# 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 2Along the path

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

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

*DA, dx = dz =*0*, x =*0*, z =*0; y varies from 2 to 0By addition, we get the closed line integral as,

Now, the curl of the vector is,