Exercise 1: The Right-Hand Rule and Stokes’ Theorem
Stokes’ Theorem concerns a surface and its boundary curve. The theorem presupposes that both the surface and its boundary curve have 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: If we from the end point of the normal vector of the surface looks down at the boundary curve, then it must be traversed in the counterclockwise 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 rule demands that the equator is traversed towards the East.
The mathematically precise formulation of the right-hand rule from eNote 29 is: “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 3D space the circular disc $\,F\,$ given by
$$\,x^2+y^2\leq 4\quad\text{and}\quad z=0\,.$$
A
Choose a parametric representation of $\,F\,$ and a parametric representation of its boundary curve $\,\partial F\,$, such that corresponding orientations of $\,F\,$ and $\,\partial F\,$ fulfill the right-hand rule.
hint
The “orientation” of a surface is given by its normal vector. The orientation of a curve is given by its tangent vector.
hint
Apply your right hand by imagining grapping the normal vector with your right hand with the thumb up along with it. Now look at your other fingers. If they curl along with the direction of the tangent vector, then the right-hand rule is fulfilled.
hint
Plot the vectors for a clear visual view on their directions. In Maple, use the arrow command.
The following question gives a mathematical approach for checking the fulfillment of the right-hand rule – the mentioned cross product must point away from the surface.
B
Let $\mN$ denote the normal vector of $\,F\,$ genererated by the parametric representation of $\,F\,$, and let $\mT$ denote the tangent vector of $\,\partial F,$ generated by the parametric representation of $\,\partial F\,$. Show that their cross product $\,\mT \times \mN\,$ points away from the surface along the boundary.
We are now 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 $\,\mV\,$. Then apply the flux formula from eNote 25 on this curl field.
answer
$$4\pi$$
D
A student has accidentally chosen the parametric representation of $\,F\,$ and $\,\partial F\,$, such that the right-hand rule is not fulfilled. But otherwise the computations are correct. Discuss with a friend, which answer this student got.
answer
The student got the answer:
$$\,-4\pi\,.$$
When the right-hand rule is not fulfilled, then applying Stokes’ Theorem will result in the correct numerical value with a wrong result.
Exercise 2: Stokes and the Right-Hand Rule
In 3D space a triangular surface $\,T\,$ with the vertices $\,A(0,0,1)\,,$$\,B(1,0,0)\,$ and $\,C(0,1,0)\,$ is given together with the vector field
$$\,\mV(x,y,z)=(z,x,y)\,.$$
A
Provide a parametric representation $\,\mr\,$ of $\,T\,$, and plot the triangle using Maple.
hint
You can start with a parametric representation of the line segment $\,BC\,$. From this, build the parametric representation of the triangle by parametrizing the line segment $\,AP\,$, where $\,P\,$ is an arbitrary point on $\,BC\,$.
Choose a positive orientation of the boundary curve $\,\partial T\,$, and show it on a figure. Does it fulfill the right-hand rule with respect to $\,\mr\,$?
hint
For the parametric representation given in the answer above (and the normal vector $\,\mN\,$ generated by that parametric representation) the orientation $ACBA$ fulfills the right-hand rule.
C
Compute using Stokes’ Theorem the circulation of $\,\mV\,$ along $\,\partial T\,.$
hint
You need a parametrization of a surface that fills in the closed curve.
hint
You have chosen a positive orientation of the curve in the previous question. Ensure that your parametrization of the surface has a normal vector that with this orientation fulfills the right-hand rule. If it doesn’t, then take proper precautions.
hint
Proper precautions to a non-fulfilled right-hand rule is either
to change the surface parametrization (while ensuring that it still decsribes the same geometry) so that its normal vector changes direction or
to simply keep in mind that the sign is wrong, and then change the sign of the result when you are done.
hint
Determine the curl of $\,\mV\,$, and use the flux formula from eNote 25.
answer
$$-\frac{3}{2}$$
Exercise 3: Surfaces with a Given Curve as Boundary Curve
Make a sketch of $\,\mathcal K\,$, and draw positive orientation of your choice.
B
Choose two different surfaces $\,\mF_1\,$ and $\,\mF_2\,$ that have $\,\mathcal K\,$ as their boundary curve. State for each of the surfaces a parametric representation that fulfills the right-hand rule with respect to your chosen orientation of $\,\mathcal K\,$.
answer
There is an infinity of possibilities.
We could e.g. choose as $\mF_1$ a unit circular disc centered at $(0,0,1)$ that is parallel to the $\,(x,y)\,$ plane.
As $\,\mF_2\,$ we could choose the upper half of a unit sphere centered at $\,(0,0,1)\,$.
A parametric representation of $\,\mF_1\,$ could be
where we have chosen a counterclockwise positive orientation when viewing the circle $\,\mathcal K\,$ from the positive end of the $\,z\,$ axis.
We now want to calculate the circulation of $\,\mV\,$ along $\,\mathcal K\,$ using Stokes’ Theorem. For the theorem we will be working with a surface that has $\,\mathcal K\,$ as its boundary curve.
C
We have two surfaces ready that we can use in Stokes’ Theorem that both have $\,\mathcal K\,$ as their boundary curve: $\,\mF_1\,$ and $\,\mF_2\,$. Discuss with a fellow student what difference it would make if you choose one instead of the other, and decide which one you would choose.
D
Compute the circulation of $\,\mV\,$ along $\,\mathcal K\,$ using Stokes’ Theorem with surface $\,\mF_1\,.$ Then repeat the calculation with surface $\,\mF_2\,$.
The result is the same regardless of the chosen surface – only the boundary curve matters when we are working within a smooth vector field. So why not always choose the surface that is simplest to parametrize and integrate over.
E
Discuss with a fellow student when it is advantageous to use Stokes’ Theorem instead of finding the circulation the ordinary way with the tangential line integral formula.
Exercise 4: Verification of Stokes’ Theorem through an Example
A cylinder of revolution is given by the equation
$$\,(x-1)^2+y^2=1\,,$$
a plane is given by the equation
$$\,z=2-x\,,$$
and finally a vector field is given by
$$\,\mV(x,y,z)=(y,z,x)\,.$$
A
Provide a parametric representation of the closed curve of intersection $\,\mathcal K\,$ between the cylinder and the plane.
hint
You can get the first and second coordinates of the parametric representation from the circle that is given in the $\,(x,y)$ plane. Then determine the third coordinate via the other information that was provided.
Provide a parametric representation of the surface $\,\mathcal F\,$ within the given plane $\,z=2-x\,$, that spans the curve of intersection (i.e. that has this curve as its boundary).
Compute the circulation of $\,\mV\,$ along the curve of intersection, this time using Stokes’ Theorem.
hint
Use the parametrization that you just created of the surface bounded by $\mathcal K$. But remember to check if it fulfills the right-hand rule with respect to the chosen parametric representation of $\,\mathcal K\,.$
hint
If we use the parametric representation of $\,\mathcal F$ mentioned in the previous answer, then a normal vector is
Explain why $\,\mathcal F\,$ is a hemi-sphere and sketch its boundary curve $\,\partial F\,$ including an indication of an orientation of $\,\partial F\,$.
B
Determine the flux of the curl of $\,\mV\,$ through $\,F\,$.
answer
$$a^2\pi$$
This flux is equal to the circulation of $\,\mV\,$ along $\,\partial F$.
Exercise 6: The Potential of a Divergence Free Vector Field
If we in $\,\reel^3\,$ are given a smooth vector field $\,\mV(x,y,z)\,$ that is divergence-free, i.e. $\,\mathrm{Div}(\mV)(x,y,z)=0\,$ in all of $\,\reel^3\,,$ then $\,\mV\,$ has a potential (also known as a vector potential) $\,\mW\,$ about which it applies that
$$\,\mathrm{Curl}(\mW)(x,y,z)=\mV(x,y,z)\,.$$
For every smooth vector field $\,\mV\,$ we introduce the star vector field$\,\mW^*(x,y,z)\,$ by the formula
The formula should be read in the following way: First three integrals are computed and afterwards the cross product is computed. The following theorem applies: “$\,\mV\,$ is divergence free if and only if the curl of $\,\mW^*\,$ equals $\,\mV\,.$”
Describe and sketch $\mathcal F$ and its boundary curve $\partial \mathcal F\,$.
answer
$\mathcal F$ is the upper half of the unit sphere centered at $(0,0,1)\,$.
B
Determine using Stokes’ Theorem the flux of $\mU$ through $\mathcal F\,$, when $\mathcal F\,$ is thought to be oriented with a unit normal vector field pointing away from the origin.
hint
Use the potential $\mU$ that you found in the previous exercise.
and the square curve $\,\mathcal K\,$ that connects the points $\,(0,0,0),(1,0,0),(1,1,0)\,$ and $\,(0,1,0)\,$, and whose orientation is determined by the order given by this list of points.
Determine the flux of $\,\mV\,$ through an arbitrary surface that has $\,\mathcal K\,$ as its boundary curve, by computing the flux as the circulation of $\,\mW\,$ along $\,\partial K\,$.
answer
$$1-\cos(1)$$
C
Explain how the above result could also be found via a usual flux computation with the flux formula. Then compute the result in this way.
hint
Since $\mV$ has a potential, then the fluxes through different surfaces that have $\mathcal K$ as their boundary curve, are identical.
