Gradient and Divergence in Orthonormal Curvilinear Coordinates

Gradient and Divergence in Orthonormal Curvilinear Coordinates

Gradient and Divergence in Orthonormal Curvilinear Coordinates
Swapnil Sunil Jain
Aug 7, 2006

Gradient and Divergence in Orthonormal Curvilinear Coordinates

Gradient in Curvilinear Coordinates

In

rectangular coordinates

(where

f

=

f

⁢

(

x

,

y

,

z

)

), an

infinitesimal

length vector

d

⁢

l

→

is given by

d

⁢

l

→

=

d

⁢

x

⁢

x

^

+

d

⁢

y

⁢

y

^

+

d

⁢

z

⁢

z

^

the

gradient

is given by

∇

=

∂

∂

⁡

x

⁢

x

^

+

∂

∂

⁡

y

⁢

y

^

+

∂

∂

⁡

z

⁢

z

^

and the

differential

change in the output is given by

d

⁢

f

=

∇

⁡

f

∘

d

⁢

l

→

=

∂

⁡

f

∂

⁡

x

⁢

d

⁢

x

+

∂

⁡

f

∂

⁡

y

⁢

d

⁢

y

+

∂

⁡

f

∂

⁡

z

⁢

d

⁢

z

Similarly in orthonormal

curvilinear coordinates

( where

f

=

f

⁢

(

q

1

,

q

2

,

q

3

)

), the infinitesimal length vector is given by

1

1

See my article

Unit Vectors in Curvilinear Coordinates

for an insight into this

expression

.

d

⁢

l

→

=

h

1

⁢

d

⁢

q

1

⁢

q

^

1

+

h

2

⁢

d

⁢

q

2

⁢

q

^

2

+

h

3

⁢

d

⁢

q

3

⁢

q

^

3

where

h

i

=

∑

k

(

∂

⁡

x

→

k

∂

⁡

q

i

)

2

⁢

and

⁢

q

^

i

=

1

h

i

⁢

(

∂

⁡

x

→

k

∂

⁡

q

i

)

⁢

for

⁢

i

∈

1

,

2

,

3

So if

∇

=

α

⁢

∂

∂

⁡

q

1

⁢

q

^

1

+

β

⁢

∂

∂

⁡

q

2

⁢

q

^

2

+

γ

⁢

∂

∂

⁡

q

3

⁢

q

^

3

then since we know that

d

⁢

f

=

∂

⁡

f

∂

⁡

q

1

⁢

d

⁢

q

1

+

∂

⁡

f

∂

⁡

q

2

⁢

d

⁢

q

2

+

∂

⁡

f

∂

⁡

q

3

⁢

d

⁢

q

3

and

d

⁢

f

=

∇

⁡

F

∘

d

⁢

l

→

=

α

⁢

h

1

⁢

∂

⁡

f

∂

⁡

q

1

⁢

d

⁢

q

1

+

β

⁢

h

2

⁢

∂

⁡

f

∂

⁡

q

2

⁢

d

⁢

q

2

+

γ

⁢

h

3

⁢

∂

⁡

f

∂

⁡

q

3

⁢

d

⁢

q

3

this implies that

α

=

1

h

i

;

β

=

1

h

2

;

γ

=

1

h

3

Hence,

∇

=

1

h

1

⁢

∂

∂

⁡

q

1

⁢

q

^

1

+

1

h

2

⁢

∂

∂

⁡

q

2

⁢

q

^

2

+

1

h

3

⁢

∂

∂

⁡

q

3

⁢

q

^

3

=

∑

i

1

h

i

⁢

∂

∂

⁡

q

i

⁢

q

^

i

Divergence in Curvilinear Coordinates

In the previous

section

we concluded that in curvilinear coordinates, the gradient

operator

∇

is given by

∇

=

∑

i

1

h

i

⁢

∂

∂

⁡

q

i

⁢

q

^

i

Then for

F

→

=

F

1

⁢

q

^

1

+

F

2

⁢

q

^

2

+

F

3

⁢

q

^

3

, the

divergence

of

F

→

is given by

∇

∘

F

→

=

(

∑

i

1

h

i

⁢

∂

∂

⁡

q

i

⁢

q

^

i

)

∘

F

→

which is not equal to

(

∑

i

1

h

i

⁢

∂

∂

⁡

q

i

⁢

q

^

i

)

∘

F

→

≠

∑

i

1

h

i

⁢

∂

⁡

F

i

∂

⁡

q

i

as one would think! The real expression can be derived the following way,

=

∑

i

[

(

1

h

i

⁢

∂

∂

⁡

q

i

⁢

q

^

i

)

∘

F

→

]

=

∑

i

[

(

1

h

i

⁢

q

^

i

)

∘

(

∂

⁡

F

→

∂

⁡

q

i

)

]

=

∑

i

[

(

1

h

i

⁢

q

^

i

)

∘

(

∂

∂

⁡

q

i

⁢

(

∑

j

F

j

⁢

q

^

j

)

)

]

=

∑

i

[

(

1

h

i

⁢

q

^

i

)

∘

(

∑

j

∂

∂

⁡

q

i

⁢

(

F

j

⁢

q

^

j

)

)

]

=

∑

i

[

(

1

h

i

⁢

q

^

i

)

∘

(

∑

j

q

^

j

⁢

∂

⁡

F

j

∂

⁡

q

i

+

F

j

⁢

∂

⁡

q

^

j

∂

⁡

q

i

)

]

=

∑

i

[

(

1

h

i

⁢

q

^

i

)

∘

(

∑

j

q

^

j

⁢

∂

⁡

F

j

∂

⁡

q

i

+

∑

j

F

j

⁢

∂

⁡

q

^

j

∂

⁡

q

i

)

]

=

∑

i

[

(

1

h

i

⁢

q

^

i

)

∘

∑

j

q

^

j

⁢

∂

⁡

F

j

∂

⁡

q

i

+

(

1

h

i

⁢

q

^

i

)

∘

∑

j

F

j

⁢

∂

⁡

q

^

j

∂

⁡

q

i

]

=

∑

i

[

(

1

h

i

⁢

q

^

i

)

∘

∑

j

q

^

j

⁢

∂

⁡

F

j

∂

⁡

q

i

]

⏟

call it A

+

∑

i

[

(

1

h

i

⁢

q

^

i

)

∘

∑

j

F

j

⁢

∂

⁡

q

^

j

∂

⁡

q

i

]

⏟

call it B

A

=

∑

i

[

(

1

h

i

⁢

q

^

i

)

∘

∑

j

q

^

j

⁢

∂

⁡

F

j

∂

⁡

q

i

]

=

∑

i

[

1

h

i

⁢

∑

j

(

q

^

i

∘

q

^

j

)

⏟

δ

i

⁢

j

⁢

∂

⁡

F

j

∂

⁡

q

i

]

=

∑

i

1

h

i

⁢

∂

⁡

F

i

∂

⁡

q

i

B

=

∑

i

[

(

1

h

i

⁢

q

^

i

)

∘

∑

j

F

j

⁢

∂

⁡

q

^

j

∂

⁡

q

i

]

Using the following equality

2

2

The proof of this

identity

is left as an exercise for the reader.

∂

⁡

q

^

j

∂

⁡

q

i

=

q

^

i

h

j

⁢

∂

⁡

h

i

∂

⁡

q

j

∀

i

≠

j

we can write

B

as

B

=

∑

i

[

(

1

h

i

⁢

q

^

i

)

∘

∑

j

F

j

⁢

(

q

^

i

⁢

1

h

j

⁢

∂

⁡

h

i

∂

⁡

q

j

)

]

∀

i

≠

j

=

∑

i

[

1

h

i

⁢

∑

j

F

j

⁢

(

q

^

i

∘

q

^

i

)

⏟

1

⁢

1

h

j

⁢

∂

⁡

h

i

∂

⁡

q

j

]

∀

i

≠

j

=

∑

i

[

1

h

i

⁢

∑

j

F

j

⁢

1

h

j

⁢

∂

⁡

h

i

∂

⁡

q

j

]

∀

i

≠

j

=

∑

i

≠

j

F

j

h

j

⁢

h

i

⁢

∂

⁡

h

i

∂

⁡

q

j

=

∑

i

≠

1

F

1

h

1

⁢

h

i

⁢

∂

⁡

h

i

∂

⁡

q

1

+

∑

i

≠

2

F

2

h

2

⁢

h

i

⁢

∂

⁡

h

i

∂

⁡

q

2

+

∑

i

≠

3

F

3

h

3

⁢

h

i

⁢

∂

⁡

h

i

∂

⁡

q

3

⇒

∇

∘

F

→

=

A

+

B

=

∑

i

1

h

i

⁢

∂

⁡

F

i

∂

⁡

q

i

+

∑

i

≠

1

F

1

h

1

⁢

h

i

⁢

∂

⁡

h

i

∂

⁡

q

1

+

∑

i

≠

2

F

2

h

2

⁢

h

i

⁢

∂

⁡

h

i

∂

⁡

q

2

+

∑

i

≠

3

F

3

h

3

⁢

h

i

⁢

∂

⁡

h

i

∂

⁡

q

3

=

[

1

h

1

⁢

∂

⁡

F

1

∂

⁡

q

1

+

1

h

2

⁢

∂

⁡

F

2

∂

⁡

q

2

+

1

h

3

⁢

∂

⁡

F

3

∂

⁡

q

3

]

+

[

F

1

h

1

⁢

h

2

⁢

∂

⁡

h

2

∂

⁡

q

1

+

F

1

h

1

⁢

h

3

⁢

∂

⁡

h

3

∂

⁡

q

1

]

+

[

F

2

h

2

⁢

h

1

⁢

∂

⁡

h

1

∂

⁡

q

2

+

F

2

h

2

⁢

h

3

⁢

∂

⁡

h

3

∂

⁡

q

2

]

+

[

F

3

h

3

⁢

h

1

⁢

∂

⁡

h

1

∂

⁡

q

3

+

F

3

h

3

⁢

h

2

⁢

∂

⁡

h

2

∂

⁡

q

3

]

Collecting

similar

terms together we get,

∇

∘

F

→

=

[

1

h

1

⁢

∂

⁡

F

1

∂

⁡

q

1

+

F

1

h

1

⁢

h

2

⁢

∂

⁡

h

2

∂

⁡

q

1

+

F

1

h

1

⁢

h

3

⁢

∂

⁡

h

3

∂

⁡

q

1

]

+

[

1

h

2

⁢

∂

⁡

F

2

∂

⁡

q

2

+

F

2

h

2

⁢

h

1

⁢

∂

⁡

h

1

∂

⁡

q

2

+

F

2

h

2

⁢

h

3

⁢

∂

⁡

h

3

∂

⁡

q

2

]

+

[

1

h

3

⁢

∂

⁡

F

3

∂

⁡

q

3

+

F

3

h

3

⁢

h

1

⁢

∂

⁡

h

1

∂

⁡

q

3

+

F

3

h

3

⁢

h

2

⁢

∂

⁡

h

2

∂

⁡

q

3

]

If we define

Ω

≡

Π

i

⁢

h

i

, we can further write the above expression as

∇

∘

F

→

=

[

h

2

⁢

h

3

Ω

⁢

∂

⁡

F

1

∂

⁡

q

1

+

F

1

⁢

h

3

Ω

⁢

∂

⁡

h

2

∂

⁡

q

1

+

F

1

⁢

h

2

Ω

⁢

∂

⁡

h

3

∂

⁡

q

1

]

+

[

h

1

⁢

h

3

Ω

⁢

∂

⁡

F

2

∂

⁡

q

2

+

F

2

⁢

h

3

Ω

⁢

∂

⁡

h

1

∂

⁡

q

2

+

h

1

⁢

F

2

Ω

⁢

∂

⁡

h

3

∂

⁡

q

2

]

+

[

h

1

⁢

h

2

Ω

⁢

∂

⁡

F

3

∂

⁡

q

3

+

h

2

⁢

F

3

Ω

⁢

∂

⁡

h

1

∂

⁡

q

3

+

h

1

⁢

F

3

Ω

⁢

∂

⁡

h

2

∂

⁡

q

3

]

=

1

Ω

(

[

∂

⁡

F

1

∂

⁡

q

1

h

2

h

3

+

F

1

h

3

∂

⁡

h

2

∂

⁡

q

1

+

F

1

h

2

∂

⁡

h

3

∂

⁡

q

1

]

+

[

h

1

⁢

∂

⁡

F

2

∂

⁡

q

2

⁢

h

3

+

∂

⁡

h

1

∂

⁡

q

2

⁢

F

2

⁢

h

3

+

h

1

⁢

F

2

⁢

∂

⁡

h

3

∂

⁡

q

2

]

+

[

h

1

h

2

∂

⁡

F

3

∂

⁡

q

3

+

∂

⁡

h

1

∂

⁡

q

3

h

2

F

3

+

h

1

∂

⁡

h

2

∂

⁡

q

3

F

3

]

)

=

1

Ω

⁢

(

∂

∂

⁡

q

1

⁢

(

F

1

⁢

h

2

⁢

h

3

)

+

∂

∂

⁡

q

2

⁢

(

h

1

⁢

F

2

⁢

h

3

)

+

∂

∂

⁡

q

3

⁢

(

h

1

⁢

h

2

⁢

F

3

)

)

Hence,

∇

∘

F

→

=

1

Ω

⁢

∑

i

∂

∂

⁡

q

i

⁢

(

Ω

h

i

⁢

F

i

)

where

⁢

Ω

=

Π

i

⁢

h

i

Title

Gradient and Divergence in Orthonormal Curvilinear Coordinates

Canonical

name

GradientAndDivergenceInOrthonormalCurvilinearCoordinates1

Date of creation

2013-03-11 19:26:23

Last modified on

2013-03-11 19:26:23

Owner

swapnizzle (13346)

Last modified by

(0)

Numerical id

1

Author

swapnizzle (0)

Entry type

Definition

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