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Problem 216

a) Find the value of the line integral of the vector field \(\mathrm{F}^{-}(\mathrm{x}, \mathrm{y})=(\mathrm{y}, \mathrm{x})\) over the curve $\mathrm{C}^{-}(\mathrm{t})=(\mathrm{r} \cos \mathrm{t}, \mathrm{r} \sin \mathrm{t})$, \(0 \leq \mathrm{t} \leq(\pi / 4) ;\) both directly and by finding a potential function. b) Repeat for $\mathrm{C}^{-}(\mathrm{t})=(3 \cos \mathrm{t}, 3 \sin \mathrm{t}), 0 \leq \mathrm{t} \leq(\pi / 6)$.

Expert verified

In summary, for part (a), we found that the line integral of the vector field F(x, y) = (y, x) over the curve C(t) = (r*cos(t), r*sin(t)) with 0 ≤ t ≤ π/4 is \((r^2 \frac{2 - \sqrt{2}}{4})\), and for part (b) with the curve C(t) = (3*cos(t), 3*sin(t)) and 0 ≤ t ≤ π/6, the line integral is \(\frac{9(\sqrt{3} - 1)}{4}\). These results were consistent between the direct calculation method and the potential function method.

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Chapter 8

Evaluate the following line integrals a) $\int_{C}\left[\left(1+y^{2}\right) / x^{3}\right] d x-\left[\left(y+x^{2} y\right) / x^{2}\right]\( dy from \)(1,0)\( to \)(5,2)$ \((3,5)\) to \((5,13)\).

Chapter 8

Find the values of: (a) \(\int_{C}\left(x y+y^{2}-x y z\right) d x\) (b) \(\int_{\mathrm{C}}\left(\mathrm{x}^{2}-\mathrm{xy}\right)\) if \(\mathrm{C}\) is the arc of the parabola $\mathrm{y}=\mathrm{x}^{2}, \mathrm{z}=0\( from \)(-1,1,0)\( to \)(2,4,0)$.

Chapter 8

A force \(\mathrm{F}\) is called conservative if it is exact. Show that the force (vector field) \(F^{-}(x, y)=(y \cos x y, x \cos x y)\) is conservative. Then find the work done by this force in moving a particle from the origin to the point \((3,8)\).

Chapter 8

Evaluate the following line integrals: a) $^{(3,4)} \int_{(1,-2)}\left[(\mathrm{ydx}-\mathrm{xdx}) / \mathrm{x}^{2}\right]\( on the line \)\mathrm{y}=3 \mathrm{x}-5$ b) $^{(1,3)} \int_{(0,2)}\left(3 \mathrm{x}^{2} / \mathrm{y}\right) \mathrm{dx}-\left(\mathrm{x}^{3} / \mathrm{y}^{2}\right)$ dy on the parabola \(\mathrm{y}=2+\mathrm{x}^{2}\) c) \(^{(2,8)} \int_{(0,0)} \nabla^{-} \mathrm{f} \cdot \mathrm{dc} \rightarrow\) where \(\nabla^{\rightarrow} \mathrm{f}\) is grad \(\mathrm{f}\) and \(\mathrm{f}\) is the function $\mathrm{f}(\mathrm{x}, \mathrm{y})=\mathrm{x}^{2}-\mathrm{y}^{2} \cdot \mathrm{C}\( is the curve \)\mathrm{y}=\mathrm{x}^{3}$

Chapter 8

Verify Green's Theorem for \(\int_{C}\left(x^{2}-y^{2}\right) d x+2 x y d y\), where \(\mathrm{C}\) is the clockwise boundary of the square formed by the lines \(\mathrm{x}=0, \mathrm{x}=2, \mathrm{y}=0\), and \(\mathrm{y}=2\)

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