Exercise 1: The Right-Hand Rule and Stokes’ Theorem
Introduction:
Stokes’ Theorem is about a surface and its boundary curve. The theorem presuppose that both the surface and its boundary curve has been given an orientation and that the relation between the two orientations fulfills what is popularly known as the right-hand rule: The direction of the unit normal vector should form a right-hand ‘screw’ with the direction of the boundary curve. Or in other words: When one from the endpoint of the unit normal vector of the surface looks at the boundary curve, it is traversed in the counter-clockwise direction.
Let us e.g. consider the northern hemisphere of the Earth. If the unit normal vector field is chosen such that the unit normal vectors point away from the center of the Earth the right-hand rules demands that the equator is traversed towards the East.
In eNote 29the right-hand rule is formulated in a formally more correct way like this: When Stokes’ Theorem is used ‘‘the orientation (given by the direction of unit tangential vector field $\,\me_{\partial F}\,$) of the boundary shall be chosen so that the cross product $\,\me_{\partial F}\times \mn_F\,$ points away from the surface along the boundary’’.
The following exercise should be solved by hand.
Consider in space the circular disc $\,F\,$ given by $\,x^2+y^2\leq 4\,$ and $\,z=0\,.$
A
Chose a parametric representation for $\,F\,$ and a paramtric representation for the boundary curve $\,\partial F\,$, such that corresponding orientations of $\,F\,$ and $\,\partial F\,$ fulfill the right-hand rule.
B
Let $\mN$ denote the normal vector for $\,F\,$ genererated tby the parametric representation for $\,F\,$, and let $\mT$ denote the tangential vector for $\,\partial F,$ generated by the parametric representation for $\,\partial F\,$. Show that the cross product $\,\mT \times \mN\,$ points away from the surface along the boundary.
Given the vector field $\,\mV(x,y,z)=(x^2-y,-yz,xz)\,$.
C
Determine using Stokes’ Theorem the circulation of $\,\mV\,$ along $\,\partial F\,$.
hint
Determine the curl of $\,\mV\,$, and use the the theorem about the flux (the orthogonal surface integral) from eNote 25.
answer
The resultat is $4\pi\,$.
D
A student has accidentally chosen the parametric representations for $\,F\,$ and $\,\partial F\,$, such that the right hand rule are not fulfilled. But otherwise the computations are correct. What will then be the answer of the student?
answer
The student has got the answer $\,-4\pi\,.$ On the one hand not that bad, but on the other hand totally wrong.
Exercise 2: Stokes and the Right-Hand Rule
In the space a triangular surface $\,T\,$ with the vertices $\,A(0,0,1)\,,$$\,B(1,0,0)\,$ and $\,C(0,1,0)\,$ is given together with a vector field $\,\mV(x,y,z)=(z,x,y)\,.$
A
Determine a parametric representation $\,\mr\,$ for $\,T\,$, and plot the triangle using Maple.
hint
E.g you can start with a parametric representation for line segment $\,BC\,$. Now build the parametric representation for the triangle by parametrizing the line segment $\,AP\,$, where $\,P\,$ is an arbitrary point on $\,BC\,$.
Choose an orientation of the boundary curve $\,\partial T\,$, and show it on a figure. Does it fulfill the right-hand rule with repspect to $\,\mr\,$?
hint
For the parametric representation given in the the answer above (and the normal vector $\,\mN\,$ genereated by the parametric representation) the orientation ACBA fulfill the right-hand rule.
C
Determine using Stokes’ Theorem the circulation of $\,\mV\,$ along $\,\partial T\,.$
hint
Determine the curl of $\,\mV\,$, and use the theorem about flux (the orthogonal surface integral) from eNote 25.
answer
$-\frac{3}{2}\,.$
Exercise 3: Surfaces with a Given Curve as Boundary Curve
The exercise is about determining the circulation of the vector field along the curve using Stokes’ Theorem, with a self-chosen orientation.
A
Make a sketch of $\,\mathcal K\,$, and mark the chosen orientation.
answer
Well, there are two possibilities for the orientation. Here we chose the counter-clockwise one, when the circle $\,\mathcal K\,$ is seen from the positive end of the $\,z\,$-axis.
B
Choose two different surfaces $\,\mF_1\,$ and $\,\mF_2\,$ that have $\,\mathcal K\,$ as boundary curve. State for each of the surfaces a parametric representation that fulfill the right-hand rule with respect to the chosen orientation of $\,\mathcal K\,$.
answer
There is an infinity of possibilities. Here we choose as $\mF_1$ the unit circular disc, centered in $(0,0,1)$ and parallel to the $\,(x,y)\,$-plane. As $\,\mF_2\,$ we choose the upper half of the unit sphere that is centered in $\,(0,0,1)\,$.
Determine the cirkulation of $\,\mV\,$ langs $\,\mathcal K\,$ using Stokes’ Theorem, using both $\,\mF_1\,$ and $\,\mF_2\,,$ as surfaces with $\,\mathcal K\,$ as boundary curve.
Why is it advantageous to use Stokes’ Theorem instead of finding the circulation using the ordinary way of computing a tangential curve integral?
Exercise 4: Verification of Stokes’ Theorem through an Example
A cylinder of revolution is given by the equation $\,(x-1)^2+y^2=1\,,$ a plane is given by equation $\,z=2-x\,$, and finally a vector field is given by $\,\mV(x,y,z)=(y,z,x)\,.$
A
Determine a parametric representation for the closed curve of intersection $\,\mathcal K\,$ between the cylinder and the plane.
hint
The first and the second coordinate to the parametric representation you get from the circle that $\,x\,$ and $\,y\,$ describe in the $\,(x,y)$-plane. Then determine the third coordinate.
Now determine a parametric representation for the surface $\,\mathcal F\,$ in the given plane $\,z=2-x\,$, that span the curve of intersection (i.e. that has this curve as the boundary curve), and again determine the circulation of $\,\mV\,$ along the curve of intersection, this time using Stokes’ Theorem.
Explain that $\,\mF\,$ is a hemi-sphere and sketch its boundary curve $\,\partial F\,$ including a marking of the orientation for $\,\partial F\,$.
A
Determine the flux of the curl of $\,\mV\,$ through $\,F\,$.
answer
The wanted flux is equal to the circulation $\,\mV\,$ along $\,\partial F$ - that is $a^2\pi\,.$
Exercise 6: The Potential for a Divergence Free Vector Field
Introduction:
If there in $\,\reel^3\,$ is given a smooth vector field $\,\mV(x,y,z)\,$ that is divergence free, i.e. $\,\mathrm{Div}(\mV)(x,y,z)=0\,$ in all of $\,\reel^3\,,$ then $\,\mV\,$ has a potential (also known as the vector potential) $\,\mW\,$ about which it applies that
$$\,\mathbf{Curl}(\mW)(x,y,z)=\mV(x,y,z)\,.$$
For every smooth vector field $\,\mV\,$ we introduce the star vector field$\,\mW^*(x,y,z)\,$ by the formula
The formula shall be read in the following way: First three integrals are computed and afterwards the cross product is computed. The following theorem applies: $\,\mV\,$ is divergence free if and only if the curl of $\,\mW^*\,$ equals $\,\mV\,.$
Describe and sketch $\mathcal F$ and its boundary curve $\partial \mathcal F\,$.
answer
$\mathcal F$ is the upper half of the unit sphere centered in $(0,0,1)\,$.
B
Determine using Stokes’ Theorem the flux of $\mU$ through $\mathcal F\,$, when $\mathcal F\,$ is thought to be oriented with the unit normal vector field pointing away from the origin.
hint
Use the potential $\mU$ you found in the previous exercise.
and the square curve $\,\mathcal K\,$ that connects the points $\,(0,0,0),(1,0,0),(1,1,0)\,$ and $\,(0,1,0)\,$, and whose orientation is determined by the order given by this list of points.
A
Show that $\,\mW(x,y,z)=\big(\,\sin(xz),\sin(xy),\sin(yz)\,\big)\,$ is a potential for $\mV$.
hint
Is $\,\mathbf{Curl}(\mW)=\mV\,$?
B
Determine the flux of $\,\mV\,$ through an arbitrary surface that has $\,\mathcal K\,$ as boundary curve, by computing the flux as the circulation of $\,\mW\,$ along $\,\partial K\,$.
answer
The circulation mentioned and hence the wanted flux is $\,1-\cos(1)\,$.
C
Explain that the result in the previous question can also be found by an ordinary flux computation. And compute the result in this way.
hint
Since $\mV$ has a potential, the fluxes through different surfaces that have $\mathcal K$ as the boundary curve, are identical.
