# Gradients : Find a Function and Evaluate an Integral

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(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).

https://brainmass.com/math/integrals/gradients-function-evaluate-integral-20187

#### Solution Preview

a.)

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)

and,

(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.

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