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Gradients : Find a Function and Evaluate an Integral

Please see the attached file for the fully formatted problems.

(a) Find a function f so that grad(f) = yi + (x + 3y^2)j
(b) Use part (a) to evaluate Sc grad(f) dt where C is the path starting at (0,2) goes down the y-axis to (0,0), along the x-axis to (2,0).


Solution Preview

grad(f) = (del(f)/del(x)).i + (del(f)/del(y)).j = (y).i + (x + 3y^2).j

Equate x and y terms separately,
=> del(f)/del(x) = y
=> f = x*y + pure terms of y ....(1)

(del(f)/del(y) = x + 3*y^2
=> f = x*y + y^3 + pure terms of x ...

Solution Summary

Gradients are involved in the evalution of an integral. The solution is explained.