Chapter 16

Green's first formula Suppose that \(f\) and \(g\) are scalar functions with continuous first- and second-order partial derivatives throughout a region \(D\) that is bounded by a closed piecewise smooth surface \(S .\) Show that $$\iint_{S} f \nabla g \cdot \mathbf{n} d \sigma=\iiint_{D}\left(f \nabla^{2} g+\nabla f \cdot \nabla g\right) d V \quad \quad\quad(10)$$ Equation \((10)\) is Green's first formula. (Hint: Apply the Divergence Theorem to the field \(\mathbf{F}=f \nabla g . )\)

\begin{equation} \begin{array}{c}{\text { a. Exact differential form How are the constants } a, b, \text { and } c} \\ {\text { related if the following differential form is exact? }} \\ {\left(a y^{2}+2 c z x\right) d x+y(b x+c z) d y+\left(a y^{2}+c x^{2}\right) d z} \\ {\text { b. Gradient field For what values of } b \text { and } c \text { will }} \\ {\mathbf{F}=\left(y^{2}+2 c z x\right) \mathbf{i}+y(b x+c z) \mathbf{j}+\left(y^{2}+c x^{2}\right) \mathbf{k}} \\\ {\text { be a gradient field? }}\end{array} \end{equation}

a. Parametrization of an ellipsoid The parametrization \(x=a \cos \theta, \quad y=b \sin \theta, \quad 0 \leq \theta \leq 2 \pi\) gives the ellipse \(\left(x^{2} / a^{2}\right)+\left(y^{2} / b^{2}\right)=1 .\) Using the angles \(\theta\) and \(\phi\) in spherical coordinates, show that $$ \mathbf{r}(\theta, \phi)=(a \cos \theta \cos \phi) \mathbf{i}+(b \sin \theta \cos \phi) \mathbf{j}+(c \sin \phi) \mathbf{k} $$ is a parametrization of the ellipsoid \(\left(x^{2} / a^{2}\right)+\left(y^{2} / b^{2}\right)+\) \(\left(z^{2} / c^{2}\right)=1\) b. Write an integral for the surface area of the ellipsoid, but do not evaluate the integral.

Green's Theorem Area Formula Area of \(R=\frac{1}{2} \oint_{C} x d y-y d x\) Use the Green's Theorem area formula given above to find the areas of the regions enclosed by the curves. The astroid \(\mathbf{r}(t)=\left(\cos ^{3} t\right) \mathbf{i}+\left(\sin ^{3} t\right) \mathbf{j}, \quad 0 \leq t \leq 2 \pi\)

Let \(R\) be a region in the \(x y\) -plane that is bounded by a piecewise smooth simple closed curve \(C\) and suppose that the moments of inertia of \(R\) about the \(x\) - and \(y\) -axes are known to be \(I_{x}\) and \(I_{y}\) . Evaluate the integral $$\oint_{C} \nabla\left(r^{4}\right) \cdot \mathbf{n} d s$$ where \(r=\sqrt{x^{2}+y^{2}},\) in terms of \(I_{x}\) and \(I_{y}\)

Mass of a wire Find the mass of a wire that lies along the curve \(\mathbf { r } ( t ) = \left( t ^ { 2 } - 1 \right) \mathbf { j } + 2 t \mathbf { k } , 0 \leq t \leq 1 ,\) if the density is \(\delta = ( 3 / 2 ) t\)

In Exercises \(31-34,\) find the circulation and flux of the field \(\mathbf{F}\) around and across the closed semicircular path that consists of the semicircular arch \(\mathbf{r}_{1}(t)=(a \cos t) \mathbf{i}+(a \sin t) \mathbf{j}, 0 \leq t \leq \pi,\) followed by the line segment \(\mathbf{r}_{2}(t)=t \mathbf{i},-a \leq t \leq a\) $$\mathbf{F}=-y \mathbf{i}+x \mathbf{j}$$

Gradient of a line integral Suppose that \(\mathbf{F}=\nabla f\) is a conservative vector field and $$g(x, y, z)=\int_{(0,0,0)}^{(x, y z)} \mathbf{F} \cdot d \mathbf{r}$$ Show that \(\nabla g=\mathbf{F}\).

Hyperboloid of one sheet $$ \begin{array}{l}{\text { a. Find a parametrization for the hyperboloid of one sheet }} \\ {x^{2}+y^{2}-z^{2}=1 \text { in terms of the angle } \theta \text { associated with }} \\ {\text { the circle } x^{2}+y^{2}=r^{2} \text { and the hyperbolic parameter } u} \\ {\text { associated with the hyperbolic function } r^{2}-z^{2}=1} \\ {\text { (Hint: } \cosh ^{2} u-\sinh ^{2} u=1 . )}\\\\{\text { b. Generalize the result in part (a) to the hyperboloid }} \\\ {\left(x^{2} / a^{2}\right)+\left(y^{2} / b^{2}\right)-\left(z^{2} / c^{2}\right)=1}\end{array} $$

Zero curl, yet the field is not conservative Show that the curl of $$\mathbf{F}=\frac{-y}{x^{2}+y^{2}} \mathbf{i}+\frac{x}{x^{2}+y^{2}} \mathbf{j}+z \mathbf{k}$$ is zero but that $$\oint_{C} \mathbf{F} \cdot d \mathbf{r}$$ is not zero if \(C\) is the circle \(x^{2}+y^{2}=1\) in the \(x y\) -plane. Theorem 7 does not apply here because the domain of \(\mathbf{F}\) is not simply connected. The field \(\mathbf{F}\) is not defined along the \(z\) -axis so there is no way to contract \(C\) to a point without leaving the domain of F.)

In Exercises \(31-34,\) find the circulation and flux of the field \(\mathbf{F}\) around and across the closed semicircular path that consists of the semicircular arch \(\mathbf{r}_{1}(t)=(a \cos t) \mathbf{i}+(a \sin t) \mathbf{j}, 0 \leq t \leq \pi,\) followed by the line segment \(\mathbf{r}_{2}(t)=t \mathbf{i},-a \leq t \leq a\) $$\mathbf{F}=-y^{2} \mathbf{i}+x^{2} \mathbf{j}$$

Center of mass of a curved wire \(A\) wire of density \(\delta ( x , y , z ) = 15 \sqrt { y + 2 }\) lies along the curve \(\mathbf { r } ( t ) = \left( t ^ { 2 } - 1 \right) \mathbf { j } +\) \(2 t \mathbf { k } , - 1 \leq t \leq 1 .\) Find its center of mass. Then sketch the curve and center of mass together.

Green's Theorem Area Formula Area of \(R=\frac{1}{2} \oint_{C} x d y-y d x\) Use the Green's Theorem area formula given above to find the areas of the regions enclosed by the curves. One arch of the cycloid \(x=t-\sin t, \quad y=1-\cos t\)

Conservation of mass Let \(\mathbf{v}(t, x, y, z)\) be a continuously differentiable vector field over the region \(D\) in space and let \(p(t, x)\) \(y, z )\) be a continuously differentiable scalar function. The variable \(t\) represents the time domain. The Law of Conservation of Mass asserts that $$\frac{d}{d t} \iiint_{D} p(t, x, y, z) d V=-\iint_{S} p \mathbf{v} \cdot \mathbf{n} d \sigma$$ where \(S\) is the surface enclosing \(D\) a. Give a physical interpretation of the conservation of mass law if \(\mathbf{v}\) is a velocity flow field and \(p\) represents the density of the fluid at point \((x, y, z)\) at time \(t\) b. Use the Divergence Theorem and Leibniz's Rule, $$\frac{d}{d t} \iiint_{D} p(t, x, y, z) d V=\iiint_{D} \frac{\partial p}{\partial t} d V$$ to show that the Law of Conservation of Mass is equivalent to the continuity equation, $$\nabla \cdot p \mathbf{v}+\frac{\partial p}{\partial t}=0$$ (In the first term \(\nabla \cdot p \mathbf{v},\) the variable \(t\) is held fixed, and in the second term \(\partial p / \partial t,\) it is assumed that the point \((x, y, z)\) in \(D\) is held fixed.)

(Continuation of Exercise \(34 . )\) Find a Cartesian equation for the plane tangent to the hyperboloid \(x^{2}+y^{2}-z^{2}=25\) at the point \(\left(x_{0}, y_{0}, 0\right),\) where \(x_{0}^{2}+y_{0}^{2}=25\)

Flow integrals Find the flow of the velocity field \(\mathbf{F}=\) \((x+y) \mathbf{i}-\left(x^{2}+y^{2}\right) \mathbf{j}\) along each of the following paths from \((1,0)\) to \((-1,0)\) in the \(x y\) -plane. \begin{equation} \begin{array}{l}{\text { a. The upper half of the circle } x^{2}+y^{2}=1} \\\ {\text { b. The line segment from }(1,0) \text { to }(-1,0)} \\ {\text { c. The line segment from }(1,0) \text { to }(0,-1) \text { followed by the line }} \\ {\text { segment from }(0,-1) \text { to }(-1,0)}\end{array} \end{equation}

Mass of wire with variable density Find the mass of a thin wire lying along the curve \(\mathbf { r } ( t ) = \sqrt { 2 } t \mathbf { i } + \sqrt { 2 } t \mathbf { j } + \left( 4 - t ^ { 2 } \right) \mathbf { k }\) \(0 \leq t \leq 1 ,\) if the density is (a) \(\delta = 3 t\) and (b) \(\delta = 1\)

Let \(C\) be the boundary of a region on which Green's Theorem holds. Use Green's Theorem to calculate a. \(\oint_{C} f(x) d x+g(y) d y\) b. \(\oint_{C} k y d x+h x d y \quad(k\) and \(h\) constants \().\)

A revealing experiment By experiment, you find that a force field \(\mathbf{F}\) performs only half as much work in moving an object along path \(C_{1}\) from \(A\) to \(B\) as it does in moving the object along path \(C_{2}\) from \(A\) to \(B\) . What can you conclude about \(\mathbf{F}\) ? Give reasons for your answer.

Hyperboloid of two sheets Find a parametrization of the hyperboloid of two sheets \(\left(z^{2} / c^{2}\right)-\left(x^{2} / a^{2}\right)-\left(y^{2} / b^{2}\right)=1\)

Integral dependent only on area Show that the value of $$\oint_{C} x y^{2} d x+\left(x^{2} y+2 x\right) d y$$ around any square depends only on the area of the square and not on its location in the plane.

Center of mass of wire with variable density Find the center of mass of a thin wire lying along the curve \(\mathbf { r } ( t ) = t \mathbf { i } + 2 t \mathbf { j } +\) \(( 2 / 3 ) t ^ { 3 / 2 } \mathbf { k } , 0 \leq t \leq 2 ,\) if the density is \(\delta = 3 \sqrt { 5 } + t\)

Work by a constant force Show that the work done by a constant force field \(\mathbf{F}=a \mathbf{i}+b \mathbf{j}+c \mathbf{k}\) in moving a particle along any path from \(A\) to \(B\) is \(W=\mathbf{F} \cdot \overrightarrow{A B} .\)

Find the area of the surface cut from the paraboloid \(x^{2}+y^{2}-z=\) 0 by the plane \(z=2\) .

Evaluate the integral $$\oint_{C} 4 x^{3} y d x+x^{4} d y$$ for any closed path \(C .\)

Find the flux of the field \(\mathbf{F}(x, y, z)=z^{2} \mathbf{i}+x \mathbf{j}-3 z \mathbf{k}\) outward through the surface cut from the parabolic cylinder \(z=4-y^{2}\) by the planes \(x=0, x=1,\) and \(z=0\) .

Moment of inertia of wire hoop A circular wire hoop of constant density \(\delta\) lies along the circle \(x ^ { 2 } + y ^ { 2 } = a ^ { 2 }\) in the \(x y\) -plane.Find the hoop's moment of inertia about the \(z\) -axis.

The flow of a gas with a density of \(\delta=0.001 \mathrm{kg} / \mathrm{m}^{2}\) over the closed curve \(\mathbf{r}(t)=(-\sin t) \mathbf{i}+(\cos t) \mathbf{j}, 0 \leq t \leq 2 \pi,\) is given by the vector field \(\mathbf{F}=\delta \mathbf{v},\) where \(\mathbf{v}=x \mathbf{i}+y^{2} \mathbf{j}\) is a velocity field measured in meters per second. Find the flux of \(\mathbf{F}\) across the curve \(\mathbf{r}(t) .\)

Gravitational field \begin{equation}\begin{array}{c}{\text { a. Find a potential function for the gravitational field }} \\ {\mathbf{F}=-G m M \frac{x \mathbf{i}+y \mathbf{j}+z \mathbf{k}}{\left(x^{2}+y^{2}+z^{2}\right)^{3 / 2}}} \\ {(G, m, \text { and } M \text { are constants })}\\\\{\text { b. Let } P_{1} \text { and } P_{2} \text { be points at distance } s_{1} \text { and } s_{2} \text { from the origin. }} \\ {\text { Show that the work done by the gravitational field in part (a) }} \\ {\text { in moving a particle from } P_{1} \text { to } P_{2} \text { is }}\end{array}\end{equation} \begin{equation}\operatorname{GmM}\left(\frac{1}{s_{2}}-\frac{1}{s_{1}}\right).\end{equation}

Find the area of the band cut from the paraboloid \(x^{2}+y^{2}-z=\) 0 by the planes \(z=2\) and \(z=6 .\)

Find the flux of the field \(\mathbf{F}(x, y, z)=4 x \mathbf{i}+4 y \mathbf{j}+2 \mathbf{k}\) outward (away from the \(z\) -axis) through the surface cut from the bottom of the paraboloid \(z=x^{2}+y^{2}\) by the plane \(z=1\)

The flow of a gas with a density of \(\delta=0.3 \mathrm{kg} / \mathrm{m}^{2}\) over the closed curve \(\mathbf{r}(t)=(\cos t) \mathbf{i}+(\sin t) \mathbf{j}, 0 \leq t \leq 2 \pi,\) is given by the vector field \(\mathbf{F}=\delta \mathbf{v},\) where \(\mathbf{v}=x^{2} \mathbf{i}-y \mathbf{j}\) is a velocity field measured in meters per second. Find the flux of \(\mathbf{F}\) across the curve \(\mathbf{r}(t) .\)

Area as a line integral Show that if \(R\) is a region in the plane bounded by a piecewise smooth, simple closed curve \(C,\) then $$\begin{array}{l}{\text {Area of}}\end{array}R=\oint_{C} x d y=-\oint_{C} y d x$$

Find the area of the region cut from the plane \(x+2 y+2 z=5\)by the cylinder whose walls are \(x=y^{2}\) and \(x=2-y^{2}\)

Let \(S\) be the portion of the cylinder \(y=e^{x}\) in the first octant that projects parallel to the \(x\) -axis onto the rectangle \(R_{y z} : 1 \leq y \leq 2\) \(0 \leq z \leq 1\) in the \(y z\) -plane (see the accompanying figure). Let \(\mathbf{n}\) be the unit vector normal to \(S\) that points away from the \(y z\) -plane. Find the flux of the field \(\mathbf{F}(x, y, z)=-2 \mathbf{i}+2 y \mathbf{j}+z \mathbf{k}\) across \(S\) in the direction of \(\mathbf{n} .\)

Two springs of constant density A spring of constant density \(\delta\) lies along the helix $$ \mathbf { r } ( t ) = ( \cos t ) \mathbf { i } + ( \sin t ) \mathbf { j } + t \mathbf { k } , \quad 0 \leq t \leq 2 \pi $$ a. Find \(I _ { z }\) b. Suppose that you have another spring of constant density \(\delta\) that is twice as long as the spring in part (a) and lies along the helix for \(0 \leq t \leq 4 \pi .\) Do you expect \(I _ { z }\) for the longer spring to be the same as that for the shorter one, or should it be different? Check your prediction by calculating \(I _ { z }\) for the longer spring.

Definite integral as a line integral Suppose that a nonnegative function \(y=f(x)\) has a continuous first derivative on \([a, b] .\) Let \(C\) be the boundary of the region in the \(x y\) -plane that is bounded below by the \(x\) -axis, above by the graph of \(f,\) and on the sides by the lines \(x=a\) and \(x=b .\) Show that $$\int_{a}^{b} f(x) d x=-\oint_{C} y d x.$$

Find the area of the portion of the surface \(x^{2}-2 z=0\) that lies above the triangle bounded by the lines \(x=\sqrt{3}, y=0,\) and \(y=x\) in the \(x y\) -plane.

Let \(S\) be the portion of the cylinder \(y=\ln x\) in the first octant whose projection parallel to the \(y\) -axis onto the \(x z\) -plane is the rectangle \(R_{x :} : 1 \leq x \leq e, 0 \leq z \leq 1 .\) Let \(n\) be the unit vector normal to \(S\) that points away from the \(x z\) -plane. Find the flux of \(\mathbf{F}=2 y \mathbf{j}+z \mathbf{k}\) through \(S\) in the direction of \(\mathbf{n} .\)

Find the outward flux of the field \(\mathbf{F}=2 x y \mathbf{i}+2 y z \mathbf{j}+2 x z \mathbf{k}\) across the surface of the cube cut from the first octant by the planes \(x=a, y=a, z=a .\)

Find the work done by the force \(\mathbf{F}=y^{2} \mathbf{i}+x^{3} \mathbf{j},\) where force is measured in newtons, in moving an object over the curve \(\mathbf{r}(t)=2 t \mathbf{i}+t^{2} \mathbf{j}, \quad 0 \leq t \leq 2,\) where distance is measured in meters.

Find the area of the surface \(x^{2}-2 y-2 z=0\) that lies above the triangle bounded by the lines \(x=2, y=0,\) and \(y=3 x\) in the \(x y\) -plane.

Find the outward flux of the field \(\mathbf{F}=x z \mathbf{i}+y z \mathbf{j}+\mathbf{k}\) across the surface of the upper cap cut from the solid sphere \(x^{2}+y^{2}+z^{2} \leq 25\) by the plane \(z=3\)

Find the work done by the force \(\mathbf{F}=e^{y \mathbf{i}}+(\ln x) \mathbf{j}+3 z \mathbf{k}\), where force is measured in newtons, in moving an object over the curve \(\mathbf{r}(t)=e^{t} \mathbf{i}+(\ln t) \mathbf{j}+t^{2} \mathbf{k}, 1 \leq t \leq e,\) where distance is measured in meters.

Center of mass and moments of inertia for wire with variable density Find the center of mass and the moments of inertia about the coordinate axes of a thin wire lying along the curve $$ \mathbf { r } ( t ) = t \mathbf { i } + \frac { 2 \sqrt { 2 } } { 3 } t ^ { 3 / 2 } \mathbf { j } + \frac { t ^ { 2 } } { 2 } \mathbf { k } , \quad 0 \leq t \leq 2 $$ if the density is \(\delta = 1 / ( t + 1 )\)

Find the area of the cap cut from the sphere \(x^{2}+y^{2}+z^{2}=2\) by the cone \(z=\sqrt{x^{2}+y^{2}} .\)

Green's Theorem and Laplace's equation Assuming that all the necessary derivatives exist and are continuous, show that if \(f(x, y)\) satisfies the Laplace equation $$\frac{\partial^{2} f}{\partial x^{2}}+\frac{\partial^{2} f}{\partial y^{2}}=0,$$ then $$\oint_{C} \frac{\partial f}{\partial y} d x-\frac{\partial f}{\partial x} d y=0$$ for all closed curves \(C\) to which Green's Theorem applies. (The converse is also true: If the line integral is always zero, then \(f\) satisfies the Laplace equation.)

Centroid Find the centroid of the portion of the sphere \(x^{2}+y^{2}+z^{2}=a^{2}\) that lies in the first octant.

Find the flow of the velocity field \(\mathbf{F}=\frac{x}{y+1} \mathbf{i}+\frac{y}{x+1} \mathbf{j}\) where velocity is measured in meters per second, over the curve \(\mathbf{r}(t)=t^{2} \mathbf{i}+t \mathbf{j}, 0 \leq t \leq 1.\)

In Exercises \(43 - 46 ,\) use a CAS to perform the following steps to evaluate the line integrals. $$ \begin{array} { l } { \text { a. Find } d s = | \mathbf { v } ( t ) | d t \text { for the path } \mathbf { r } ( t ) = g ( t ) \mathbf { i } + h ( t ) \mathbf { j } + k ( t ) \mathbf { k } \text { . } } \\ { \text { b. Express the integrand } f ( g ( t ) , h ( t ) , k ( t ) ) | \mathbf { v } ( t ) | \text { as a function of the parameter } t . } \\ { \text { c. Evaluate } \int _ { C } f d s \text { using Equation } ( 2 ) \text { in the text. } } \end{array} $$ $$ \begin{array} { l } { f ( x , y , z ) = \sqrt { 1 + 30 x ^ { 2 } + 10 y } ; \quad \mathbf { r } ( t ) = t \mathbf { i } + t ^ { 2 } \mathbf { j } + 3 t ^ { 2 } \mathbf { k } } \\ { 0 \leq t \leq 2 } \end{array} $$