Spherical Coordinates

Coordinate Map

We use the spherical coordinates defined by

where

Equivalently, the position vector is

Differential Geometry of the Coordinates

Differentiate with respect to each coordinate:

These are the tangent vectors to the coordinate curves. Their lengths are

Therefore the spherical scale factors are

Dividing each tangent vector by its length gives the orthonormal basis vectors:

Since

we get

Taking the squared norm,

Time Derivative of Basis Vectors

Suppose the spherical coordinates depend on time:

Then the spherical basis vectors also depend on time through and :

Notice that none of these basis vectors depends on . Therefore

So only derivatives with respect to and contribute to the time derivatives.

Partial Derivatives of the Basis Vectors

First compute the derivatives of .

With respect to ,

With respect to ,

Now compute the derivatives of .

With respect to ,

With respect to ,

Finally compute the derivatives of .

With respect to ,

since does not depend on .

With respect to ,

To rewrite this in the spherical basis, express it as a linear combination of and :

Hence

Collecting all partial derivatives:

Time Derivatives

Now apply the chain rule. Since the basis vectors depend only on and ,

for each spherical basis vector .

For :

For :

For :

Therefore the time derivatives of the spherical basis vectors are

Jacobian and Volume Element

From

the differential relation in Cartesian coordinates is

So the Jacobian matrix is

Its determinant is

Now expand along the third column:

Compute the two cofactors:

Substitute these back:

Hence the Jacobian is

Therefore the volume element is

The same result also follows immediately from the scale factors:

Inverse Differential Relations

Starting from

take dot products with the orthonormal basis vectors.

Because

we obtain

Now substitute

For :

For :

so

For :

so

Collecting the three formulas,

Gradient, Divergence, and Curl

For any orthogonal coordinate system with scale factors ,

and

In spherical coordinates,

Write the vector field as

Gradient

Substitute the scale factors into the general formula:

Therefore

Divergence

Start from the orthogonal-coordinate formula:

Now use

Then

Because does not depend on , and does not depend on or , this simplifies to

Curl

Substitute into the determinant formula:

Expanding the determinant along the first row gives the component:

For the component:

For the component:

Therefore

Equivalently, the components are

These formulas are valid away from the coordinate singularities and .