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