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 \((2,4,0)\).

Evaluate the following line integrals: a) \(\oint_{C} y^{2} d x+x^{2}\) dy where \(C\) is the triangle with vertices \((1,0)\), \((1,1),(0,0)\) (Figure 1). (1) b) \(\int_{C} x^{2} d x+x y d y\) where \(C\) is the straight line segment from \((1,0)\) to \((2,3)\). (2)

Evaluate the following line integrals: a) \(\oint_{C} F^{~} \cdot d C^{-}\) where \(F^{\rightarrow}\) is the vector field \(F^{-}(x, y)=\left(x^{2}, x y\right)\) and \(\mathrm{C}\) consists of the segment of the parabola \(\mathrm{y}=\mathrm{x}^{2}\) between \((0,0)\) and \((1,1)\) and the line segment from \((1,1)\) to \((0,0) .\) (Figure 1) b) \(\int_{C} F^{-} \cdot d C^{\rightarrow}\) where \(F^{-}=\left(x\left[\left(1-y^{2}\right) /\left(y^{2}+z^{2}\right)\right]^{1 / 2}, 0,0\right)\) and \(\mathrm{C}\) is the portion of the curve (in the first octant) of the intersection of the plane \(\mathrm{x}=\mathrm{y}\) and the cylinder \(2 \mathrm{y}^{2}+\mathrm{z}^{2}=1\) from \((0,0,1)\) to \([(\sqrt{2} / 2),(\sqrt{2} / 2), 0]\)

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.

Find the integral of the vector field \(\mathrm{F}^{\rightarrow}(\mathrm{x}, \mathrm{y})=\left[\left\\{-\mathrm{y} /\left(\mathrm{x}^{2}+\mathrm{y}^{2}\right)\right\\}\right.\), \(\left.\left\\{\mathrm{x} /\left(\mathrm{x}^{2}+\mathrm{y}^{2}\right)\right\\}\right]\) around the circle of radius 2 counterclockwise from the point \((2,0)\) to the point \((\sqrt{2}, \sqrt{2})\) (Figure 1). Repeat for the circle of radius 1 counterclockwise from \((1,0)\) to \([(\sqrt{2} / 2),(\sqrt{2} / 2)]\) (Figure 2).

a) Let \(\mathrm{F}^{-}\) be a vector field on some open set \(\mathrm{V}\). Assume that for some function \(\Phi\) on \(\mathrm{V}, \mathrm{F}^{\rightarrow}=\operatorname{grad} \phi^{-} \cdot\) Let \(\mathrm{P}\) and \(\mathrm{Q}\) be two points in \(\mathrm{V}\) and let \(\mathrm{C}\) be a curve joining these two points. Prove $$ \int_{\mathrm{C}} \mathrm{F}^{-} \cdot \mathrm{dc}^{-}=\Phi(\mathrm{Q})-\Phi(\mathrm{P}) $$ b) Evaluate \(\int_{C}\left[x /\left(x^{2}-y^{2}\right)\right] d x+\left[y /\left(y^{2}-x^{2}\right)\right]\) dy where \(C\) is a curve from \((1,0)\) to \((5,3)\) and lies between the lines \(\mathrm{y}=\mathrm{x}\) and \(\mathrm{y}=-\mathrm{x}\), by the above method.

Does the vector field \(F^{-\infty}(x, y)=\left[\left\\{-y /\left(x^{2}+y^{2}\right)\right\\},\left\\{x /\left(x^{2}+y^{2}\right)\right\\}\right]\) have a potential function? Explain why or why not. Is this vector field exact?

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 \(\mathrm{C}^{-}(\mathrm{t})=(3 \cos \mathrm{t}, 3 \sin \mathrm{t}), 0 \leq \mathrm{t} \leq(\pi / 6)\).

a) Find the integral of the vector field \(\mathrm{F}^{-}(\mathrm{x}, \mathrm{y})=\left[\left(\mathrm{x} / \mathrm{r}^{3}\right),\left(\mathrm{y} / \mathrm{r}^{3}\right)\right.\), where \(\mathrm{r}=\left(\mathrm{x}^{2}+\mathrm{y}^{2}\right)^{1 / 2}\), along the curve \(\mathrm{C}^{\rightarrow}(\mathrm{t})=\left(\mathrm{e}^{\mathrm{t}} \cos \mathrm{t}\right.\), \(\mathrm{e}^{\mathrm{t}}\) sin \(\mathrm{t}\) ) from the point \((1,0)\) to the point \(\left(\mathrm{e}^{2 \pi}, 0\right)\). b) Does \(\mathrm{F}^{-}\) admit a potential function? If so, compute the integral in a) by this method.

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 \(\mathrm{y}=\mathrm{x}^{3}\)

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 \((5,13)\).

Let \(\mathrm{F}^{\rightarrow}\left(\mathrm{X}^{\rightarrow}\right)=\left(\mathrm{kX}^{-} / \mathrm{r}^{3}\right)\) where \(\mathrm{r}=\left\|\mathrm{X}^{-}\right\| ; \mathrm{X}^{-}=(\mathrm{x}, \mathrm{y}, \mathrm{z})\) and \(\mathrm{k}\) is a constant. Find \(\mathrm{Q} \int_{\mathrm{P}} \mathrm{F}^{\rightarrow} \cdot \mathrm{dc}^{-}\) where \(\mathrm{P}=(1,1,1)\) and \(\mathrm{Q}=(1,2,-1)\)

Let \(\mathrm{r}=\left(\mathrm{x}^{2}+\mathrm{y}^{2}+\mathrm{z}^{2}\right)^{1 / 2}\). Compute \(\mathrm{Q} \int_{\mathrm{P}}[(\cos \mathrm{r}) / \mathrm{r}](\mathrm{x} \mathrm{dx}+\mathrm{ydy}+\mathrm{zdz})\) where \(\mathrm{P}\) is the point \((\pi,-\pi, \pi / 2)\) and \(\mathrm{Q}\) is the point \([(2 \pi / 3),(2 \pi / 3),(-\pi / 3)]\).

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 \(\left.\Phi\right)\).

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) \((x, y) f_{(0,0)} \sin y d x+x \cos y d y\)

Let \(\mathrm{F}^{-7}\) be the following vector fields a) \(\mathrm{F}^{\rightarrow}{\underline{\phantom{xx}}}_{1}(\mathrm{x}, \mathrm{y}, \mathrm{z})=(1-\mathrm{yz}, 1-\mathrm{zx},-\mathrm{xy})\) b) \(\mathrm{F}^{\rightarrow}{\underline{\phantom{xx}}}_{2}(\mathrm{x}, \mathrm{y}, \mathrm{z})=(\mathrm{yz}, \mathrm{xz}, \mathrm{xy})\) c) \(\mathrm{F}^{\rightarrow}{\underline{\phantom{xx}}}_{3}(\mathrm{x}, \mathrm{y}, \mathrm{z})=(\log (\mathrm{xy}), \mathrm{x}, \mathrm{y})\) Determine, for each vector field, if \(\int \mathrm{F}^{\rightarrow} \cdot \mathrm{dc}^{\rightarrow}\) is independent of the path and if so integrate the vector field over the curve from the point \(\mathrm{P}=(1,6,5)\) to the point \(\mathrm{Q}=(4,3,2)\).

Let \(F\) be the vector field \(F^{-}(x, y, z)=(y z, x z, x y) .\) Compute \(\mathrm{Q} \int_{\mathrm{P}} \mathrm{F}^{-} \cdot \mathrm{dc}^{-}\) where \(\mathrm{Q}\) is the point \((1,2,3)\) and \(\mathrm{P}\) is the point \((1,0,-1)\), first by choosing a broken line path with segments parallel to the axes. Then find a function \(\phi(\mathrm{x}, \mathrm{y}, \mathrm{z})\) such that \(\mathrm{F}^{\rightarrow}=\operatorname{grad} \Phi\).

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 \((3,8)\).

Prove Green's Theorem in the plane if the region \(\mathrm{R}\) is representable in both of the forms $$ \mathrm{a} \leq \mathrm{x} \leq \mathrm{b}, \mathrm{f}_{1}(\mathrm{x}) \leq \mathrm{y} \leq \mathrm{f}_{2}(\mathrm{x}) $$ $$ \mathrm{C} \leq \mathrm{y} \leq \mathrm{D}, \mathrm{g}_{1}(\mathrm{y}) \leq \mathrm{x} \leq \mathrm{g}_{2}(\mathrm{y}) $$ as in Fig. 1 . Let \(\mathrm{R}\) be the region in \(\mathrm{R}^{2}\) and let \(\mathrm{C}\) be the curve bounding \(\mathrm{R}\) as given in the figure. \(\mathrm{C}\) is oriented in such a way that \(\mathrm{R}\) is always to the left of \(\mathrm{C}\). Let \(\mathrm{P}(\mathrm{x}, \mathrm{y}), \mathrm{Q}(\mathrm{x}, \mathrm{y})\) be real-valued differentiable functions defined on \(\mathrm{R}\). Then Green's Theorem in the plane states that $$ \oint_{C} P d x+Q d y=\iint_{R}[(\partial Q / \partial x)-(\partial P / \partial y)] d x d y . $$

Let \(\mathrm{C}\) be the ellipse \(\mathrm{x}^{2}+4 \mathrm{y}^{2}=4\). Compute \(\oint_{C}(2 x-y) d x+(x+3 y) d y\) by Green's Theorem.

a) Verify Green's Theorem for the vector field \(F^{-}(x, y)=(-y, x)\) where \(\mathrm{C}^{-}(\mathrm{t})=(\mathrm{r} \cos \mathrm{t}, \mathrm{r} \sin \mathrm{t})-\pi \leq \mathrm{t} \leq \pi\) and \(\mathrm{R}=(\mathrm{P}:|\mathrm{P}| \leq \mathrm{r})\). b) Evaluate \(\oint_{\mathrm{C}}\left(\mathrm{x}^{2}+2 \mathrm{y}^{2}\right) \mathrm{dx}\) where \(\mathrm{C}\) is the square with vertices \((1,1),(1,-1),(-1,-1),(-1,1)\)

Use Green's Theorem to find: a) \(\mathrm{y}^{2} \mathrm{~d} \mathrm{x}-\mathrm{xdy}\) clockwise around the triangle whose vertices are at \((0,0),(0,1),(1,0)\). b) The integral of the vector field \(\mathrm{F}^{\rightarrow}(\mathrm{x}, \mathrm{y})=(\mathrm{y}+3 \mathrm{x}, 2 \mathrm{y}-\mathrm{x})\) counterclockwise around the ellipse \(4 \mathrm{x}^{2}+\mathrm{y}^{2}=4\)

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 \(\mathrm{y}=2\)

Use Green's Theorem to deduce the integral formula: \(\iint_{R}\left[\left(\partial^{2} u / \partial x^{2}\right)+\left(\partial^{2} u / \partial y^{2}\right)\right] d x d y=\int_{C}(\partial u / \partial n) d x\) where s is the arc length along \(\mathrm{C}\) and \(\mathrm{n}\) is the outer normal to \(\mathrm{C}\).

Show that if the flow \(\mathrm{F}^{\rightarrow}\) is defined in a rectangle \(\mathrm{R}\), and has zero divergence at every point of \(R\), then the rate of flow across every closed path in \(\mathrm{R}\) is zero.