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Solved Problems-13

Evaluate the line integral of the vector function Description: 81702.png around the square contour ABCD in the xy-plane as shown in the figure. Also integrate Description: 81695.png over the surface bounded by ABCD and verify that Stokes’ theorem holds good.
Description: 81686.png
Line integral
Here, Description: 81652.png
Description: 81644.png
Description: 81636.png
The closed line integral is given as,
Description: 81624.png
Along the path AB, dy = dz = 0, y = 0, z = 0; x varies from 0 to 2
Description: 81614.png
Along the path BC, dx = dz = 0, x = 2, z = 0; y varies from 0 to 2
Description: 81607.png
Along the path CD, dy = dz = 0, y = 2, z = 0; x varies from 2 to 0
Description: 81598.png
Along the path DA, dx = dz = 0, x = 0, z = 0; y varies from 2 to 0
Description: 81591.png
By addition, we get the closed line integral as,
Description: 81582.png
Now, the curl of the vector is,
Description: 81575.png
Description: 102911.png
Thus, Description: 81559.png; Stokes’ theorem is verified.

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