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Exercise 1: Flux through Parametric Surfaces. By Hand

This exercise should be solved by hand.

Given a vector field

$$\mV(x,y,z)=(\cos(x),\cos(x)+\cos(z),0)$$

together with a surface $\mathcal F$ that is given by the parametric representation

$$\mr(u,v)=(u,0,v), \quad u\in\left[ 0,\pi\right] ,\, v\in\left[ 0,2\right] .$$
A

Determine the parametric representation corresponding to the normal vector $\,\mathbf n_{\mathcal F}\,$ and compute the flux of the vector field through the surface.

B

What is the meaning of the sign of the flux? Can you change the sign of the flux by changing the parametric representation of the surface?

Given a vector field

$$\mV(x,y,z)=(yz,-xz,x^2+y^2)$$

together with a surface$\,\mathcal F\,$ that is given by the parametric representation

$$\mr(u,v)=(u\sin(v),-u\cos( v),uv), \quad u\in\left[ 0,1\right] ,\, v\in\left[ 0,1\right] .$$
C

Determine the parametric representation corresponding to the normal vector $\,\mathbf n_{\mathcal F}\,$ and compute the flux of the vector field through the surface.

Exercise 2: 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 3: Optimization of Flux. Maple

This exercise is solved using Maple.

The vector field

$$\mV(x,y,z)=(xyz\,,x+y+z\,,\frac{z}2\,)\,.$$

is given together with the plane $\,\alpha\,$ with the equation $\,z+x=2\,.$

A

Determine a parameteric representation for the part of $\,\alpha\,$ that lies (vertically) above the square spanned by the points $\,(1,1,0),(-1,1,0),(-1,-1,0)\,$ and $\,(1,-1,0)\,$. The parametric representation should be chosen so that its corresponding normal vector has a positive $\,z$-coordinate.

B

Determine the flux through the parametrized part of the $\,\alpha\,.$

A surface $\mathcal F$ consists of two parts: $\,\mathcal F_1$ that is that part of $\alpha$ that lies (vertically) above the circular disc $x^2+y^2\leq 1\,$ in the $(x,y)$-plane. $\mathcal F_2$ is the (vertical) cylindrical surface that is bounded from below by the unit circle $x^2+y^2=1$ in the $(x,y)$-plane and above by the plane $\alpha\,.$

Cyl2.png

Open surface consisting of two parts

C

Determine a parametric representation for $\,\mathcal F\,$, such that the $\,z-$coordinate for the normal vector corresponding to $\,\mathcal F_1\,$ is positive and such that normal vector corresponding to $\,\mathcal F_2\,$ points away from the $\,z$-axis.

D

Determine the flux of $\,\mV\,$ through $\,\mathcal F\,$.

$\mathcal F\,$ is now rotated the angle $\,w\,$ around the $\,z\,$-axis counter-clockwise as seen from the top of the positive part of the $\,z$-axis.

E

Determine a value of $\,w\,$ that yields the maximum flux, and a value that yields the minimal flux. State the maximum value and the minimum value.

Exercise 4: Flux Using Gauss’ Theorem. By Hand

This exercises should be solved by hand.

A spatial field fills the cube

$$\Omega=\left\{\,(x,y,z)\,|\,\,x \in\left[ 0,1\right],\,y\in \left[ 0,1\right],\,z\in \left[ 0,1\right]\,\right\}$$

equipped with an outward pointing unit normal vector field.

A

Determine the flux out of the surface of $\Omega$ of the vector field

$$\mV(x,y,z)=(2x-\sqrt{1+z^2}\,,\,x^2y\,,\,-xz^2)\,.$$

B

Determine the flux out through the surface of $\Omega$ of the vector field

$$\mW(x,y,z)=(2x-\sqrt[3]{y^2+z^2}\,,\,xz-\cos(y)\,,\,\sin(xy)+2z)\,.$$

C

Given that

$$\displaystyle{\int_0^1\int_0^1\int_0^1(x+y+z)\,\mathrm dx\,\mathrm dy\,\mathrm dz=\frac 32}\,.$$

Determine a vector field whose flux out through the surface of $\,\Omega$ is $\,\displaystyle{\frac 32}\,.$

Exercise 5: 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 6: 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?