Leibniz notation for vector fields

Leibniz notation for vector fields

Consider a

vector field

V

,
such as

V

⁢

(

x

,

y

)

=

(

y

,

-

x

)

for

(

x

,

y

)

∈

ℝ

2

.

Oftentimes

V

will be expressed in the following Leibnizian notation

y

⁢

∂

∂

⁡

x

-

x

⁢

∂

∂

⁡

y

,

which is extremely strange the first time one encounters it. After all, why
not just
write

V

⁢

(

x

,

y

)

=

y

⁢

e

1

-

x

⁢

e

2

where

e

1

=

(

1

,

0

)

,

e

2

=

(

0

,

1

)

?
What do

partial derivatives

have to do with anything?
(That was the question I asked.)

However, if viewed in the right way, it is not so strange.
For if a vector field

V

is given, then one natural thing that can be done
with it is to

differentiate

a scalar-valued function

f

in the

direction

of

V

.
In functional notation, this would be

D

⁡

f

⁢

(

p

)

⋅

V

⁢

(

p

)

where

p

is a point,

V

⁢

(

p

)

is gives the particular vector
in the vector field at the point

p

, and

D

⁡

f

⁢

(

p

)

⋅

v

denotes the

directional derivative

of

f

at

p

with respect
to the direction

v

.

For our example,

D

⁡

f

⁢

(

p

)

⋅

V

⁢

(

p

)

(where

p

=

(

x

,

y

)

) equals

D

1

⁡

f

⁢

(

x

,

y

)

⋅

y

+

D

2

⁡

f

⁢

(

x

,

y

)

⋅

(

-

x

)

=

∂

⁡

f

∂

⁡

x

⁢

y

-

∂

⁡

f

∂

⁡

y

⁢

x

=

y

⁢

∂

⁡

f

∂

⁡

x

-

x

⁢

∂

⁡

f

∂

⁡

y

=

(

y

⁢

∂

∂

⁡

x

-

x

⁢

∂

∂

⁡

y

)

⁢

[

f

]

=

V

⁢

[

f

]

.

We have written out the steps explicitly to show
where the partial derivative notation for

V

comes from.
At the second last step we considered

∂

/

∂

⁡

x

and

∂

/

∂

⁡

y

as

operators

acting on the

function

f

.

The Leibniz notation on manifolds

But there is more. We can consider a more general situation,
where

V

is a vector field on a

manifold

.
In this case, because the

tangent space

to a manifold varies
with each point

p

, we cannot fix certain basis vectors

e

i

to describe our vector field anymore.
Loosely speaking, the basis vectors now have to vary smoothly.

Suppose we have a

coordinate system

{

x

i

}

i

=

1

,

…

,

n

on the manifold.
Then the

tangent vector

on the manifold corresponding to an

infinitesimal

change in

x

i

is often written

∂

∂

⁡

x

i

.

This makes total sense, because one of the favorite ways to

define

tangent vectors on abstract manifolds is to identify them
with directional derivatives. The basis

∂

/

∂

⁡

x

i

(which
varies smoothly with

p

)
then becomes the replacement for the fixed basis

e

i

in

Euclidean space

.
Note that this is

consistent

with the

calculus

notation

∂

⁡

f

/

∂

⁡

x

i

for the partial derivative of

f

with respect to the

x

i

variable,
because partial derivatives are merely directional derivatives
with respect to the direction

e

i

.

If

{

y

j

}

is another coordinate system for the manifold,
then we have the

formula

(for the

derivation

, see the entry on
vector fields (

http://planetmath.org/VectorField

))

∂

∂

⁡

y

j

=

∂

⁡

x

i

∂

⁡

y

j

⁢

∂

∂

⁡

x

i

,

(1)

We have used the

Einstein
summation convention

above to emphasize the mnemonic cancelling of fractions.

The quantity

∂

⁡

x

i

/

∂

⁡

y

j

is the directional derivative in the direction

∂

/

∂

⁡

y

j

of the

i

th

coordinate

function

,

mapping

a point

p

on the manifold to
the coordinate

x

i

. Notice the subtle subtle change of
viewpoint here:
we are not considering

x

i

as mere “variables”,
but as

functions

of the point

p

.

Formula (

1

), which is a

linear combination

of

vectors

in a tangent space,
of course looks like the

chain rule

learned in

elementary

multivariate
calculus

,
but it is much more than that:
we are saying that the formula holds for general

curvilinear coordinate

systems on manifolds!
This is definitely one of the virtues of the Leibnizian notation — making advanced

concepts

look simple.

As a simple example to get used to this notation,
consider the function

f

:

ℝ

3

→

ℝ

defined by

f

⁢

(

x

,

y

,

z

)

=

x

2

+

y

2

+

z

2

.
The Euclidean space

ℝ

3

can also be thought of as a manifold,
and suppose we use a

spherical coordinate system

(

r

,

θ

,

ϕ

)

on it.

Let us compute

∂

⁡

f

/

∂

⁡

r

. If

f

is “viewed as a function of

(

r

,

θ

,

ϕ

)

”,
then

f

⁢

(

r

,

θ

,

ϕ

)

=

r

2

,
so we certainly hope that

∂

⁡

f

/

∂

⁡

r

=

2

⁢

r

with our definition of directional derivatives.
This easily follows from the formula for

differential forms

on manifolds:

d

⁢

f

=

∂

⁡

f

∂

⁡

r

⁢

d

⁢

r

+

∂

⁡

f

∂

⁡

θ

⁢

d

⁢

θ

+

∂

⁡

f

∂

⁡

ϕ

⁢

d

⁢

ϕ

.

But let us see this by calculating from (

1

) too.
We have

x

=

r

⁢

sin

⁡

θ

⁢

cos

⁡

ϕ

,

y

=

r

⁢

sin

⁡

θ

⁢

sin

⁡

ϕ

,

z

=

r

⁢

cos

⁡

θ

.

So

∂

⁡

x

∂

⁡

r

=

sin

⁡

θ

⁢

cos

⁡

ϕ

,

∂

⁡

y

∂

⁡

r

=

sin

⁡

θ

⁢

sin

⁡

ϕ

,

∂

⁡

z

∂

⁡

r

=

cos

⁡

θ

,

and substituting in (

1

),

∂

⁡

f

∂

⁡

r

=

∂

⁡

x

∂

⁡

r

⁢

∂

⁡

f

∂

⁡

x

+

∂

⁡

y

∂

⁡

r

⁢

∂

⁡

f

∂

⁡

y

+

∂

⁡

z

∂

⁡

r

⁢

∂

⁡

f

∂

⁡

z

=

(

sin

⁡

θ

⁢

cos

⁡

ϕ

)

⁢

(

2

⁢

x

)

+

(

sin

⁡

θ

⁢

sin

⁡

ϕ

)

⁢

(

2

⁢

y

)

+

(

cos

⁡

θ

)

⁢

(

2

⁢

z

)

=

2

⁢

r

-

1

⁢

(

x

⁢

r

⁢

sin

⁡

θ

⁢

cos

⁡

ϕ

+

y

⁢

r

⁢

sin

⁡

θ

⁢

sin

⁡

ϕ

+

z

⁢

r

⁢

cos

⁡

θ

)

=

2

⁢

r

-

1

⁢

(

x

2

+

y

2

+

z

2

)

=

2

⁢

r

.

Needless to say, this calculation can be done with the usual functional notation,
but it will be somewhat clumsy. We have to say: let

α

be the

spherical coordinate

chart; and then

D

1

⁡

(

f

∘

α

)

⁢

(

r

,

θ

,

ϕ

)

=

D

⁡

f

⁢

(

α

⁢

(

r

,

θ

,

ϕ

)

)

⋅

D

1

⁡

α

⁢

(

r

,

θ

,

ϕ

)

would be the quantity

∂

⁡

f

/

∂

⁡

r

.
That is not to say Leibnizian notation has no disadvantages.
For example,
one of the typical objections to the Leibnizian formula

d

⁢

f

d

⁢

x

=

d

⁢

f

d

⁢

y

⁢

d

⁢

y

d

⁢

x

is that the function

f

means something different on the two sides of the equation.
However, formula (

1

) partly escapes this objection:
we can consider

f

to be a function on a

manifold

, ignoring the

vector space

structure

of

ℝ

3

.

Cartesian coordinates

(

x

,

y

,

z

)

simply become another coordinate chart.
Spherical coordinates constitute another.
Then

∂

⁡

f

/

∂

⁡

r

(i.e. the directional derivative

∂

/

∂

⁡

r

applied to

f

) is a natural quantity to consider,
rather than “the

derivative

of

f

∘

α

’ with respect to the first variable”
(which is what the functional notation says).

Physicists seem to grasp the Leibnizian

formalism

very readily,
and it is a shame that many calculus textbooks do not fully explain the logic
behind the formalism — probably because it looks to be unrigorous —
but the point we are trying to drive here is that when

differentials

are suitably
interpreted, they

are

rigorous

.

The dual to the tangent vectors

∂

/

∂

⁡

x

i

There is a subtle ambiguity in the

Leibniz notation

that we should also discuss here.
Suppose we are on a two-dimensional manifold with coordinates

u

and

v

. The notation

∂

⁡

f

/

∂

⁡

u

is ambiguous, because it implicitly depends on the

v

coordinate as well. What we really mean when we refer to

∂

⁡

f

/

∂

⁡

u

is a displacement where

u

changes at a uniform rate of 1, and where

v

does not change at all.

Say, for some bizarre reason, I decided to use a coordinate system on the

Euclidean plane

made up of the

Euclidean

x

coordinate and the radial

r

coordinate. Now when I write

∂

⁡

f

/

∂

⁡

x

, I mean something quite different than when I write

∂

⁡

f

/

∂

⁡

x

relative to the Euclidean coordinates.
In the first

instance

the derivative is with respect to the vector field

e

1

-

(

y

/

x

)

⁢

e

2

.

In the second instance, the derivative is with respect to the vector field

e

1

+

0

⁢

e

2

.

On closer thought, we can see that

∂

/

∂

⁡

x

in the elementary calculus

interpretation

has the same ambiguity, but the problem is so

trivial

that
we often forget that it exists.
For instance, if we have a function

f

=

x

⁢

y

⁢

z

, and we stipulate
that also

y

=

x

2

,
then obviously

∂

⁡

f

/

∂

⁡

x

≠

y

⁢

z

, because

y

is changing
at the same time as

x

.
The definition of a partial derivative with respect
to

x

is the derivative when

x

changes and all the other variables are held

fixed

. So this rule should be applied when
working with the tangent vectors

∂

/

∂

⁡

x

i

on a manifold too.

If we agree to use different letters for each coordinate system,
and not mix them up (always a reasonable thing to do),
then we will not make any mistakes arising from this ambiguity
with the Leibniz notation.

Another way to understand the ambiguity is as follows.
In a vector space, there is no natural isomorphism between
it and its

dual space

(unless we involve the

inner product

or something like that). On manifolds, the role of the dual space
is taken by the space of differential

one-forms

.
(See differential forms (

http://planetmath.org/DifferentialForms

)
for the rigorous details.) A basis for this dual space
is

d

⁢

x

j

for

j

=

1

,

…

,

n

.
There is a basis in the tangent space that is dual to

d

⁢

x

i

:
namely, these are the

∂

/

∂

⁡

x

i

:

d

⁢

x

j

⁢

(

∂

∂

⁡

x

i

)

=

δ

i

j

(Kronecker delta).

(2)

But if we are given a lone

element

d

⁢

x

j

, we
cannot produce
a unique vector

∂

/

∂

⁡

x

j

from it (i.e. there is no

isomorphism

) —
we need to be given the

entire

basis

{

d

⁢

x

j

}

j

=

1

,

…

,

n

.
So it is not surprising why the vectors

∂

/

∂

⁡

x

i

should depend
on each other.

Motivation for the notation of differential forms

Incidentally, the formula (

2

) explains the following

identity

involving differential forms (often seen in calculus textbooks with hardly
any explanation of what it means):

d

⁢

f

=

∂

⁡

f

∂

⁡

x

j

⁢

d

⁢

x

j

.

(3)

The various

d

’s floating around obscures the essential idea somewhat,
but the derivation of this formula is basic

linear algebra

. Since

d

⁢

f

is a

linear functional

on the tangent space,

defined

by

d

⁢

f

⁢

(

V

)

=

V

⁢

[

f

]

, it can be written
as a linear combination of the

dual basis

d

⁢

x

j

.
That is, for some

a

j

, we have

d

⁢

f

=

a

j

⁢

d

⁢

x

j

.

And these

a

j

are solved for by evaluating at the tangent
vectors

V

=

∂

/

∂

⁡

x

i

:

∂

⁡

f

∂

⁡

x

i

=

∂

∂

⁡

x

i

⁢

[

f

]

=

d

⁢

f

⁢

(

∂

∂

⁡

x

i

)

=

a

j

⁢

d

⁢

x

j

⁢

(

∂

∂

⁡

x

i

)

=

a

j

⁢

δ

i

j

=

a

i

,

giving formula (

3

).

For those who are not familiar with the

language

of differential forms,
the definition

d

⁢

f

⁢

(

V

)

=

V

⁢

[

f

]

just used
might seem to be somewhat artificial, designed solely to make
the classical formula (

3

) work out.
The following comment by Spivak

[

2

]

might help clarify matters:

Classical differential geometers (and classical analysts)
did not hesitate to talk about “infinitely small” changes

d

⁢

x

i

of the coordinates

x

i

, just as Leibniz had.
No one wanted to admit that this was nonsense, because true
results were obtained when these infinitely small quantities
were divided into each other (provided one did it in the right way).
Eventually it was realized that the closest one can come to describing an
infinitely small change is to describe a direction in which this change
is supposed to occur, i.e. a tangent vector. Since

d

⁢

f

is supposed
to be the infinitesimal change of

f

under an infinitesimal change
of the point,

d

⁢

f

must be a function of this change,
which means that

d

⁢

f

should be a function on tangent vectors.
The

d

⁢

x

i

themselves then metamorphosed into functions,
and it became clear that they must be distinguished from the tangent
vectors

∂

/

∂

⁡

x

i

.
Once this realization came, it was only a matter of making new definitions,
which preserved the

old

notation, and waiting for everybody
to catch up. In short, all classical notions involving infinitely
small quantities became functions on tangent vectors, like

d

⁢

f

,
except for quotients of infinitely small
quantities, which became tangent vectors, like

d

⁢

c

/

d

⁢

t

.

We can also give an

analogy

as follows (also from

[

2

]

).
Suppose

f

is a
function on a manifold. Let “

x

i

=

x

i

⁢

(

t

)

” be a
curve on this manifold (this is the classical notation).
Then by the chain rule,

d

⁢

f

d

⁢

t

=

∂

⁡

f

∂

⁡

x

i

⁢

d

⁢

x

i

d

⁢

t

,

where

f

on the left side really means

f

⁢

(

x

i

⁢

(

t

)

)

.
The formal identity obtained by multiplying both sides by

d

⁢

t

,

d

⁢

f

=

∂

⁡

f

∂

⁡

x

i

⁢

d

⁢

x

i

,

means that “true results are obtained by dividing by

d

⁢

t

again,

no matter what the functions

x

i

⁢

(

t

)

are.”
Also, the left- and right-hand sides individually
do not depend on any particular curve

x

i

⁢

(

t

)

at all, but
really just the tangent vectors

d

⁢

x

i

/

d

⁢

t

to that curve.
Again, this leads us to realization that

d

⁢

f

and

d

⁢

x

i

should be treated as functions of a tangent vector.

References

1

Vladimir I. Arnol’d (trans. Roger Cooke).

Ordinary Differential Equations

. Springer-Verlag, 1992.

2

Michael Spivak.

A Comprehensive

Introduction

to

Differential Geometry

,
Volume I. Publish or Perish, 1979.

3

Michael Spivak.

Calculus on Manifolds

. Perseus, 1965.

Title

Leibniz notation for vector fields

Canonical name

LeibnizNotationForVectorFields

Date of creation

2013-03-22 15:26:54

Last modified on

2013-03-22 15:26:54

Owner

stevecheng (10074)

Last modified by

stevecheng (10074)

Numerical id

13

Author

stevecheng (10074)

Entry type

Topic

Classification

msc 58A10

Classification

msc 34-00

Classification

msc 53-00

Related topic

VectorField

Related topic

DerivativeNotation

Related topic

DifferentialForms

Related topic

FirstOrderOperatorsInRiemannianGeometry

Related topic

LieDerivative

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