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Exercise 1: Volume Expansion Rate and Flux. Maple

This exercise is solved using Maple.

Consider in the $\,(x,y,z)$-space the vector field

$$\mV(x,y,z)=\left(\frac x2\,, \frac y2\,,2z\,\right)\,.$$
A

Determine the flow curve $\,\mr(t)\,$ for $\,\mV\,$ that fulfills the initial condition $\,\mr(0)=(1,1,1)\,$.

The surface $\,\mathcal S_0\,$ consists of the part of the unit sphere with centre at the origin that lies on or above the plane given by the equation $\,\displaystyle{z=\frac 12\,.}$

B

Give a parametric representation for $\,\mathcal S_0\,$, and for the surface $\,\mathcal S_t\,$ that $\,\mathcal S_0\,$ is deformed into, at time $\,t\,$, when it floats with the flow curves of $\,\mV\,$. Plot $\,\mathcal S_0\,$ using Maple together with $\,\mathcal S_t\,$ for selected values of $\,t\,$.

C

Explain that $\,\mathcal S_0\,$ does not have common points with $\,\mathcal S_t\,$ for $\,t>0\,.$

D

Determine a parametric representation for the spatial field $\,\Omega_t\,$ that $\,\mathcal S_t\,$ has passed since it left $\,\mathcal S_0\,$ at time $\,t=0\,,$ and determine the volume Vol$(t)\,$ of $\,\Omega_t\,.$

E

Determine Vol$’(t)\,$ and Vol$’(0)\,,$ and compare the result to the flux of $\,\mV\,$ through $\,\mathcal S_0\,$. Why is there this link?

Exercise 2: Gauss’with Constant and Variable Divergence

A parametrized spatial field $\Omega_{\mathbf r}$ in the $(x,y,z)$-space has the parametric representation

$$\mr(u,v,w)=(u\cos(v),u\sin(v),w)\,,\,\,u\in\left[0,2\right]\,,\,\,v\in\left[0,\frac{\pi}{2}\right]\,,\,\,w\in\left[0,5\right]\,.$$
A

$\Omega_{\mathbf r}$ is a parametrization of a simple geometrical object. Describe which, and find its volume by simple mental computation.

About the vector fields $\mU$ and $\mV$ it is given that

$$\mathrm{Div}(\mU)(x,y,z)=\pi\,\,\,\,\mathrm{and}\,\,\,\,\mathrm{Div}(\mV)(x,y,z)=yz\,.$$
B

Determine the fluxes

$$\,\displaystyle{\int_{\partial \Omega_{\mathbf r}}\mU\cdot \mathbf n_{\partial \Omega_{\mathbf r}}\,\mathrm du}\,\,\,\,\textrm{and}\,\,\,\, \,\displaystyle{\int_{\partial\Omega_{\mathbf r}}\mV\cdot \mathbf n_{\partial \Omega_{\mathbf r}}\,\mathrm du}\,.$$

Exercise 3: Verification of Gauss’ Theorem

In this exercise we shall check Gauss’ Theorem in an example with a flux that is computed by the usual method and as a space integral of a divergence.

Given the vector field

$$\mV(x,y,z)=(-8x,8,4z^3)$$

and a spatial field

$$\Omega=\lbrace (x,y,z)\,\vert\, x^2+y^2+z^2\leq a^2\,\, \mathrm{og}\,\, z\geq 0\rbrace\,,\,a>0\,,$$

whose surface $\,\partial \Omega\,$ has an orientation with an outward pointing unit normal vector field $\,\mathbf n_{\,\partial \Omega}\,.$

A

Determine the space integralet

$$\int_{\Omega}\mathrm{Div(\mV)}\,d\mu\,.$$

B

Determine the orthogonal surface integral

$$\int_{\partial\,\Omega}\,\mV \mathbf{\cdot}\mathbf n_{\,\partial \Omega}\,d\mu\,.$$

C

For which $\,a\,$ is the Flux($\mV,\partial\,\Omega$), with the given unit normal vector field $\,\mathbf n_{\,\partial \Omega}\,$ positive ( ‘‘outflow through $\partial \Omega$ larger than inflow’’).

D

Which characteristic equality is there between Gauss’ Theorem about the relation between the divergence integral and the orthogonal surface integral on the one hand and the identity known from highschool:

$$\left[ F(x)\right] _a^b=\int_a^b F'(x)dx\,?$$

Exercise 4: Gauss’ Theorem Applied on an Open Surface!

Given the vector field

$$\mV(x,y,z)=(\e^y+\cos(yz),\e^z+\sin (xz),x^2z^2),\, (x,y,z)\in\reel^3$$

together with a hemi-spherical surface $F,$ that is given by

$$x^2+y^2+z^2-4z=0\,\,\mathrm{og}\,\,z\leq 2\,.$$
A

Draw a sketch of $F$ using pen and paper.

$F$ is thought to be oriented with a unit normal vector field with negative $z$-coordinate. We wish to determine the flux though $F,$ but it turns out to be rather difficult to integrate over the surface $F\,,$ since the vector field is a bit complicated. On the other hand it is not difficult to find Div$(\mV)(x,y,z)\,,$ therefore we will tune the problem, so it can be solved using Gauss’ Theorem. We start by integrating the divergence of $\mV$ over the solid hemisphere $\Omega$ that fills $F$.

B

Compute the flux of $\mV$ out through the surface $\partial \Omega$ of $\Omega$, by computing the flux as

$$\int_{\Omega}\mathrm{Div} (\mV)\, \mathrm d\mu.$$

But, the hemi-spherical surface is open on the top side, but we have computed the flux through the closed surface!

C

Find a parametric representation of the circular disc that can cover the top side of the hemisphere.

D

Compute the flux through the circular disc.

E

Now find the flux through the spherical.

Exercise 5: The Coulomb Vector Field

Coulomb (1736-1806) worked with electro magnetism. From his work we know the so-called Coulomb Vector Field:

$$\mV(x,y,z)= \left(\frac{x}{\left(x^2+y^2+z^2\right)^{\frac32}}\,,\,\frac{y}{\left(x^2+y^2+z^2\right)^{\frac32}}\,,\,\frac{z}{\left(x^2+y^2+z^2\right)^{\frac32}}\right)\,.$$

A solid cylinder of revolution $\Omega$ is given by the parametric representation

$$\mr(u,v,w)=\left(u\cos(w)\,,\,u\sin(w)\,,\,v\right)\,,\,\,u\in\left[0,a\right] \,,\,\,v\in[-h,h]\,,\,\,w\in \left[-\pi\,,\,\pi\right]\,,$$

where $\,a\,$ and $\,h\,$ are positive real numbers. In the following we shall compute the flux out of the surface of $\,\Omega\,$ in two different ways. Just follow the steps below.

A

Draw a sketch of $\,\Omega\,$ using paper and pencil and determine a parametric representation for each of the three parts that the surface $\,\partial\Omega\,$ of $\,\Omega\,$ consists of: The bottom, the top and the tubular shaped part.

B

Determine the flux of $\,\mV\,$ out through $\,\partial\Omega\,$ by computing the flux out of each of the three parts that $\,\partial\Omega\,$ consists of. What does in fact the size of the cylinder mean for strength of the flux? And in addition: What is the limit value of the strength of the flux when $\,a\,$ and $\,h\,$ tend towards 0?

C

Determine the flux of $\,\mV\,$ out through $\,\partial\Omega\,$ using Gauss’ Theorem. Readily use Maple to compute the divergence of $\,\mV\,.$

D

Maybe you find out that something is terribly wrong! What seems to be the problem?

Exercise 6: Special Divergence

Given a vector field

$$\mV(x,y,z)=(x^3+xy^2,4yz^2-2x^2y,-z^3)$$

and a solid spatial field

$$\Omega=\left\{ (x,y,z)\,|\, x^2+y^2+z^2\leq a^2\right\}\,.$$
A

Determine the flux of $\mV$ out through the surface $\partial \Omega$ of $\Omega\,.$

Exercise 7: Flux out of an Ellipsoid

Given a vector field $\mV(x,y,z)=(2x,3y,-z)$ and a solid spatial field

$$\Omega=\left\{ (x,y,z)\,|\,\left( \frac{x}{a}\right) ^2+\left( \frac{y}{b}\right) ^2+\left( \frac{z}{c}\right) ^2\leq 1\right\}\,.$$
A

Determine the flux of $\mV$ out through the surface $\partial \Omega$ of $\Omega\,.$