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Problem 209

a) Let \(F^{\rightarrow}\) be a vector field on an open set \(V\) and \(C\) a curve in V defined on the interval \([a, b]\). Prove $\int_{(C)-} F^{\rightarrow}=-\int_{C} F^{-}\(, where \)C^{-}$ is the reverse path of the curve \(C\). b) Then evaluate $\int_{\mathrm{C}} \mathrm{F}^{-} \cdot \mathrm{d} \mathrm{C}^{-}\( where \)\mathrm{F}^{-}(\mathrm{x}, \mathrm{y})=\left(\mathrm{x}^{2}, \mathrm{xy}\right)$ along the line segment from the point \((1,1)\) to \((0,0)\) using the reverse path.

Expert verified

The short version of the answer based on the provided step-by-step solution would be:
We first proved the relationship between the line integrals of curve \(C\) and its reverse path \(C^-\) by parametrizing both paths and comparing the results. Then, we found the parametric representation of the reverse path of the line segment from point (1,1) to point (0,0) as \(\vec{r}^-(t) = (1-t,1-t)\) and evaluated the line integral of the given vector field \(\vec{F}(x, y) = (x^2, xy)\) along this path. The result of the integral is \(\int_{\mathrm{C}} \mathrm{F}^{-}\cdot \mathrm{d}
\mathrm{C}^{-} = -\frac{2}{3}\).

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Chapter 8

Prove in an open connected set \(\mathrm{U}\) that $\mathrm{Q} \int_{\mathrm{P}, \mathrm{C}} \mathrm{F}^{-} \cdot \mathrm{dc}^{-}$ is independent of the path \(\mathrm{C}\) if \(\mathrm{F}^{-}\) has a potential function \(\left(\mathrm{F}^{-}=\operatorname{grad} \Phi\right.\) for some scalar function \(\left.\Phi\right)\).

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 \(\mathrm{y}=2\)

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

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

Show that the following functions are independent of the path in the \(\mathrm{xy}\) -plane and evaluate them: a) \(^{(x, y)} f_{(1,1,)} 2 x y d x+\left(x^{2}-y^{2}\right) d y\) b) \((x, y) f_{(0,0)} \sin y d x+x \cos y d y\)

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