Compute the gradient and Laplacian of the function T = r (cos θ + sin θ cos ϕ). Check the Laplacian by converting T to Cartesian coordinates and using Eq. 1.42. Test the gradient theorem for this function, using the path shown in Fig. 1.41, from (0, 0, 0) to (0, 0, 2).

Eq. 1.42

Problem 41P

Solution 41P

Step 1 of 5:

In this question, we need to find gradient and Laplacian of function

Laplacian needs to be checked in terms of cartesian coordinates

Gradient of the function has be checked using the path given in the figure

Data given

Cartesian coordinates are given as

The figure give

Step 2 of 5:

We shall find the gradient of the function

We have gradient as

Using this on function

We get

Thus we have obtained the gradient of the function