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Problem 14

(a) Find a function $ f $ such that $ \textbf{F} …

Problem 13

(a) Find a function $ f $ such that $ \textbf{F} = \nabla f $ and (b) use part (a) to evaluate $ \int_C \textbf{F} \cdot d \textbf{r} $ along the given curve $ C $.

$ \textbf{F}(x, y) = x^2y^3 \, \textbf{i} + x^3y^2 \, \textbf{j} $,
$ C $: $ \textbf{r}(t) = \langle t^3 - 2t, t^3 + 2t \rangle $, $ 0 \leqslant t \leqslant 1 $


(a) $f(x, y)=\frac{1}{3} x^{3} y^{3}$
(b) -9


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Video Transcript

find the function final potential function value is they grow by the fundament of serum. So we first find a function f by definition of radiant. We know I have has to satisfy this That's cube y squared. And that should gives us f because one over three excuse Why too, plus any constant which is function off why similarly f should be by taking anti their route to with respect or why f should be this and they in order for them to be equal the only way it can be true. This f equals one over three excuse like you, plus a constant and, uh, a sphere out. Start starting point at one point. Your problem t go zero you kinda starting point, which is zero zero. Andi, we problem t hose once we get em point, which should be one minus twos. Nephew for the one plus two is three. So this integral is simply f off negative one three minus off zero zero. So f off negative one three should be one over three times death one times three to which twenty seven as net fifty nine and proceed with the matter is called can so zero zero F zero zero zero So minus zero, which is negative nine

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