Math 214 - Multivariable Calculus - 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}]

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 $\bar z$.
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.

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.
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.

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 $\vec F$?
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.

Big View
Prep Day 36 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}] Consider the solid domain $D$ in space which is above the cone $z=\sqrt{x^2+y^2}$ and below the paraboloid $z=6-x^2-y^2$. Sketch the region by hand. Explain why an equation of the cone in cylindrical coordinates is $z=r$. Then obtain an equation of the paraboloid in cylindrical coordinates. 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. 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 $\bar z$. 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 $x^2+y^2+z^2=25$. Provide a sketch of the domain $D$. Explain why an equation of the cone in spherical coordinates is $\phi = \pi/4$. Then given an equation of the sphere in spherical coordinates. 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. 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. 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. 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 $\vec F_1(x,y,z)$. 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. 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. 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. The gravitational vector field is directly related to the radial field $\ds\vec F(x,y,z) = \frac{\left(-x,-y,-z\right)}{(x^2+y^2+z^2)^{3/2}}$. 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 $\vec F$. 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. Suppose $f(x,y,z)$ is twice continuously differentiable. Compute the curl of the gradient of $f$, so compute $\vec \nabla \times \vec \nabla f$. Simplify the result as much as possible. If a vector field $\vec F = (M,N,P)$ has a potential, then what is the curl of $\vec F$? 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. 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. 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. 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Day 36

Task 36.1

Task 36.2

Task 36.3

Task 36.4