied by real numbers.

Vectors in real vector space have real coefficients. Given a vector ***x*** = *x*1 e1 + *x*2 e2 + *x*3 e3 of real numbers, one can construct its tensor representation.

x
1

σ
x

x
2

σ
y

x
3

σ
z

=

(

x
3

x
1

−
i

x
2

x
1

i

x
2

−

x
3

)

=
vector

{\displaystyle x\_1 \sigma\_x + x\_2 \sigma\_y + x\_3 \sigma\_z =
\left(\begin{matrix}x\_3&x\_1-ix\_2\\x\_1+ix\_2&-x\_3\end{matrix}\right) = \text{vector}
}

But by allowing the coefficients (*x*1, *x*2, *x*3) to be complex numbers we create a complex vector space.

Doing so requires us to multiply Pauli matrices by i. Multiplying any Pauli matrix with i results in a bivector called a quaternion.

i

σ
x

=

σ
y

σ
z

=

(

0

i

i

0

)

=

k

i

σ
y

=

σ
z

σ
x

=

(

0

1

−
1

0

)

=

j

i

σ
z

=

σ
x

σ
y

=

(

i

0

0

−
i

)

=

i

{\displaystyle \begin{align}
 i \sigma\_x = \sigma\_y \sigma\_z &=
 \begin{pmatrix}
 0&i \\
 i&0
 \end{pmatrix} = \mathbf{k}\\
 i \sigma\_y = \sigma\_z \sigma\_x &=
 \begin{pmatrix}
 0&1 \\
 -1&0
 \end{pmatrix} = \mathbf{j}\\
 i \sigma\_z = \sigma\_x \sigma\_y &=
 \begin{pmatrix}
 i&0 \\
 0&-i
 \end{pmatrix} = \mathbf{i}
\end{align}}

and adding in the identity matrix (a scalar):

i
(

x
0

σ
0

x
1

σ
x

x
2

σ
y

x
3

σ
z

# )

(

x
0

i

x
3

i

x
1

x
2

i

x
1

−

x
2

x
0

−
i

x
3

)

=
quaternion = spinor

{\displaystyle i(x\_0 \sigma\_0 + x\_1 \sigma\_x + x\_2 \sigma\_y + x\_3 \sigma\_z) =
\left(\begin{matrix}x\_0 + ix\_3&ix\_1 + x\_2\\ix\_1-x\_2&x\_0 -ix\_3\end{matrix}\right) = \text{quaternion = spinor}
}

If the vector is a unit vector then the quaternion is a versor and the spinor is a rotor.

A complex vector therefore consists of a real vector plus an imaginary vector (a pseudovector) which in 3 dimensions is a bivector.

In complex vector space the dot product is replaced with an inner product.

The geometric product of complex vectors:

(
1
i
)

e
x

(
1
)

e
y

=
(

e
x

i

e
x

)
(

e
y

)

=
(

e
x

e

y
z

)
(

e
y

)

=

e

x
y

e

y
z
y

=

e

x
y

−

e

z

=
−

e

z

i

e

x
y

=
(
−
1
i
)

e
z

{\displaystyle
\begin{align}
(1 + i)e\_x (1)e\_y &=
(e\_x + ie\_x) (e\_y) \\ &=
(e\_x + e\_{yz}) (e\_y) \\ &=
e\_{xy} + e\_{yzy} \\ &=
e\_{xy} - e\_{z} \\ &=
-e\_{z} + i e\_{xy} \\ &=
(-1 + i) e\_z
\end{align}
}

(
1
i
)

e
x

(
1
i
)

e
y

=
(

e
x

i

e
x

)
(

e
y

i

e
y

)

=
(

e
x

e

y
z

)
(

e
y

e

z
x

)

=

e

x
y

e

x
z
x

e

y
z
y

e

y
z
z
x

=

e

x
y

−

e

z

−

e

z

e

y
x

=
−
2

e

z

{\displaystyle
\begin{align}
(1 + i)e\_x (1 + i)e\_y &=
(e\_x + ie\_x) (e\_y + ie\_y) \\ &=
(e\_x + e\_{yz}) (e\_y + e\_{zx}) \\ &=
e\_{xy} + e\_{xzx} + e\_{yzy} + e\_{yzzx} \\ &=
e\_{xy} - e\_{z} - e\_{z} + e\_{yx} \\ &=
 - 2e\_{z}
\end{align}
}

See also:  Linear complex structure

By replacing complex numbers with their matrix representations one can often eliminate the complex and imaginary components of a complex vector space. The resulting Clifford algebra is called geometric algebra and is often more intuitive with obvious geometric interpretations of its components.

#### Minkowski "inner product"

From Wikipedia:Minkowski\_space#Minkowski\_metric

The simplest example of a Lorentzian manifold is flat spacetime, which can be given as **R**4 with coordinates

(
t
,
x
,
y
,
z
)

{\displaystyle (t, x, y, z)}

 and the metric

d

s
2

=
−

c
2

d

t
2

d

x
2

d

y
2

d

z
2

=

η

μ
ν

d

x

μ

d

x

ν

.

{\displaystyle ds^2 = -c^2 dt^2 + dx^2 + dy^2 + dz^2 = \eta\_{\mu\nu} dx^{\mu} dx^{\nu}. \,}

Note that these coordinates actually cover all of **R**4. The flat space metric (or Minkowski metric) is often denoted by the symbol *η* and is the metric used in special relativity.

A standard basis for Minkowski space is a set of four mutually orthogonal vectors { *e*0, *e*1, *e*2, *e*3 }  such that

−
η
(

e
0

,

e
0

# )
η
(

e
1

,

e
1

# )
η
(

e
2

,

e
2

# )
η
(

e
3

,

e
3

# )
1
.

{\displaystyle -\eta(e\_0, e\_0) = \eta(e\_1, e\_1) = \eta(e\_2, e\_2) = \eta(e\_3, e\_3) = 1 .}

These conditions can be written compactly in the form

η
(

e
μ

,

e
ν

# )

η

μ
ν

.

{\displaystyle \eta(e\_\mu, e\_\nu) = \eta\_{\mu \nu}.}

Relative to a standard basis, the components of a vector *v* are written (*v*0, *v*1, *v*2, *v*3) where the Einstein summation convention is used to write *v* = *v**μ**e**μ*. The component *v*0 is called the **timelike component** of *v* while the other three components are called the **spatial components**. The spatial components of a 4-vector *v* may be identified with a 3-vector **v** = (*v*1, *v*2, *v*3).

In terms of components, the Minkowski inner product between two vectors *v* and *w* is given by

η
(
v
,
w
# )

η

μ
ν

v
μ

w
ν

=

v
0

w
0

v
1

w
1

v
2

w
2

v
3

w
3

=

v
μ

w
μ

=

v
μ

w
μ

,

{\displaystyle \eta(v, w) = \eta\_{\mu \nu} v^\mu w^\nu = v^0 w\_0 + v^1 w\_1 + v^2 w\_2 + v^3 w\_3 = v^\mu w\_\mu = v\_\mu w^\mu,}

and

η
(
v
,
v
# )

η

μ
ν

v
μ

v
ν

=

v
0

v
0

v
1

v
1

v
2

v
2

v
3

v
3

=

v
μ

v
μ

.

{\displaystyle \eta(v, v) = \eta\_{\mu \nu} v^\mu v^\nu = v^0v\_0 + v^1 v\_1 + v^2 v\_2 + v^3 v\_3 = v^\mu v\_\mu.}

Here **lowering of an index** with the metric was used.

The Minkowski metric η is the metric tensor of Minkowski space. It is a pseudo-Euclidean metric, or more generally a *constant* pseudo-Riemannian metric in Cartesian coordinates. As such it is a nondegenerate symmetric bilinear form, a type (0,2) tensor. It accepts two arguments *u*, *v*.

The definition

u
⋅
# v
η
(
u
,
v
)

{\displaystyle u \cdot v =\eta(u, v)}

yields an inner product-like structure on *M*, previously and also henceforth, called the *Minkowski inner product*, similar to the Euclidean inner product, but it describes a different geometry. It is also called the *relativistic dot product*. If the two arguments are the same,

u
⋅
# u
η
(
u
,
u
)
≡

u

2

≡

u
2

,

the resulting quantity will be called the *Minkowski norm squared*.

This bilinear form can in turn be written as

u
⋅
# v

u

T

[
η
]
v
,

{\displaystyle u \cdot v = u^{\mathrm T}[\eta] v,}

where [*η*] is a 4×4 matrix associated with η. Possibly confusingly, denote [*η*] with just η as is common practice. The matrix is read off from the explicit bilinear form as

# η
±

(

−
1

0

0

0

0

1

0

0

0

0

1

0

0

0

0

1

)

,

{\displaystyle \eta = \pm \begin{pmatrix}-1&0&0&0\\0&1&0&0\\0&0&1&0\\0&0&0&1\end{pmatrix},}

and the bilinear form

u
⋅
# v
η
(
u
,
v
)
,

{\displaystyle u \cdot v =\eta(u, v),}

with which this section started by assuming its existence, is now identified.

## References

1. ↑ The Minkowski inner product is not an inner product, since it is not positive-definite, i.e. the quadratic form *η*(*v*, *v*) need not be positive for nonzero v. The positive-definite condition has been replaced by the weaker condition of non-degeneracy. The bilinear form is said to be *indefinite*.