Determine the normal vector $\,\mathbf n_{\mathcal F}\,$ corresponding to the parametric representation and compute the flux of the vector field through the surface.
answer
Flux($\mV,\mathcal F$)= $-\sin(2)\pi.$
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 suface?
Given a vectorfield
$$\mV(x,y,z)=(yz,-xz,x^2+y^2)$$
and a surface $\,\mathcal F\,$ given by the parametric representation
Determine the normal vector $\,\mathbf n_{\mathcal F}\,$ corresponding to the parametric representation and compute the flux of the vector field through the surface.
answer
Flux($\mV,\mathcal F$)$=\frac 38\,.$
Exercise 2: Fluxthrough an open and a closed surface
A function $\,h:\reel^2 \rightarrow \reel\,$ is given by the expression
$$\,h(x,y)=1-x^3\,.$$
We consider a rectangle in the $(x,y)$-plane given by $\,0\leq x\leq 1\,$ and $\,-\frac{\pi}2\leq y\leq \frac{\pi}2\,.$ Let the surface $\,\mathcal F\,$ be the part of the graph for $\,h\,$ that lies vertically above the rectangle.
A
Determine a parametric representation for $\mathcal F\,.$
hint
$\mathcal F$ is a so-called graph surface.
answer
$\mathbf s(u,v)=(u,v,1-u^3)\,$ hvor $u\in \left[0,1\right]$ and $v\in \left[-\pi/2,\pi/2\right]\,.$
Now let $\Omega$ denote the solid spatial region that lies vertically between the rectangle $(x,y)$-plane and $\mathcal F\,.$
C
Determine a parametric representaion for $\Omega\,.$
hint
You can probably reuse the parametric representation you made for $\mathcal F\,.$ You just have to fiddle a bit with the 3rd coordinate.
answer
$\mr(u,v,w)=(u,v,w(1-u^3))\,$ hvor $u\in \left[0,1\right]\,,$$v\in \left[-\pi/2,\pi/2\right]\,$ and $w\in \left[0,1\right]\,.$
D
Ues Gauss’ theorem to determine the flux of $\mV$ out through the surface of $\Omega\,.$
hint
In fact, the surface of $\Omega$ consists of 5 surface segments. The beauty of Gauss’s theorem is that we only need one integral, but it is a space integral. Which one?
hint
You shall find the space integral of the of divergence $\mV$ over $\Omega\,.$ Find the integrand, that is
$$\textrm{Div}\mV(\mr(u,v,w))\mathrm{Jacobi}_{\mathbf r }(u,v,w)\,$$
and determine the triple integral, possibly using Maple.
answer
The integrand is
$$\,u^6w-2u^3w+w+u^4\sin(v)-u\sin(v)\,$$
and the answer is $\,\displaystyle{\frac{9}{28}}\,\pi\,.$
Exercise 3: Optimization of a flux. Maple
This exercise is solved using Maple.
Given the vector field
$$\mV(x,y,z)=(xyz\,,x+y+z\,,\frac{z}2\,)\,.$$
and the plane $\,\alpha\,$ with the equation $\,z+x=2\,.$
E
Determine a paramteric representation for the part of $\,\alpha\,$ that lies vertically above the the squater spanned byt the points $\,(1,1,0),(-1,1,0),(-1,-1,0)\,$ and $\,(1,-1,0)\,$. The parametric representation is chosen such that the corresponding normal vector has a positive $\,z$-coordinate.
hint
Maybe you will find the parametrization immediately. Or maybe you wil think about $\alpha$ as the graph for the height functionn $z=h(x,y)=2-x\,$ such that the questio0n is about a graph surface.
F
Determine the flux through the parametrized part of $\,\alpha\,.$
answer
Fluxen = 4.
A surface $\mathcal F$ consists of two parts: $\,\mathcal F_1$ that is the part of $\alpha$ that lies (vertically) above the $(x,y)$-planen circular disc $x^2+y^2\leq 1\,$ in the $(x,y)$-plane. $\mathcal F_2$ is the (vertical) cylindrical surface that is bounded below by the unit circle $x^2+y^2=1$ in the $(x,y)$-plane and above by the plane $\alpha\,.$
Open surface consisting of two parts
G
Determine a parametric representation for $\,\mathcal F\,$, cuch that the$\,z$coordinate for the normal vector corresponding to $\,\mathcal F_1\,$ has a positive $\,z$-coordinate and such the the normal vector corresponding to $\,\mathcal F_2\,$ is pointing away from the $\,z$-axis.
hint
$\,\mathcal F_1\,$ can be perceived as a graph surface. The circular disc in the $\,(x,y)$-plane can be parametriced as
with obvious intervals for $u$ and $v\,.$ Then insert into the default parameterization for a graph surface. Finally, check the direction of the normal vector (crucial for the sign of the flux).
For $\mathcal F_2$ start by parametrization of the periphery of the circle in the $(x,y)$-plane:
with obvious intervals for $u\,.$ Then we just need to get $z$ parameterized –– $z$ runs from 0 to $2-x\,$ –– and check the direction of the normal vector.
hint
Hey, try one more time before you check the answer.
