Investigation Let (a) Show that where and (b)Let for Find (c)Let for Find (d)Let for

Chapter 15, Problem 53

(choose chapter or problem)

Let \(\mathbf{F}(x, y)=\frac{y}{x^{2}+y^{2}} \mathbf{i}-\frac{x}{x^{2}+y^{2}} \mathbf{j}\).

(a) Show that

\(\frac{\partial N}{\partial x}=\frac{\partial M}{\partial y}\)

where

\(M=\frac{y}{x^{2}+y^{2}}\) and \(N=\frac{-x}{x^{2}+y^{2}}\)

(b) Let \(\mathbf{r}(t)=\cos t \mathbf{i}+\sin t \mathbf{j}\) for \(0 \leq t \leq \pi\). Find \(\int_{C} \mathbf{F} \cdot d \mathbf{r}\).

(c) Let \(\mathbf{r}(t)=\cos t \mathbf{i}-\sin t \mathbf{j}\) for \(0 \leq t \leq \pi\). Find \(\int_{C} \mathbf{F} \cdot d \mathbf{r}\).

(d) Let \(\mathbf{r}(t)=\cos t \mathbf{i}+\sin t \mathbf{j}\) for \(0 \leq t \leq 2 \pi\). Find \(\int_{C} \mathbf{F} \cdot d \mathbf{r}\). Why doesn't this contradict Theorem 15.7?

(e) Show that \(\nabla\left(\arctan \frac{x}{y}\right)=\mathbf{F}\).

Text Transcription:

F(x, y) = y / x^{2} + y^{2} i - x / x^{2} + y^{2} j

partial N / partial x = partial M / partial y

M = y / x^2 + y^2

N= -x / x^2 + y^2

r(t) = cos ti + sin tj

0 leq t leq pi

int_C F cdot dr

0 leq t leq 2 pi

nabla (arctan x / y) = F