Chapter 15

Find the mass of the lamina that is the portion of the surface \(y^{2}=4-z\) between the planes \(x=0, x=3, y=0,\) and \(y=3\) if the density is \(\delta(x, y, z)=y\).

Use Green’s Theorem to find the work done by the force field F on a particle that moves along the stated path. \(\mathbf{F}(x, y)=x y \mathbf{i}+\left(\frac{1}{2} x^{2}+x y\right) \mathbf{j} ;\) the particle starts at \((5,0)\) traverses the upper semicircle \(x^{2}+y^{2}=25,\) and returns to its starting point along the \(x\) -axis.

Suppose that \(C\) is a circle in the domain of a conservative vector field in the \(x y-\) plane whose component functions are continuous. Explain why there must be at least two points on \(C\) at which the vector field is normal to the circle.

Let $$ \mathbf{F}(x, y, z)=a^{2} x \mathbf{i}+(y / a) \mathbf{j}+a z^{2} \mathbf{k} $$ and let \(\sigma\) be the sphere of radius 1 centered at the origin and oriented outward. Use a CAS to find all values of \(a\) such that the flux of \(\mathbf{F}\) across \(\sigma\) is \(3 \pi .\)

Find the mass of the lamina that is the portion of the cone \(z=\sqrt{x^{2}+y^{2}}\) between \(z=1\) and \(z=4\) if the density is \(\delta(x, y, z)=x^{2} z\)

Use Green’s Theorem to find the work done by the force field F on a particle that moves along the stated path. \(\mathbf{F}(x, y)=\sqrt{y} \mathbf{i}+\sqrt{x} \mathbf{j} ;\) the particle moves counterclockwise one time around the closed curve given by the equations \(y=0, x=2,\) and \(y=x^{3} / 4\)

Prove the identity, assuming that F, ?, and G satisfy the hypotheses of the Divergence Theorem and that all necessary differentiability requirements for the functions f(x, y, z) and g(x, y, z) are met. $$ \begin{array}{l}{\iint_{\sigma}(f \nabla g-g \nabla f) \cdot \mathbf{n} d S=\iiint_{G}\left(f \nabla^{2} g-g \nabla^{2} f\right) d V} \\ {[\text {Hint: Interchange } f \text { and } g \text { in } 29 .]}\end{array} $$

If a curved lamina has constant density \(\delta_{0},\) what relationship must exist between its mass and surface area? Explain your reasoning.

Prove: If $$ \mathbf{F}(x, y, z)=f(x, y, z) \mathbf{i}+g(x, y, z) \mathbf{j}+h(x, y, z) \mathbf{k} $$ is a conservative field and \(f, g,\) and \(h\) are continuous and have continuous first partial derivatives in a region, then $$ \frac{\partial f}{\partial y}=\frac{\partial g}{\partial x}, \quad \frac{\partial f}{\partial z}=\frac{\partial h}{\partial x}, \quad \frac{\partial g}{\partial z}=\frac{\partial h}{\partial y} $$ in the region.

Evaluate \(\oint_{C} y d x-x d y,\) where \(C\) is the cardioid $$ r=a(1+\cos \theta) \quad(0 \leq \theta \leq 2 \pi) $$

Let \(k\) be a constant, \(\mathbf{F}=\mathbf{F}(x, y, z), \mathbf{G}=\mathbf{G}(x, y, z),\) and \(\phi=\phi(x, y, z) .\) Prove the following identities, assuming that all derivatives involved exist and are continuous. $$ \operatorname{div}(k \mathbf{F})=k \operatorname{div} \mathbf{F} $$

Let \(k\) be a constant, \(\mathbf{F}=\mathbf{F}(x, y, z), \mathbf{G}=\mathbf{G}(x, y, z),\) and \(\phi=\phi(x, y, z) .\) Prove the following identities, assuming that all derivatives involved exist and are continuous. $$ \operatorname{curl}(k \mathbf{F})=k \operatorname{curl} \mathbf{F} $$

Writing Write a paragraph explaining the concept of flux to someone unfamiliar with its meaning.

The centroid of a surface \(\sigma\) is defined by $$ \quad \bar{x}=\frac{\iint_{\sigma} x d S}{\operatorname{arcaof} \sigma}, \quad \bar{y}=\frac{\iint_{\sigma} y d S}{\operatorname{arcaof} \sigma}, \quad \bar{z}=\frac{\iint_{\sigma} z d S}{\operatorname{arcaof} \sigma} $$ Find the centroid of the surface. The portion of the paraboloid \(z=\frac{1}{2}\left(x^{2}+y^{2}\right)\) below the plane \(z=4\)

Find a nonzero function \(h\) for which $$ \begin{aligned} \mathbf{F}(x, y)=h(x)[x \sin y+y \cos y] \mathbf{i} \\\\+h(x)[x \cos y-y \sin y] \mathbf{j} \end{aligned} $$ is conservative.

Determine whether the vector field F(x, y, z) is free of sources and sinks. If it is not, locate them. $$ \mathbf{F}(x, y, z)=(y+z) \mathbf{i}-x z^{3} \mathbf{j}+\left(x^{2} \sin y\right) \mathbf{k} $$

Let \(k\) be a constant, \(\mathbf{F}=\mathbf{F}(x, y, z), \mathbf{G}=\mathbf{G}(x, y, z),\) and \(\phi=\phi(x, y, z) .\) Prove the following identities, assuming that all derivatives involved exist and are continuous. $$ \operatorname{div}(\mathbf{F}+\mathbf{G})=\operatorname{div} \mathbf{F}+\operatorname{div} \mathbf{G} $$

Evaluate \(\int_{C} \mathbf{F} \cdot d \mathbf{r}\) along the curve \(C\) $$ \begin{array}{l}{\mathbf{F}(x, y)=x^{2} \mathbf{i}+x y \mathbf{j}} \\ {C: \mathbf{r}(t)=2 \cos t \mathbf{i}+2 \sin t \mathbf{j} \quad(0 \leq t \leq \pi)}\end{array} $$

The centroid of a surface \(\sigma\) is defined by $$ \quad \bar{x}=\frac{\iint_{\sigma} x d S}{\operatorname{arcaof} \sigma}, \quad \bar{y}=\frac{\iint_{\sigma} y d S}{\operatorname{arcaof} \sigma}, \quad \bar{z}=\frac{\iint_{\sigma} z d S}{\operatorname{arcaof} \sigma} $$ Find the centroid of the surface. The portion of the sphere \(x^{2}+y^{2}+z^{2}=4\) above the plane \(z=1\)

Determine whether the vector field F(x, y, z) is free of sources and sinks. If it is not, locate them. $$ \mathbf{F}(x, y, z)=x y \mathbf{i}-x y \mathbf{j}+y^{2} \mathbf{k} $$

Let \(k\) be a constant, \(\mathbf{F}=\mathbf{F}(x, y, z), \mathbf{G}=\mathbf{G}(x, y, z),\) and \(\phi=\phi(x, y, z) .\) Prove the following identities, assuming that all derivatives involved exist and are continuous. $$ \operatorname{curl}(\mathbf{F}+\mathbf{G})=\operatorname{curl} \mathbf{F}+\operatorname{curl} \mathbf{G} $$

Evaluate \(\int_{C} \mathbf{F} \cdot d \mathbf{r}\) along the curve \(C\) $$ \begin{array}{l}{\mathbf{F}(x, y)=x^{2} y \mathbf{i}+4 \mathbf{j}} \\ {C: \mathbf{r}(t)=e^{t} \mathbf{i}+e^{-t} \mathbf{j} \quad(0 \leq t \leq 1)}\end{array} $$

Evaluate the integral \(\iint_{\sigma} f(x, y, z) d S\) over the surface \(\sigma\) represented by the vector-valued function \(\mathbf{r}(u, v) .\) $$ \begin{array}{l}{f(x, y, z)=x y z ; \mathbf{r}(u, v)=u \cos v \mathbf{i}+u \sin v \mathbf{j}+3 u \mathbf{k}} \\ {(1 \leq u \leq 2,0 \leq v \leq \pi / 2)}\end{array} $$

Determine whether the vector field F(x, y, z) is free of sources and sinks. If it is not, locate them. $$ \mathbf{F}(x, y, z)=x^{3} \mathbf{i}+y^{3} \mathbf{j}+z^{3} \mathbf{k} $$

Let \(k\) be a constant, \(\mathbf{F}=\mathbf{F}(x, y, z), \mathbf{G}=\mathbf{G}(x, y, z),\) and \(\phi=\phi(x, y, z) .\) Prove the following identities, assuming that all derivatives involved exist and are continuous. $$ \operatorname{div}(\phi \mathbf{F})=\phi \operatorname{div} \mathbf{F}+\nabla \phi \cdot \mathbf{F} $$

Evaluate \(\int_{C} \mathbf{F} \cdot d \mathbf{r}\) along the curve \(C\) $$ \begin{array}{l}{\mathbf{F}(x, y)=\left(x^{2}+y^{2}\right)^{-3 / 2}(x \mathbf{i}+y \mathbf{j})} \\ {C: \mathbf{r}(t)=e^{t} \sin t \mathbf{i}+e^{t} \cos t \mathbf{j} \quad(0 \leq t \leq 1)}\end{array} $$

Evaluate the integral \(\iint_{\sigma} f(x, y, z) d S\) over the surface \(\sigma\) represented by the vector-valued function \(\mathbf{r}(u, v) .\) $$ \begin{array}{l}{f(x, y, z)=\frac{x^{2}+z^{2}}{y} ; \mathbf{r}(u, v)=2 \cos v \mathbf{i}+u \mathbf{j}+2 \sin v \mathbf{k}} \\ {(1 \leq u \leq 3,0 \leq v \leq 2 \pi)}\end{array} $$

Determine whether the vector field F(x, y, z) is free of sources and sinks. If it is not, locate them. $$ \mathbf{F}(x, y, z)=\left(x^{3}-x\right) \mathbf{i}+\left(y^{3}-y\right) \mathbf{j}+\left(z^{3}-z\right) \mathbf{k} $$

Let \(k\) be a constant, \(\mathbf{F}=\mathbf{F}(x, y, z), \mathbf{G}=\mathbf{G}(x, y, z),\) and \(\phi=\phi(x, y, z) .\) Prove the following identities, assuming that all derivatives involved exist and are continuous. $$ \operatorname{curl}(\phi \mathbf{F})=\phi \operatorname{curl} \mathbf{F}+\nabla \phi \times \mathbf{F} $$

Evaluate \(\int_{C} \mathbf{F} \cdot d \mathbf{r}\) along the curve \(C\) $$ \begin{array}{l}{\mathbf{F}(x, y, z)=z \mathbf{i}+x \mathbf{j}+y \mathbf{k}} \\\ {C: \mathbf{r}(t)=\sin t \mathbf{i}+3 \sin t \mathbf{j}+\sin ^{2} t \mathbf{k} \quad(0 \leq t \leq \pi / 2)}\end{array} $$

Evaluate the integral \(\iint_{\sigma} f(x, y, z) d S\) over the surface \(\sigma\) represented by the vector-valued function \(\mathbf{r}(u, v) .\) $$ \begin{array}{l}{f(x, y, z)=\frac{1}{\sqrt{1+4 x^{2}+4 y^{2}}}} \\\ {\mathbf{r}(u, v)=u \cos v \mathbf{i}+u \sin v \mathbf{j}+u^{2} \mathbf{k}} \\\ {(0 \leq u \leq \sin v, 0 \leq v \leq \pi)}\end{array} $$

Let \(\sigma\) be the surface of the solid \(G\) that is enclosed by the paraboloid \(z=1-x^{2}-y^{2}\) and the plane \(z=0 .\) Use a CAS to verify Formula ( 1) in the Divergence Theorem for the vector field $$ \mathbf{F}=\left(x^{2} y-z^{2}\right) \mathbf{i}+\left(y^{3}-x\right) \mathbf{j}+(2 x+3 z-1) \mathbf{k} $$ by evaluating the surface integral and the triple integral.

Let \(k\) be a constant, \(\mathbf{F}=\mathbf{F}(x, y, z), \mathbf{G}=\mathbf{G}(x, y, z),\) and \(\phi=\phi(x, y, z) .\) Prove the following identities, assuming that all derivatives involved exist and are continuous. $$ \operatorname{div}(\operatorname{curl} \mathbf{F})=0 $$

Find the mass of a thin wire shaped in the form of the cir- cular are \(y=\sqrt{9}-x^{2}(0 \leq x \leq 3)\) if the density function is \(\delta(x, y)=x \sqrt{y}\)

Evaluate the integral \(\iint_{\sigma} f(x, y, z) d S\) over the surface \(\sigma\) represented by the vector-valued function \(\mathbf{r}(u, v) .\) $$ \begin{array}{l}{f(x, y, z)=e^{-z}} \\ {\mathbf{r}(u, v)=2 \sin u \cos v \mathbf{i}+2 \sin u \sin v \mathbf{j}+2 \cos u \mathbf{k}} \\ {(0 \leq u \leq \pi / 2,0 \leq v \leq 2 \pi)}\end{array} $$

(a) Let \(C\) be the line segment from a point \((a, b)\) to a point \((c, d) .\) Show that $$ \int_{C}-y d x+x d y=a d-b c $$ (b) Use the result in part (a) to show that the area \(A\) of a triangle with successive vertices \(\left(x_{1}, y_{1}\right),\left(x_{2}, y_{2}\right),\) and \(\left(x_{3}, y_{3}\right)\) going counterclockwise is $$ \begin{aligned} A=\frac{1}{2}\left[\left(x_{1} y_{2}\right.\right.&\left.-x_{2} y_{1}\right) \\ &\left.+\left(x_{2} y_{3}-x_{3} y_{2}\right)+\left(x_{3} y_{1}-x_{1} y_{3}\right)\right] \end{aligned} $$ (c) Find a formula for the area of a polygon with successive vertices \(\left(x_{1}, y_{1}\right),\left(x_{2}, y_{2}\right), \ldots,\left(x_{n}, y_{n}\right)\) going counterclockwise. (d) Use the result in part (c) to find the area of a quadrilateral with vertices \((0,0),(3,4),(-2,2),(-1,0)\)

Writing Discuss what it means to say that the divergence of a vector field is independent of a coordinate system. Explain how we know this to be true.

Let \(k\) be a constant, \(\mathbf{F}=\mathbf{F}(x, y, z), \mathbf{G}=\mathbf{G}(x, y, z),\) and \(\phi=\phi(x, y, z) .\) Prove the following identities, assuming that all derivatives involved exist and are continuous. $$ \operatorname{curl}(\nabla \phi)=0 $$

Find the mass of a thin wire shaped in the form of the curve \(x=e^{t} \cos t, y=e^{t} \sin t(0 \leq t \leq 1)\) if the density function \(\delta\) is proportional to the distance from the origin.

Use a CAS to approximate the mass of the curved lamina \(z=e^{-x^{2}-y^{2}}\) that lies above the region in the \(x y\) -plane enclosed by \(x^{2}+y^{2}=9\) given that the density function is \(\delta(x, y, z)=\sqrt{x^{2}+y^{2}}\).

Evaluate the integral \(\int_{C} \mathbf{F} \cdot d \mathbf{r},\) where \(C\) is the boundary of the region \(R\) and \(C\) is oriented so that the region is on the left when the boundary is traversed in the direction of its orientation. \(\mathbf{F}(x, y)=\left(x^{2}+y\right) \mathbf{i}+(4 x-\cos y) \mathbf{j} ; C\) is the boundary of the region \(R\) that is inside the square with vertices \((0,0)\) \((5,0),(5,5),(0,5)\) but is outside the rectangle with vertices \((1,1),(3,1),(3,2),(1,2)\)

Writing Describe some geometrical and physical applications of the Divergence Theorem.

Find the mass of a thin wire shaped in the form of the helix \(x=3 \cos t, y=3 \sin t, z=4 t(0 \leq t \leq \pi / 2)\) if the density function is \(\delta=k x /\left(1+y^{2}\right)(k>0)\)

Evaluate the integral \(\int_{C} \mathbf{F} \cdot d \mathbf{r},\) where \(C\) is the boundary of the region \(R\) and \(C\) is oriented so that the region is on the left when the boundary is traversed in the direction of its orientation. \(\mathbf{F}(x, y)=\left(e^{-x}+3 y\right) \mathbf{i}+x \mathbf{j} ; C\) is the boundary of the region \(R\) inside the circle \(x^{2}+y^{2}=16\) and outside the circle \(x^{2}-2 x+y^{2}=3\)

Find the mass of a thin wire shaped in the form of the curve \(x=2 t, y=\ln t, z=4 \sqrt{t}(1 \leq t \leq 4)\) if the density function is proportional to the distance above the \(x y\) -plane.

Discuss the similarities and differences between the definition of a surface integral and the definition of a double integral.

Discuss some of the ways that you can show a vector field is not conservative.

Writing Discuss the role of the Fundamental Theorem of Calculus in the proof of Green's Theorem.

Verify that the radius vector r = xi + y j + zk has the stated property. $$ \text { (a) curl } \mathbf{r}=\mathbf{0} \quad \text { (b) } \nabla\|\mathbf{r}\|=\frac{\mathbf{r}^{*}}{\|\mathbf{r}\|} $$

Find the work done by the force field \(\mathbf{F}\) on a particle that moves along the curve \(C\). $$ \begin{array}{l}{\mathbf{F}(x, y)=x y \mathbf{i}+x^{2} \mathbf{j}} \\ {C: x=y^{2} \text { from }(0,0) \text { to }(1,1)}\end{array} $$