Chapter 14

Compute the work done by the force field \(\mathbf{F}\) along the curve \(C.\) \(\mathbf{F}(x, y, z)=\langle x y, 3 z, 1\rangle . C\) is the helix \(x=\cos t, y=\sin t\) \(z=2 t\) from (1,0,0) to \((0,1, \pi)\)

Use Gauss' Law \(\nabla \cdot \mathbf{E}=\frac{\rho}{\epsilon_{0}}\) for an electric field E, charge density \(\rho\) and permittivity \(\epsilon_{0}\) The integral form of Gauss' Law is \(\iint_{S} \mathbf{E} \cdot \mathbf{n} d S=\frac{q}{\epsilon_{0}},\) where \(\mathbf{E}\) is an electric field, \(q\) is the total charge enclosed by \(S\) and \(\epsilon_{0}\) is the permittivity constant. Use equation (7.1) to derive the differential form of Gauss' Law: \(\nabla \cdot \mathbf{E}=\frac{\rho}{\epsilon_{0}},\) where \(\rho\) is the charge density.

Compute the work done by the force field \(\mathbf{F}\) along the curve \(C.\) \(\mathbf{F}(x, y, z)=\left\langle z, 0,3 x^{2}\right\rangle, C\) is the quarter-ellipse \(x=2 \cos t\) \(y=3 \sin t, z=1\) from (2,0,1) to (0,3,1)

Evaluate the flux integral \(\iint_{S} \mathbf{F} \cdot \mathbf{n} d S\) \(\mathbf{F}=\langle x, y, z\rangle, S\) is the portion of \(z=4-x^{2}-y^{2}\) above the \(x y\) -plane ( \(\mathbf{n}\) upward)

Find equations for the flow lines. $$\left\langle 2 y, 3 x^{2}\right\rangle$$

If \(\mathbf{F}\) and \(\mathbf{G}\) are vector fields, prove that $$ \nabla \cdot(\mathbf{F} \times \mathbf{G})=\mathbf{G} \cdot(\nabla \times \mathbf{F})-\mathbf{F} \cdot(\nabla \times \mathbf{G}) $$

Where is \(\mathbf{F}(x, y)=\left\langle\frac{2 x}{x^{2}+y^{2}}, \frac{2 y}{x^{2}+y^{2}}\right\rangle\) defined? Show that \(M_{y}=N_{x}\) everywhere the partial derivatives are defined. If \(C\) is a simple closed curve enclosing the origin, does Green's Theorem guarantee that \(\oint_{C} \mathbf{F} \cdot d \mathbf{r}=0 ?\) Explain.

Evaluate the flux integral \(\iint_{S} \mathbf{F} \cdot \mathbf{n} d S\) \(\mathbf{F}=\langle y,-x, 1\rangle, S\) is the portion of \(z=x^{2}+y^{2}\) below \(z=4\) (n downward)

If \(\mathbf{F}\) is a vector field, prove that \(\nabla \cdot(\nabla \times \mathbf{F})=0\).

Show that the line integral is not independent of path by finding two paths that give different values of the integral. \(\int_{C} 2 d x+x d y,\) where \(C\) goes from \((1,4)\) to \((2,-2)\)

Evaluate the flux integral \(\iint_{S} \mathbf{F} \cdot \mathbf{n} d S\) \(\mathbf{F}=(y,-x, z), S\) is the portion of \(z=\sqrt{x^{2}+y^{2}}\) below \(z=3\) (n downward)

If \(\mathbf{F}\) is a vector field, prove that $$ \nabla \times(\nabla \times \mathbf{F})=\nabla(\nabla \cdot \mathbf{F})-\nabla^{2} \mathbf{F} $$

If \(\mathbf{F}(x, y)=\left\langle\frac{2 x}{x^{2}+y^{2}}, \frac{2 y}{x^{2}+y^{2}}\right\rangle\) and \(C\) is a simple closed curve in the fourth quadrant, does Green's Theorem guarantee that \(\oint_{C} \mathbf{F} \cdot d \mathbf{r}=0 ?\) Explain.

Evaluate the flux integral \(\iint_{S} \mathbf{F} \cdot \mathbf{n} d S\) \begin{aligned} &\mathbf{F}=\langle 0,1, y\rangle, S \text { is the portion of } z=-\sqrt{x^{2}+y^{2}} \text { inside }\\\ &x^{2}+y^{2}=4 \text { (n upward) } \end{aligned}

Find equations for the flow lines. $$e^{-x} \mathbf{i}+2 x \mathbf{j}$$

If \(A\) is a constant vector and \(r=\langle x, y, z\rangle,\) prove that $$\nabla \times(\mathbf{A} \times \mathbf{r})=2 \mathbf{A}$$

Show that the line integral is not independent of path by finding two paths that give different values of the integral. \(\int_{C} y^{2} d x+x^{2} d y,\) where \(C\) goes from \((0,0)\) to \((1,1)\)

Evaluate the flux integral \(\iint_{S} \mathbf{F} \cdot \mathbf{n} d S\) \(\mathbf{F}=\left\langle x y, y^{2}, z\right\rangle, S\) is the boundary of the unit cube with \(0 \leq x \leq 1,0 \leq y \leq 1,0 \leq z \leq 1\) (n outward)

Find equations for the flow lines. $$\left\langle y, y^{2}+1\right\rangle$$

Evaluate the flux integral \(\iint_{S} \mathbf{F} \cdot \mathbf{n} d S\) \(\mathbf{F}=\langle y, z, 0\rangle, S\) is the boundary of the box with \(0 \leq x \leq 2\) \(0 \leq y \leq 3,0 \leq z \leq 1\) (n outward)

Find equations for the flow lines. $$\left\langle 2, y^{2}+1\right\rangle$$

Label each statement as True or False and briefly explain. If \(\int_{C} \mathbf{F} \cdot d \mathbf{r}\) is independent of path, then \(\mathbf{F}\) is conservative.

Evaluate the flux integral \(\iint_{S} \mathbf{F} \cdot \mathbf{n} d S\) \(\mathbf{F}=(1,0, z), S\) is the boundary of the region bounded above by \(z=4-x^{2}-y^{2}\) and below by \(z=1\) (n outward)

Prove Green's first identity: For \(C=\partial R\) $$\iint_{R} f \nabla^{2} g d A=\int_{C} f(\nabla g) \cdot \mathbf{n} d s-\iint_{R}(\nabla f \cdot \nabla g) d A$$ [Hint: Use the vector form of Green's Theorem in (5.3) applied to \(\mathbf{F}=f \nabla g .]\)

Label each statement as True or False and briefly explain. If \(\mathbf{F}\) is conservative, then \(\int_{\mathcal{C}} \mathbf{F} \cdot d \mathbf{r}=0\) for any closed curve \(C.\)

Use the formulas \(m=\int_{C} \rho d s, \bar{x}=\frac{1}{m} \int_{C} x \rho d s\) \(\bar{y}=\frac{1}{m} \int_{c} y \rho d s, I=\int_{C} w^{2} \rho d s.\) Compute the mass \(m\) of a rod with density \(\rho(x, y)=x\) in the shape of \(y=x^{2}, 0 \leq x \leq 3.\)

Evaluate the flux integral \(\iint_{S} \mathbf{F} \cdot \mathbf{n} d S\) \(\mathbf{F}=\langle x, y, z\rangle, S\) is the boundary of the region between \(z=0\) and \(z=-\sqrt{4-x^{2}-y^{2}}\)

Label each statement as True or False and briefly explain. If \(\mathbf{F}\) is conservative, then \(\int_{C} \mathbf{F} \cdot d \mathbf{r}\) is independent of path.

Use the formulas \(m=\int_{C} \rho d s, \bar{x}=\frac{1}{m} \int_{C} x \rho d s\) \(\bar{y}=\frac{1}{m} \int_{c} y \rho d s, I=\int_{C} w^{2} \rho d s.\) Compute the mass \(m\) of a rod with density \(\rho(x, y)=y\) in the shape of \(y=4-x^{2}, 0 \leq x \leq 2.\)

Evaluate the flux integral \(\iint_{S} \mathbf{F} \cdot \mathbf{n} d S$$\mathbf{F}=\langle y x, 1, x\rangle, S\) is the portion of \(z=2-x-y\) above the square \(0 \leq x \leq 1,0 \leq y \leq 1\) (n upward)

Use the notation \(r=\langle x, y\rangle\) and \(r=\|\mathbf{r}\|=\sqrt{x^{2}+y^{2}}\) $$\text { Show that } \nabla(r)=\frac{\mathbf{r}}{r}$$

For a vector field \(\mathbf{F}(x, y)=\left\langle F_{1}(x, y), F_{2}(x, y)\right\rangle\) and closed curve \(C\) with normal vector \(\mathbf{n}\) (that is, \(\mathbf{n}\) is perpendicular to the tangent vector to \(C\) at each point), show that \(\oint_{C} \mathbf{F} \cdot \mathbf{n} d s=\iint_{R} \nabla \cdot \mathbf{F} d A=f_{C} F_{1} d y-F_{2} d x\).

Let \(\mathbf{F}(x, y)=\frac{1}{x^{2}+y^{2}}\langle-y, x\rangle .\) Find a potential function \(f\) for F and carefully note any restrictions on the domain of \(f\). Let \(C\) be the unit circle and show that \(\int_{\mathcal{C}} \mathbf{F} \cdot d \mathbf{r}=2 \pi .\) Explain why the Fundamental Theorem for Line Integrals does not apply to this calculation. Quickly explain how to compute \(\int_{C} \mathbf{F} \cdot d \mathbf{r}\) over the circle \((x-2)^{2}+(y-3)^{2}=1.\)

Evaluate the flux integral \(\iint_{S} \mathbf{F} \cdot \mathbf{n} d S\) \(\mathbf{F}=\langle y, 3, z\rangle, S\) is the portion of \(z=x^{2}+y^{2}\) above the triangle with vertices (0,0),(0,1),(1,1) (n downward)

Use the notation \(r=\langle x, y\rangle\) and \(r=\|\mathbf{r}\|=\sqrt{x^{2}+y^{2}}\) $$\text { Show that } \nabla\left(r^{2}\right)=2 \mathbf{r}$$

If \(T(x, y, t)\) is the temperature function at position \((x, y)\) at time \(t,\) heat flows across a curve \(C\) at a rate given by \(\oint_{C}(-k \nabla T) \cdot \mathbf{n} d s,\) for some constant \(k .\) At steady-state, this rate is zero and the temperature function can be written as \(T(x, y)\) In this case, use Green's Theorem to show that \(\nabla^{2} T=0\).

Evaluate the flux integral \(\iint_{S} \mathbf{F} \cdot \mathbf{n} d S\) \(\mathbf{F}=\langle y, 0,2\rangle, S\) is the boundary of the region bounded above by \(z=\sqrt{8-x^{2}-y^{2}}\) and below by \(z=\sqrt{x^{2}+y^{2}}\) (n outward)

Use the notation \(r=\langle x, y\rangle\) and \(r=\|\mathbf{r}\|=\sqrt{x^{2}+y^{2}}\) $$\text { Find } \nabla\left(r^{3}\right)$$

If \(f\) is a scalar function and \(\mathbf{F}\) a vector field, show that $$\nabla \cdot(f \mathbf{F})=\nabla f \cdot \mathbf{F}+f(\nabla \cdot \mathbf{F})$$

Determine whether or not each region is simply-connected. (a) \(\left\\{(x, y): x^{2}+y^{2}<2\right\\}\) (b) \(\left\\{(x, y): 1

Evaluate the flux integral \(\iint_{S} \mathbf{F} \cdot \mathbf{n} d S\) \(\mathbf{F}=\langle 3, z, y\rangle, S\) is the boundary of the region between \(z=8-2 x-y\) and \(z=\sqrt{x^{2}+y^{2}}\) and inside \(x^{2}+y^{2}=1\) (n outward)

If \(f\) is a scalar function and \(\mathbf{F}\) a vector field, show that $$\nabla \times(f \mathbf{F})=\nabla f \times \mathbf{F}+f(\nabla \times \mathbf{F})$$

Determine whether or not each region is simply-connected. (a) \(\\{(x, y): 1

Find the mass and center of mass of the region. The portion of the plane \(3 x+2 y+z=6\) inside the cylinder \(x^{2}+y^{2}=4, \rho(x, y, z)=x^{2}+1\)

Use the notation \(r=\langle x, y\rangle\) and \(r=\|\mathbf{r}\|=\sqrt{x^{2}+y^{2}}\) Show that \(\frac{\langle 1,1\rangle}{r}\) is not conservative.

The Coulomb force for a unit charge at the origin and charge \(q\) at point \(P_{1}=\left(x_{1}, y_{1}, z_{1}\right)\) is \(\mathbf{F}=\frac{k q}{r^{2}} \hat{\mathbf{r}},\) where \(r=\sqrt{x^{2}+y^{2}+z^{2}}\) and \(\hat{\mathrm{r}}=\frac{\langle x, y, z\rangle}{r} .\) Show that the work done by \(\mathbf{F}\) to move the charge \(q\) from \(P_{1}\) to \(P_{2}=\left(x_{2}, y_{2}, z_{2}\right)\) is equal to \(\frac{k q}{r_{1}}-\frac{k q}{r_{2}}\) where \(r_{1}=\sqrt{x_{1}^{2}+y_{1}^{2}+z_{1}^{2}}\) and \(r_{2}=\sqrt{x_{2}^{2}+y_{2}^{2}+z_{2}^{2}}.\)

Find the mass and center of mass of the region. The portion of the plane \(x+2 y+z=4\) above the region bounded by \(y=x^{2}\) and \(y=1, \rho(x, y, z)=y\)

Use the notation \(r=\langle x, y\rangle\) and \(r=\|\mathbf{r}\|=\sqrt{x^{2}+y^{2}}\) Show that \(\frac{\langle-y, x\rangle}{r^{2}}\) is conservative on the domain \(y>0\) by finding a potential function. Show that the potential function can be thought of as the polar angle \(\theta\)

Find the mass and center of mass of the region. The hemisphere \(z=\sqrt{1-x^{2}-y^{2}}, \rho(x, y, z)=1+x\)

If \(f\) is a scalar function, \(\mathbf{r}=\langle x, y\rangle\) and \(r=\|\mathbf{r}\|,\) show that $$\nabla f(r)=f^{\prime}(r) \frac{\mathbf{r}}{r}$$