Determine the normal vector $\,\mathbf N_{\mathcal F}\,$ corresponding to the given parametric representation, and compute the flux of the vector field through the surface.
What is the meaning of the sign of the flux? Try changing the sign of the flux by changing the parametric representation of the surface.
answer
A new parametrization that represents the same geometry could be
$$\mr(u,v)=(u,0,-v)\quad , \, u \in[0,\pi],v \in [-2,0]$$
where the sign in front of the parameter $v$, and thus also the sign of its interval, has been flipped. Try recalculating the normal vector and the flux with this parametrization.
We are given the 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
Determine the normal vector $\,\mathbf N_{\mathcal F}\,$ corresponding to the given parametric representation and compute the flux of the vector field through the surface.
A surface $\,\mathcal S_0\,$ consists of the part of a unit sphere centred at the origin that is located on or above the plane given by the equation $\,\displaystyle{z=\frac 12\,.}$
B
Provide a parametric representation of $\,\mathcal S_0\,.$
answer
$$\mr(u,v)=(\sin(u)\cos(v), \sin(u)\sin(v), \cos(u))\quad , \, u \in \left[0,\frac{\pi}3\right],v\in [0,2\pi]$$
Note that this is the usual unit sphere parametrization but with an adjusted $u$ interval.
C
Compute the flux of $\,\mV\,$ through $\,\mathcal S_0\,.$
$\,\mathcal S_0\,$ deforms over time by flowing with (the flow curves of) $\,\mV\,$. After a time $t$ we will denote the new, deformed surface $\,\mathcal S_t\,$. Provide a parametric representation of $\,\mathcal S_t\,$ after time $t$. Plot $\,\mathcal S_0\,$ and $\,\mathcal S_t\,$ for selected values of $\,t\,$.
hint
Redo the flow curve calculation from Question a), but with the initial condition $\mr(0)=\mathcal S_0\,.$
answer
With the above parametrization of $\mathcal S_0$, the parametrization of $\mathcal S_t$ becomes:
where $u \in \left[0,\frac{\pi}{3}\right]$ and $v \in [0,2\pi]\,.$
E
Justify that $\,\mathcal S_0\,$ has no points in common with $\,\mathcal S_t\,$ for $\,t>0\,.$
F
Determine a parametric representation of the spatial region $\,\Omega_t\,$ that $\,\mathcal S_t\,$ has swept through after time $t$ since it left $\,\mathcal S_0\,$ at time $\,t=0\,.$
Determine $\mathrm{Vol}’(t)\,$ and then $\mathrm{Vol}’(0)\,.$ Compare this result with the flux of $\,\mV\,$ through $\,\mathcal S_0\,$. Why do we see this link?
is given together with a plane $\,\alpha\,$ with the equation $\,z+x=2\,.$
A
Provide a parametric representation of the part of $\,\alpha\,$ that is located (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.
hint
$\alpha$ represents the graph of an elevation function $z=h(x,y)=2-x\,.$ The question is thus about a graph surface.
answer
$$\mr(u,v)=(u,v,2-u)\quad,\, u \in [-1,1],v\in[-1,1]$$
This parametrization has the normal vector $\mathbf N_{\mr}=(1,0,1)\,.$
B
Compute the flux through the parametrized part of $\,\alpha\,.$
answer
$$4$$
An open surface $\mathcal F$ consists of two parts:
$\mathcal F_1$ is the part of $\alpha$ that is located (vertically) above the circular disc $x^2+y^2\leq 1\,$ in the $(x,y)$ plane.
$\mathcal F_2$ is the (vertical) cylindrical surface bounded from below by the unit circle $x^2+y^2=1$ in the $(x,y)$ plane and from above by $\alpha\,.$
C
Provide a parametric representation of $\,\mathcal F\,$ with the requirements that the normal vector corresponding to $\,\mathcal F_1\,$ has a positive $z$ coordinate and that the normal vector corresponding to $\,\mathcal F_2\,$ points away from the $\,z$ axis.
hint
Providing a parametric representation of a surface that consists of several parts is done by providing a parametric representation of each part.
hint
$\,\mathcal F_1\,$ can be interpreted as a graph surface. The circular disc in the $\,(x,y)$ plane can be parametrized by
For the third coordinate, use the elevation function retrieved from $\alpha$ that was used in Question a). You must multiply it with a parameter to fill out the cylindrical surface from bottom to top. When done, remember to check the normal vector direction.
Compute the flux of $\,\mV\,$ through $\,\mathcal F\,$.
hint
Compute the flux through $\mathcal F_1$ and through $\mathcal F_2$ and add up the results. You have already ensured that their signs agree by checking that their normal vectors are both outwards, so they can safely be added together.
answer
$$\mathrm{Flux}(\mV,\mathcal F)=3\pi$$
$\mathcal F\,$ is now being rotated by the angle $\,w\,$ about the $\,z\,$ axis counterclockwise as seen from the positive end of the $\,z$ axis.
E
Compute the value of $\,w\,$ that yields the maximum flux through the surface, and the value that yields the minimum flux. State those maximum and minimum values.
hint
The rotation matrix for rotation about the $z$ axis is
Calculate the volume expressed in terms of $w$. Then perform the usual optimization approach for a function of one variable: differentiate, set equal to zero, and solve.
answer
The minimum value $\frac{5\pi}{2}$ is attained at an angle of $w=\frac{\pi}2\,.$
The maximum value $\frac{7\pi}{2}$ is attained at an angle of $w=-\frac{\pi}2\,.$
Exercise 4: Flux Using Gauss’ Divergence Theorem. By Hand
A 3D region in the form of a cube is given by the parametric representation
out through the surface $\partial\Omega$ of $\Omega\,.$
hint
This can be solved with the usual flux formula. But we would then have to parametrize each of the six surfaces, check all six normal vector directions, and then compute a flux out through each. Maybe it can be done in an easier way?
hint
Use Gauss´ Divergence Theorem. Note that $\Omega$ is an axis-parallel cube.
In this exercise we will verify Gauss’ Theorem in an example with a flux that is computed via the usual formula versus the same flux computed via a triple integral of a divergence.
Note that the surface of $\Omega$ consists of two parts: A hemi-spherical shell and a circular base. You must parametrize each and the calculate the flux out through each after which you can add the fluxes together. Remember to ensure outwards-pointing normal vectors so that their signs match.
answer
If Gauss was right, you got the same answer to both this and the previous questions:
For which $\,a\,$ is $\mathrm{Flux}(\mV,\partial\,\Omega)$, with the given unit normal vector field $\,\mathbf n_{\,\partial \Omega}\,,$ positive?
hint
This means: for which $a$ value is the “outflow” through $\partial \Omega$ larger than the “inflow”?
answer
The flux is negative for $\,0<a<\frac{\sqrt{30}}{3}\,$, otherwise positive.
D
Discuss with a fellow student which characteristic equality there is between Gauss’ Theorem about the relation between the divergence integral and the flux on one hand and the following identity known from highschool on the other hand:
$$\left[ F(x)\right] _a^b=\int_a^b F'(x)dx\,.$$
answer
Divergence can, freely formulated, be considered as “the derivative of the vector field”. In both cases we can say that we have pushed the integration ‘‘out on the boundary’’, i.e. on the surface and the end-points of the interval, respectively.
Exercise 6: The Coulomb Vector Field
Coulomb (1736-1806) worked with electromagnetism. From his work we have the so-called Coulomb vector field:
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.
A
Draw a sketch of $\,\Omega\,$ using paper and pencil and provide a parametric representation of each of the three parts that the surface $\,\partial\Omega\,$ of $\,\Omega\,$ consists of: The bottom, the top and the tubular-shaped part.
answer
Possible parametrizations with outwards-pointing normal vectors:
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. How does the size of the cylinder influence the strength of the flux?
answer
$$4\pi$$
Note that the size parameters $a$ and $h$ do not influence the flux.
C
Compute the flux of $\,\mV\,$ out through $\,\partial\Omega\,$ using Gauss’ Divergence Theorem.
answer
$$0$$
D
Maybe (hopefully) you are now thinking that something is terribly wrong! Discuss with a fellow student what the problem is.
hint
Consider closely the Coulomb vector field. It has a problematic point.
answer
Gauss’ Theorem assumes a smooth vector field. But the Coulomb field has a singularity at $(0,0,0)$ where it is undefined and thus not smooth. Thus, Gauss’ Theorem does not apply and flux can only be calculated via the usual formula!
Note, Gauss’ Theorem does apply for regions within a Coulomb field that do not include this point. Then the field is smooth throughout the region that is being considered.
