Step by Step Solutions

Solution: (Line integrals of closed paths) Let F(x, y)=(-y)/(x^2+y^2) i+(x)/(x^2+y^2) j Let C be the counter-clockwisely oriented curve given by the polar

Question: (Line integrals of closed paths) Let

\[\mathbf{F}(x, y)=\frac{-y}{x^{2}+y^{2}} \mathrm{i}+\frac{x}{x^{2}+y^{2}} \mathrm{j}\]
  1. Let \(C\) be the counter-clockwisely oriented curve given by the polar equation \(r=2+\cos \theta, 0 \leq \theta \leq 2 \pi .\) Evaluate \(\oint_{C} \mathbf{F} \cdot d \mathbf{r} .\) Here, \(\oint\) denotes the line integral along a closed path. Hint: One may want to use Green's theorem.
  2. Let \(C\) be the counter-clockwisely oriented curve given by the polar equation \(r=2+\cos (7 \theta), 0 \leq \theta \leq 10 \pi .\) Evaluate \(\oint_{C} \mathbf{F} \cdot d \mathbf{r}\)

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