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Question

# Let$\mathbf{F}=\frac{y}{x^2+y^2} \mathbf{i}-\frac{x}{x^2+y^2} \mathbf{j}=M \mathbf{i}+N \mathbf{j}$(a) Show that $\partial N / \partial x=\partial M / \partial y$.(b) Show, by using the parametrization $x=\cos t, y=\sin t$, that $\oint_C M d x+N d y=-2 \pi$, where C is the unit circle.(c) Why doesn't this contradict Green's Theorem?

Solution

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(a) In order to find $\dfrac{\partial M}{\partial x}$, we will differentiate $M$, while treating $y$ like a constant

\begin{aligned} \dfrac{\partial M}{\partial x}&=\dfrac{\partial}{\partial x}\left[\dfrac{-x}{x^2+y^2}\right]\\ &\overset{\color{#4257b2}{(1)}}{=}\dfrac{\left(-x\right)'\left(x^2+y^2\right)-\left(-x\right)\left(x^2+y^2\right)'}{\left(x^2+y^2\right)^2}\\ &\overset{\color{#4257b2}{(2)}}{=}\dfrac{\left(-1\right)\left(x^2+y^2\right)-\left(-x\right)\left(2x^{2-1}+0\right)}{\left(x^2+y^2\right)^2}\\ &=\dfrac{-\left(x^2+y^2\right)+2x^2}{\left(x^2+y^2\right)^2}\\ &=\dfrac{x^2-y^2}{\left(x^2+y^2\right)^2}\\ \end{aligned}

Note$\color{#4257b2}{(1)}$: Use the quotient rule for differentiation

Note$\color{#4257b2}{(2)}$: Use the power rule for differentiation

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