Day 36 - Prep

Task 36.1

This task has you practice using spherical and cylindrical coordinates. Mathematica's ParametricRegion[] command allows us to plot a region, using any coordinate system, which is extremely useful for visualizing regions defined by bounds of an integral. As an example, the code below visualizes the region whose volume is given by the cylindrical coordinate iterated triple integral $$\int_{1}^{3}\int_{\pi/2}^{2\pi}\int_{0}^{r}rdz d\theta dr,$$ and then computes the triple integral.

coordinates = {r Cos[theta], r Sin[theta], z}
R = ParametricRegion[coordinates, {{r, 1, 3}, {theta, Pi/2, 2 Pi}, {z, 0, r}}];
Region[R, Axes -> True, AxesLabel -> {x, y, z}, AxesOrigin -> {0, 0, 0}]
Integrate[r, {r, 1, 3}, {theta, Pi/2, 2 Pi}, {z, 0, r}]

To compute integrals in spherical coordinates, we just update the change-of-coordinates and bounds. Here's code to plot the region whose volume is given by the spherical coordinate iterated triple integral $$\ds \int_{0}^{\pi}\int_{\pi/6}^{\pi/3}\int_{1}^{3}\rho^2\sin\phi d\rho d\phi d\theta.$$

coordinates = {rho Sin[phi] Cos[theta], rho Sin[phi] Sin[theta], rho Cos[phi]}
R = ParametricRegion[coordinates, {{theta, 0, Pi/2}, {phi, Pi/6, Pi/3}, {rho, 1, 3}}];
Region[R, Axes -> True, AxesLabel -> {x, y, z}, AxesOrigin -> {0, 0, 0}]
Integrate[rho^2 Sin[phi], {theta, 0, Pi/2}, {phi, Pi/6, Pi/3}, {rho, 1, 3}]

1. Consider the solid domain $D$ in space which is above the cone $z=\sqrt{x^2+y^2}$ and below the paraboloid $z=6-x^2-y^2$.
   1. Sketch the region by hand.
   2. Explain why an equation of the cone in cylindrical coordinates is $z=r$. Then obtain an equation of the paraboloid in cylindrical coordinates.
   3. Use cylindrical coordinates to set up an iterated triple integral that would give the volume of the region. You'll need to find where the surfaces intersect, as their intersection will help you determine the appropriate bounds. Use the ParametricRegion[] command to verify that the bounds you gave do indeed produce the correct region.
   4. By symmetry, it should be clear that for the centroid of this region, we have $\bar x = \bar y = 0$. Set up a formula involving iterated triple integrals that would give $\bar z$ for this solid, and then use software to compute $\bar z$.
2. Consider the solid domain $D$ in space that lies below the cone $z=\sqrt{x^2+y^2}$, above the $xy$-plane, and inside the sphere $x^2+y^2+z^2=25$.
   1. Provide a sketch of the domain $D$.
   2. Explain why an equation of the cone in spherical coordinates is $\phi = \pi/4$. Then given an equation of the sphere in spherical coordinates.
   3. Set up an integral in spherical coordinates that gives the volume of $D$. Use the ParametricRegion[] command to verify that the bounds you gave do indeed produce the correct region.
   4. Set up an integral in spherical coordinates that would give the $z$-coordinate of the centroid of $D$.

Task 36.2

Two important vector fields show up over and over again when studying gravity and electrostatics. In this task we will develop a common formula for these fields, show that these fields have a potential, and then practice using the fundamental theorem of line integrals to perform work computations, using the potential.

1. We need a formula for a vector field where at each point in space, the vector points towards the origin with a magnitude that proportional to 1 over the square of the distance to the origin. To obtain this field, complete the following steps.
   1. Let $P=(x,y,z)$ be a point in space. At the point $P$, let $\vec F_1(x,y,z)$ be the vector which points from $P$ to the origin. Give a formula for $\vec F_1(x,y,z)$.
   2. Give an equation of the vector field where at each point $P$ in space, the vector $\vec F_2(P)$ is a unit vector that points towards the origin.
   3. Give an equation of the vector field where at each point $P$ in space, the vector $\vec F_3(P)$ is a vector of length 7 that points towards the origin.
   4. Give an equation of the vector field where at each point $P$ in space, the vector $\vec F(P)$ points towards the origin, and has a magnitude equal to $G/d^2$ where $d = \sqrt{x^2+y^2+z^2}$ is the distance to the origin, and $G$ is a constant.
2. The gravitational vector field is directly related to the radial field $\ds\vec F(x,y,z) = \frac{\left(-x,-y,-z\right)}{(x^2+y^2+z^2)^{3/2}}$.
   1. We say that a vector field is conservative when the work done by the field is independent of the path traveled. Show that this vector field is conservative, by finding a potential for $\vec F$.
   2. Compute the work done by $\vec F$ to move an object from $(1,2,-2)$ to $(0,-3,4)$ along ANY path that avoids the origin.

Task 36.3

We need to gain some familiarity with the notation related to gradients, divergence, and curl. As you work on the tasks below, you are welcome to use subscript notation (such as $f_x$ and $M_y$) to simplify writing.

1. Suppose $f(x,y,z)$ is twice continuously differentiable.
   1. Compute the curl of the gradient of $f$, so compute $\vec \nabla \times \vec \nabla f$. Simplify the result as much as possible.
   2. If a vector field $\vec F = (M,N,P)$ has a potential, then what is the curl of $\vec F$?
2. Suppose $\vec F(x,y,z) = (M,N,P)$ is a vector field and $f(x,y,z)$ is a function, both of which are twice continuously differentiable.
   1. Compute the divergence of the curl of $\vec F$, so compute $\vec \nabla \cdot \left(\vec \nabla \times \vec F\right)$, and simplify the result as much as possible.
   2. Compute the divergence of the gradient of $f$, so compute $\vec \nabla \cdot \vec \nabla f$, and simplify the result as much as possible.

Task 36.4

Pick some problems related to the topics we are discussing from the Text Book Practice page.

Day 37 - Prep

Task 37.1

1. Consider the region that lies below the $z$-axis and between 2 spheres of radii $a$ and $b$ with $a<b$. The image below shows such a region where the inner radius is $a=3$, and the outer radius is $b=5$.
   • 
   1. Set up an iterated triple integral in spherical coordinates to compute the volume of this region.
   2. Set up an iterated triple integral formula to compute the $z$-coordinate of the centroid (symmetry gives us $\bar x = \bar y = 0$).
   3. If the temperature at points in this region is given by $T(x,y,z) = x+3$, then set up an iterated triple integral formula that would give the average temperature of the region.
2. A metal casing lies inside the cylinder $x^2+y^2=4$, outside the cylinder $x^2+y^2=1$, below the paraboloid $z=9-x^2-y^2$, and above the plane $z=0$. The region is shown below, where one quarter of the region was removed so you can see the hollow interior.
   • 
   1. Set up an iterated integral in cylindrical coordinates to compute the volume of the casing.
   2. The casing is made of a composite material and the density of the casing is more dense the further from the center. The density is given by $\delta(x,y,z) = x^2+y^2$. Set up an iterated triple integral formula to compute the $z$-coordintate of the center-of-mass of the casing.

Remember that you can verify that your bound are correct by using Mathematica to draw whatever you decide the bounds should be. Here's two examples of how to construct regions, the first in cylindrincal coordinates, and the second in spherical coordinates.

coordinates = {r Cos[theta], r Sin[theta], z}
R = ParametricRegion[coordinates, {{r, 1, 3}, {theta, Pi/2, 2 Pi}, {z, 0, r}}];
Region[R, Axes -> True, AxesLabel -> {x, y, z}, AxesOrigin -> {0, 0, 0}]
Integrate[r, {r, 1, 3}, {theta, Pi/2, 2 Pi}, {z, 0, r}]

coordinates = {rho Sin[phi] Cos[theta], rho Sin[phi] Sin[theta], rho Cos[phi]}
R = ParametricRegion[coordinates, {{theta, 0, Pi/2}, {phi, Pi/6, Pi/3}, {rho, 1, 3}}];
Region[R, Axes -> True, AxesLabel -> {x, y, z}, AxesOrigin -> {0, 0, 0}]
Integrate[rho^2 Sin[phi], {theta, 0, Pi/2}, {phi, Pi/6, Pi/3}, {rho, 1, 3}]

If the code above is extremely slow on your computer, then you can use the code below instead. The code is more complicated, as it plots the six surfaces defined by the bounds you choose, but the code used requires very minimal processing (it works fast). (The Evaluate[] command is needed for the code to work prior to Mathematica 13.)

plotRegion3D[cs_, ob_, mb_, ib_] := 
 Show[{{ParametricPlot3D[Evaluate[Table[cs /. (ib[[1]] -> ib[[i]]), {i, 2, 3}], ob, mb], AxesLabel -> {x, y, z}, Mesh -> {15, 1}], 
    ParametricPlot3D[Evaluate[Table[(cs /. (ib[[1]] -> (ib[[2]] (1 - s) + ib[[3]] s))) /. (mb[[1]] -> mb[[i]]), {i, 2, 3}], ob], {s, 0, 1}, PlotStyle -> {Red, Blue}, Mesh -> {15, 0}], 
    ParametricPlot3D[Evaluate[Table[(cs /. (ib[[1]] -> (ib[[2]] (1 - s) + ib[[3]] s))) /. (mb[[1]] -> (mb[[2]] (1 - t) + mb[[3]] t)) /. (ob[[1]] -> ob[[i]]), {i, 2, 3}]], {t, 0, 1}, {s, 0, 1}, PlotStyle -> Green, Mesh -> {0, 0}]}}, PlotRange -> All]

coordinates = {r Cos[theta], r Sin[theta], z}
plotRegion3D[coordinates, {r, 1, 3}, {theta, Pi/2, 2 Pi}, {z, 0, r}]

coordinates = {rho Sin[phi] Cos[theta], rho Sin[phi] Sin[theta], rho Cos[phi]}
plotRegion3D[coordinates, {theta, 0, Pi/2}, {phi, Pi/6, Pi/3}, {rho, 1, 3}]

Much simpler code will draw 2D regions

plotRegion[cs_, ob_, ib_] := ParametricPlot[Evaluate[cs, ob, ib], Mesh -> {10, 0}]

coordinates = {r Cos[theta], r Sin[theta]};
plotRegion[coordinates, {theta, Pi/4, Pi}, {r, 2, 5}]

Task 37.2

1. Consider the region $R$ in space satisfying $0\leq x-y\leq 4$ and $1\leq 2x+y\leq 3$. We wish to evaluate the integral $\ds\iint_R xy dA$.
   1. Draw the region $R$.
   2. Using the change-of-coordinates $u=x-y$ and $v=2x+y$, compute the Jacobian $\frac{\partial(u,v)}{\partial (x,y)}$.
   3. Find $x$ and $y$ in terms of $u$ and $v$.
   4. Use this change-of-coordinates to compute $\ds\iint_R xy dA$ by first setting up an appropriate iterated integral of the form $\ds \int_{?}^{?}\int_{?}^{?}?dudv$.
2. Consider the ellipsoid $\ds\frac{x^2}{a^2}+\frac{y^2}{b^2}+\frac{z^2}{c^2}=1$, for some positive constants $a$, $b$, and $c$.
   1. Draw the region.
   2. Using the change-of-coordinates $x = a u \sin v \cos w$, $y = b u \sin v \sin w$, $x = c u \cos v$, compute the Jacobian $\frac{\partial (x,y,z)}{\partial (u,v,w)}$. Feel free to use software to help you.
   3. Set up an iterated integral using $uvw$-coordinates to compute the volume inside the ellipsoid. Then compute the integral.
   4. Set up an iterated integral using $uvw$-coordinates to show that $\bar z = \frac{3c}{16}$ for the region inside the ellipsoid that is above the $xy$-plane.

Task 37.3

When you can use a potential to compute work, it greatly simplifies things.

1. As you complete each problem below, first ask if there is a potential.
   1. Compute the work done by the vector field $\vec F(x,y) = (y,x)$ on a object that moves along the path $\vec r(t) = (\cos t, \sin t)$ for $0\leq t\leq 2 \pi$.
   2. Compute the work done by the vector field $\vec F(x,y) = (-y,x)$ on a object that moves along the path $\vec r(t) = (\cos t, \sin t)$ for $0\leq t\leq 2 \pi$.
   3. For each vector field above, use Mathematica to construct an image that shows the vector field along with the curve in the same plot. (You'll need VectorPlot[] and ParametricPlot[], with Show[] to get them in the same plot).

We've seen that if a vector field has a potential, then the derivative is symmetric. Is the converse of this statement true, namely if the derivative of a vector field is symmetric, then does that mean the vector field has a potential?

2. Consider the vector field $\ds\vec F(x,y) = \left(\frac{-y}{x^2+y^2},\frac{x}{x^2+y^2}\right)$.
   1. Show that the derivative is symmetric.
   2. Compute the work done by the vector field $\vec F(x,y)$ on a object that moves along the path $\vec r(t) = (\cos t, \sin t)$ for $0\leq t\leq 2 \pi$.
   3. Explain why $\vec F(x,y)$ does not have a potential.
   4. Look up "simply connected region" (see section 6.3), and explain why the domain of $\ds\vec F(x,y) = \left(\frac{-y}{x^2+y^2},\frac{x}{x^2+y^2}\right)$ is NOT simply connected.

When the domain of a continuously differentiable vector field is simply connected, then the vector field has a potential if and only if the derivative is symmetric. The concept of a simply connected domain is the start of an entire branch of mathematics called algebraic topology, all stemming from the question, "under what circumstances can we guarantee that a vector field will have a potential?"

Task 37.4

Pick some problems related to the topics we are discussing from the Text Book Practice page.