Solving Divergence in Spherical Coordinates

Introduction to solving divergence in spherical coordinate:

Spherical coordinates are mostly denoted as (r, ?, F). They used in three dimensional planes. Divergence of the function in spherical coordinates is denoted as `(grad .A).` Here, A is the given function. Spherical coordinates are mostly used in three dimensional planes. In online, we solving more problems with the help of tutors. More number of tutor are available in online at any time. Divergence of spherical coordinates are having three coordinates. In this article, we see about solving divergence in spherical coordinates and their example problems.

Divergence formula for solving spherical coordinate:

`(grad.A) = (1/r^2)(del)/(delr)(r^2A_r) + (1/(rsinphi))(del)/(deltheta)(A_theta) + (1/(rsinphi))(del)/(delphi)(sinphiA_phi)`
Example Problems for Solving Divergence in Spherical Coordinate

Solving divergence in spherical coordinate example problem 1:

Find the divergence value of the given function A = 2r3 + 6?3 + 6F

Solution:

Given spherical coordinate function A = 2r3 + 6?3 + 6F

Here,

Ar = 2r3, A? = 6?3 and AF = 6F

Find the divergence of the given function A by using the formula

`(grad.A) = (1/r^2)(del)/(delr)(r^2A_r) + (1/(rsinphi))(del)/(deltheta)(A_theta) + (1/(rsinphi))(del)/(delphi)(sinphiA_phi)`

From the given

r2Ar = 2r5

Differentiate the coordinates r with respect to r, we get

`(del)/(delr) (r^2 A_r)` = 10r4

Differentiate the coordinates ? with respect to ?, we get

`(del)/(deltheta) (A_theta)` = 18?2

Differentiate the coordinates F with respect to F, we get

` (del)/(delphi) (sinphiA_phi)` = `6sinphi + 6phi cosphi`

Substitute the above differentiation values in the given formula, we get

`(grad.A)` = `(1/r^2)` 10r4 + `(1/(rsinphi))` (18?2) + `(1/(rsinphi))(6sinphi + 6phi cosphi)`

Rearrange the above step, we get

`(grad.A)` = 10r2 + `((18theta^2)/(rsinphi))` + `((6sinphi + 6phi cosphi)/(rsinphi))`

Answer:

The final divergence of the function is 10r2 + `((18theta^2)/(rsinphi))` + `((6sinphi + 6phi cosphi)/(rsinphi))`
Solving Divergence in Spherical Coordinate Example Problem 2:

Find the divergence value of the given function A = 11r2 + 30?2 + F

Solution:

Given spherical coordinate function A = 11r2 + 30?2 + F

Here,

Ar = 11r2, A? = 30?2 and AF = F

Find the divergence of the given function A by using the formula

` (grad.A) = (1/r^2)(del)/(delr)(r^2A_r) + (1/(rsinphi))(del)/(deltheta)(A_theta) + (1/(rsinphi))(del)/(delphi)(sinphiA_phi)`

From the given

r2Ar = 11r4

Differentiate the coordinates r with respect to r, we get

`(del)/(delr) (r^2 A_r)` = 44r3

Differentiate the coordinates ? with respect to ?, we get

` (del)/(deltheta) (A_theta)` = 60?

Differentiate the coordinates F with respect to F, we get

`(del)/(delphi) (sinphiA_phi)` = `sinphi + phi cosphi`

Substitute the above differentiation values in the given formula, we get

`(grad.A)` = `(1/r^2)` 44r3 + `(1/(rsinphi))`(60?) + `(1/(rsinphi))(sinphi + phi cosphi)`

Rearrange the above step, we get

`(grad.A)` = 44r + `((60theta)/(rsinphi))` + `((sinphi + phi cosphi)/(rsinphi))`

Answer:

The final divergence of the function is 44r + `((60theta)/(rsinphi))` + `((sinphi + phi cosphi)/(rsinphi))`