# Solved Problems-13

Problem-13
Evaluate the line integral of the vector function  around the square contour ABCD in the xy-plane as shown in the figure. Also integrate  over the surface bounded by ABCD and verify that Stokesâ€™ theorem holds good.
Line integral
Solution
Here,

The closed line integral is given as,
Along the path AB, dy = dz = 0, y = 0, z = 0; x varies from 0 to 2

Along the path BC, dx = dz = 0, x = 2, z = 0; y varies from 0 to 2

Along the path CD, dy = dz = 0, y = 2, z = 0; x varies from 2 to 0

Along the path DA, dx = dz = 0, x = 0, z = 0; y varies from 2 to 0

By addition, we get the closed line integral as,

Now, the curl of the vector is,

Thus, ; Stokesâ€™ theorem is verified.