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## Homework Statement

grad f = (3x

^{2}+ y

^{2})i + (2xy - 3)j

a) find f

b) evalue the integral of grad f dot dr where c is any path from 1,2 to 2,1

c)evaluate the integral of gradf dot dr where c is the curve given by 4x

^{2}+ 9y

^{2}= 36

## Homework Equations

## The Attempt at a Solution

a) f = x

^{3}+ xy

^{2}- 3j

b) integral grad f dot dr = f(Q) - f(P) = (1 + 4 - 12) - (8 + 2 - 3) = -14

c) integral grad f dot dr = integral F(r(t)) dot r'(t) is this the correct method

r(t) = 4cost i + 9sint j

r'(t) = -4sint i + 9cost j

F(r(t)) = (3(4cost)

^{2}+ (9sint)

^{2})i + (2(4cost)(9sint) - 3)j

F(r(t)) dot r'(t) = (3(-4sint)(4cost)

^{2}+ (-4sint)(9sint)

^{2}) + (2(9cost)(4cost)(9sint) - 3(9cost)

= (3(-4sint)(4cost)

^{2}+ (-4sint)(9sint)

^{2}) + (2(9cost)(4cost)(9sint) - 3(9cost)

=[tex]\int[/tex] -192(cost)

^{2}sint + 324(sint)

^{3}+ 648(cost)

^{2}(sint)

^{2}- 27cost dt from 0 to pi

is this correct thus far