Star Wars: Padmé and the Vote of No Confidence

Jeff and Todd discuss Padmé and her call for the vote of no confidence that launched Palpatine’s rise to the chancellorship and to absolute power. We look at her actions from different ethical perspectives – deontological, teleological, virtue – and at her complicated role in the narrative as both hero and pawn, as well as her dominant presence and importance in the arc of the whole saga.

Stokes’ Beautiful Theorem: Differential Forms, Boundaries, Exterior Derivatives, and Manifolds

Stokes’ Theorem in its general form is a remarkable theorem with many applications in calculus, starting with the Fundamental Theorem of Calculus. The pattern in each case is that the integral of a function over a region is equal to the integral of a related function over the boundary of the region. We can use information about the boundary of a region to get information about the entire region, which is both useful and mathematically elegant.

For the visual aids that go with this episode see the video version on YouTube.

I’ve been reviewing some calculus recently but I’ve been approaching it in a different way than I have before, like when I learned it in school to be an engineer. Instead of studying it for the purpose of solving specific problems I’m looking at it more from a high-level, trying to understand it conceptually to see the general structure and patterns. It’s in line with my philosophical penchant to take things up a level or look behind things at another level of abstraction. One of the patterns I’ve seen across calculus has been something that falls under the generalized version of Stokes’ Theorem. I find the generalized Stokes’ Theorem quite beautiful, in that way that mathematics can be beautiful. It can express concisely and compactly a concept that has broad and varied applications in its more particular forms.

I’ll go through the generalized Stokes’ Theorem and some of its special applications. Since a lot of this is better understood visually I’ve made the YouTube video to go along with this so those listening to the podcast might want to check it out as well if this stuff is hard to picture.

The generalized Stokes’ Theorem states that the integral of some differential form of dimension k-1 over the boundary of some orientable manifold of dimension k is equal to the integral of that differential form’s exterior derivative over the whole of that orientable manifold.

The four key concepts here (the ones listed in the subtitle of this episode) are:

1. Differential forms
2. Boundaries
3. Exterior derivatives
4. Manifolds

That’s very abstract, not that there’s anything wrong with that. We need to be abstract to be general. The key concept, the idea I want to drive home with this episode, is that under certain conditions information about a boundary can give you information about the entire region that it bounds. In the generalized form we’re getting information about a whole manifold from the boundary of that manifold. That will be the case in all these examples. But now let’s look at examples to illustrate. Interestingly enough this comes into play from the very beginning of calculus, with the Fundamental Theorem of Calculus.

The Fundamental Theorem of Calculus

As a very quick crash course in calculus for the uninitiated, in calculus the two most important operations are derivatives and integrals. When you take the derivative of a function it produces another function that tells you the instantaneous rate of change of the original function at any point. So for example, if you have an equation for distance from some starting point with respect to time, the derivative of that function will tell you what the velocity is at any point in time. Very useful. You can also do the opposite of that, which is an antiderivative, or integral. Say you were starting with the function of velocity with respect to time. You could take the antiderivative of that function and get a function for the position at any point in time. You’d just need to know what your starting point was and add that to it. One of the most important applications of these operations uses the Fundamental Theorem of Calculus.

The Fundamental Theorem of Calculus is probably the most recognizable thing to calculus students, even if they don’t remember that that’s what it was called. It’s the principle behind finding the area under a curve. For example, if you have a function for the velocity with respect to time plotted on a graph, the area under that curve, between the curve and the horizontal axis, sweeps out an area that will give you the value of the distance traveled between any two points in time you choose. The cool thing is that you only need to know the values of the antiderivative at your starting time and at your ending time. You don’t need to know the values in between them.

So finding the area under a curve, finding it analytically as opposed to numerically, is really quite a remarkable thing if you think about it. Because you’re basically taking some function, doing an operation on it, and then applying the resulting function only to the two points bounding the region you’re interested in. If you’re integrating from point a to point b you’re only paying attention to points a and b, not to any of the points in between them. But you’re still getting information about the whole region. You’re getting the area under all those points in between a and b. I think that’s quite remarkable. And that’s what happens in each particular version of Stokes’ Theorem. You’re able to get information about an entire region from its boundaries.

We might forget this sometimes if we’re using numerical methods more than analytical methods of integration. Using numerical methods like the trapezoidal or the rectangle method we actually do go in and add up all the regions between the boundaries that approximate the total area under the curve. But for analytical integration we don’t need to do that. We only need the antiderivative and the boundary points. And that tells us everything about the region in between those points. It seems almost magical.

Circling back to those four key concepts in the generalized Stokes’ Theorem, with the Fundamental Theorem of Calculus what we have is a differential form of dimension 0 and a manifold of dimension 1. For a function, lower case f(x), the antiderivative, upper case F, is the (0-form) differential form. The closed interval from a to b is a 1-dimensional manifold. (For the sake of simplicity think of a manifold as a surface of some dimension. A 1-dimensional manifold here being a line.) The boundaries a and b are 0-dimensional; they’re points. 

 And the exterior derivative is lower case f(x)dx.

Green’s Theorem

What other applications are there of this generalized Stokes’ Theorem? There’s also Green’s Theorem. Green’s Theorem has a similar form to the Fundamental Theorem of Calculus but instead of looking at a curve bounded by points we’re looking at a plane region bounded by a curve. With the Fundamental Theorem of Calculus we have the integral of a 0-dimensional differential form over the boundary of a 1-dimensional manifold. With Green’s Theorem we have the integral of a 1-dimensional differential form over the boundary of a 2-dimensional manifold.

As with all these theorems we’re looking at, in Green’s Theorem we are able to determine features of a region by looking at its boundaries. If we have a closed curve C that surrounds a region D we can figure out the area of D from the closed curve C. This is actually how planimeters work. And planimeters are pretty cool. A planimeter is a device that you can use to trace out, with a mechanical arm, a curve of any shape, and when you return to the position you started at it calculates the area that that shape encloses. This is exactly what Green’s theorem does.

So now to state Green’s Theorem: Say we have curve C, and functions M and N defined on a region containing D, the region enclosed by C. Green’s Theorem states that the double integral of partial derivative of M with respect to y, minus the partial derivative of L with respect to x, δM/δy – δL/δx over a region D is equal to the line integral of Ldx + Mdy over the curve C. This is that perimeter to area connection, where we can get an area, normally found using a double integral, from a perimeter, found using a line integral. The differential form is Ldx + Mdy. The region D is the two-dimensional manifold. The curve C is the 1-dimensional boundary of the manifold. And the exterior derivative is δM/δy – δL/δx.

The Divergence Theorem

We can bump all this up another dimension to get the Gauss’s Theorem, also known as the Divergence Theorem. With Green’s Theorem we have the integral of a 1-dimensional differential form over the boundary of a 2-dimensional manifold. With the Divergence Theorem we have the integral of a 2-dimensional differential form over the boundary of a 3-dimensional manifold.

With the Divergence Theorem we’re able to get information about a 3-dimensional region from it’s 2-dimensional boundary. Call the 3-dimensional region V and the 2-dimensional surface S. Then we have a vector field F. The Divergence Theorem states that the triple integral or volume integral of the divergence of vector field F over the 3-dimensional region V is equal to the surface integral of F over the surface S.

In terms of the generalized Stokes’ Theorem, with the Divergence Theorem the differential form is F·dS. The space E is the 3-dimensional manifold. The surface S is the 2-dimensional boundary of the manifold. And the exterior derivative is the divergence div F dV.

One way to understand this is that the net flux out of the region gives the sum of all sources of the field in a region. And this has many physical applications. For example, the Divergence Theorem has application to the first two of Maxwell’s four equations in physics. All four of Maxwell’s Equations have an integral form and a differential form. But the integral and differential forms are really equivalent. For the first two of Maxwell’s Equations the Divergence Theorem shows the equivalence between these two forms.

The first of Maxwell’s Equations, Gauss’s Law, relates an electric field to its source charge. This is a perfect application for the Divergence Theorem because the divergence operator gives information about sources and sinks. And an electric charge is a source. Gauss’s Law states that the net outflow of the electric field through any closed surface is proportional to the charge enclosed by the surface. In the integral form the way this is expressed is that the surface integral of electric field E over an enclosed boundary is equal to the charge divided by the permittivity of free space. In the differential form the way this is expressed is that the divergence of the electric field is equal to the charge density divided by the permittivity of free space.

The Divergence Theorem tells us that the triple integral of the divergence of electric field E over a volume is equal to the surface integral of the electric field E over the surface boundary. Since by the differential form of Gauss’s Law the divergence of the electric field is equal to the charge density divided by the permittivity of free space, if we take the triple integral of both sides we see that the triple integral of the divergence is equal to the charge divided by the permittivity of free space. By the integral form of Gauss’s Law the surface integral of electric field E is also equal to the charge divided by the permittivity of free space. So both the surface integral of the electric field E over the surface boundary and the triple integral of the divergence of the electric field E are equal to the charge divided by the permittivity of free space, and so they are equal to each other, which is exactly what the Divergence Theorem says. So these two forms are actually equivalent. 

The second of Maxwell’s Equations, Gauss’s Law for Magnetism has a similar form but demonstrates that there are no magnetic monopoles. The surface integral of a magnetic field B over some surface S is always equal to 0. Magnetic field lines neither begin nor end but make loops or extend to infinity and back. Any magnetic field line that enters a given volume must somewhere exit that volume.  In the integral form the way this is expressed is that the surface integral of magnetic field B over an enclosed boundary is equal to 0. In the differential form the way this is expressed is that the divergence of the magnetic field is equal to zero. The Divergence Theorem tells us that the triple integral of the divergence of magnetic field B over a volume is equal to the surface integral of the magnetic field B over the surface boundary. If we take the triple integral of the divergence of the magnetic field this is still equal to zero, as is the surface integral of the magnetic field over the enclosed boundary. So again these two forms are also equivalent.

Kelvin-Stokes’ Theorem

The last of the particular applications of the generalized Stokes’ Theorem is also called Stokes’ Theorem or Kelvin-Stokes’ Theorem. With Kelvin-Stokes’ Theorem as with Green’s Theorem we have the integral of a 1-dimensional differential form over the boundary of a 2-dimensional manifold, but in R3, i.e. 3-dimensional space. Given a vector field F the theorem states that the double integral or surface integral of the curl of the vector field over some surface is equal to the line integral of the vector field around the boundary of that surface. Here again, the boundary gives us information about the region inside it. 

In terms of the generalized Stokes’ Theorem, with Kelvin-Stokes’ Theorem the differential form is F·dr. The surface S is the 2-dimensional manifold. The curve C is the 1-dimensional boundary of the manifold. And the exterior derivative is curl F·dS.

Curl is another vector operator, like divergence, and it’s easier to get the gist of it from physical examples, which we can get from the other two of Maxwell’s Equations.

The third of Maxwell’s Equations is also known as Faraday’s Law of Induction. Faraday’s Law describes how a time varying magnetic field creates, or induces, an electric field, which is the reason we’re able to generate electricity from turbines. In the integral form the way this is expressed is that the line integral of electric field E is equal to the negative derivative with respect to time of the surface integral of the magnetic field B. In the differential form the way this is expressed is that the curl of the electric field E is equal to the negative derivative of the magnetic field B with respect to time. Kelvin-Stokes’ Theorem tells us that the surface integral of the curl of the electric field E over some surface is equal to the line integral of the electric field E around the boundary of that surface. If we take the surface integral of the curl of the electric field E this is equal to the surface integral of the negative partial derivative of the magnetic field B with respect to time. And by the integral form of Faraday’s Law this is also equal to the line integral of the electric field around the surface boundary. So these two forms are also equivalent.

The fourth of Maxwell’s Equations is also known as Ampère’s Law. Ampère’s Law describes how a magnetic field can be generated by (a) an electric current and (b) a changing electric field. In the integral form the way this is expressed is that the line integral of magnetic field B is equal to the permeability of free space times the surface integral of current density J, plus the permittivity of free space, times the derivative with respect to time of the surface integral of the electric field E. In the differential form the way this is expressed is that the curl of the magnetic field B is equal to the permeability of free space times the current density J plus the permittivity of free space times the partial derivative of the electric field E with respect to time. Kelvin-Stokes’ Theorem tells us that the surface integral of the curl of the electric field E over some surface is equal to the line integral of the electric field E around the boundary of that surface. If we take the surface integral of the curl of the magnetic field B this is equal to the permeability of free space times the surface integral of current density J, plus the permittivity of free space, times the derivative with respect to time of the surface integral of the electric field E. And by the integral form of Ampere’s Law this is also equal to the line integral of magnetic field B around the surface boundary. So these two forms are also equivalent.

Maxwell’s Equations and Differential Forms

All of Maxwell’s equations actually simplify considerably in the language of differential forms. I’m just going to brush over this quickly without going into detail. We can describe both the electric and magnetic fields jointly by a 2-form, F,  in a 4-dimensional spacetime manifold. And we can describe electric current by a 3-form, J. Then we’ll need the exterior derivative operator, d, and the Hodge star operator, *. And Maxwell’s Equations are just:

dF = 0
d*F = J

That’s it. And one benefit of this is that thinking of the equations in terms of differential forms lets them generalize more easily to manifolds and relativistic settings.

To summarize, the general pattern with all these forms of Stokes’ Theorem is that the integral of a function over a region is equal to the integral of a related function over the boundary of the region. We can get information about an entire region from its boundary. And this is something that applies in interesting ways at different dimensions. Mathematically it’s aesthetically satisfying and elegant.

“But Also For the Interests of Others”

In his letter to the Philippians Paul said: “Do nothing from selfishness or empty conceit, but with humility of mind regard one another as more important than yourselves; do not merely look out for your own personal interests, but also for the interests of others.” (Philippians 2:3-4)

“I gotta go home. What do I owe ya?”
“The real question, Eleanor, is what do we owe to each other?”
(The Good Place, Season 2, Episode 12)

What do we owe to each other? This is the question that runs through all of what is now one of my favorite television shows, The Good Place. This question has stuck in my head the last few days. And as a Christian it’s got me thinking about what it means to be a disciple of Christ and live in imitation of Christ.

In his letter to the Philippians Paul said: “Do nothing from selfishness or empty conceit, but with humility of mind regard one another as more important than yourselves; do not merely look out for your own personal interests, but also for the interests of others.” (Philippians 2:3-4) I think there is a radical shift in perspective, call it a new life, being born again, in coming to look out not for your own personal interests, but also for the interests of others. We could call it maturity, while noting that it’s a kind of maturity that we don’t just reach naturally. It’s not maturation of the old man into a more developed version of the old man. It’s a complete rebirth and transformation into a new creature.

For all of this Christ is the model and the means. Paul continues in his exhortation to the Philippians saying, “Have this attitude in yourselves which was also in Christ Jesus, who, although He existed in the form of God, did not regard equality with God a thing to be grasped, but emptied Himself, taking the form of a bond-servant, and being made in the likeness of men. Being found in appearance as a man, He humbled Himself by becoming obedient to the point of death, even death on a cross.” (Philippians 2:5-8) These are some of the most important verses in the New Testament and have been foundational to the development of the doctrine of Christ’s nature and Godhood. But what I’d like to focus on is the exhortation embedded in it; the ethic.

Jesus’s teachings repeatedly feature a theme of reversal, particularly in the ways that we esteem ourselves and others.

“And whoever exalts himself will be humbled, and he who humbles himself will be exalted.” (Matthew 23:12)

“For whoever wishes to save his life will lose it; but whoever loses his life for My sake will find it.” (Matthew 16:25)

There is a repeated message of shifting focus away from self and onto others. Paul said to the Philippians that this was exactly Christ did at a level fundamental to the very nature of his mortal existence. He “emptied Himself”; ἑαυτὸν ἐκένωσεν (heauton ekenōsen). That verb, κενόω (kenóō), “to empty out”, is significant in Christian theology for the “kenosis” of Christ, the ‘self-emptying’ of Jesus’ own will to become entirely receptive to the will of the Father. But this isn’t only teaching about Christ’s nature. Paul is calling for the Church to do this as well, to imitate Christ in his kenosis of self. Rather than being self-centered we are to de-center ourselves from our own circle of concerns, maybe even put ourselves on the outside edge of that circle, looking in, to center the interests of others.

Another thing Jesus said was that, “whoever wishes to become great among you shall be your servant” (Matthew 20:26). This could be understood, perhaps in the first instance, as a warning. If you try to be greater than others you’ll end up being a servant instead. But I think the positive interpretation, that we ought to act as servants to others, that this is true greatness, is also consistent with Jesus’s teachings. And it’s consistent with Paul’s message to the Philippians. Christ took “the form of a bond-servant”, a δοῦλος (doulos). And I think here of the image of Jesus kneeling and washing his own disciples’s feet.

What does it look like to “not merely look out for your own personal interests, but also for the interests of others”? A few years ago in my church’s worldwide general conference one of the church’s leaders, Dallin H. Oaks, quoted Alexandr Solzhenitsyn who said: “It is time, in the West, to defend not so much human rights as human obligations.” I come back to that idea a lot. Was Solzhenitsyn saying that we shouldn’t defend human rights? Certainly not. Having been a victim of the Soviet gulags himself he would be the last person to say that. But I think he was onto something quite astute. Rights don’t carry much force without human obligations. We have to think about others. We have to think about one another.

A community has to have people who look out for the interests of others. This is what Paul wanted to see among the Philippians. He said these things would make his “joy complete” (Philippians 2:2).

The author of comparative religion Karen Armstrong has focused a lot on kenosis in her writing and it is clearly on view in the 2009 Charter for Compassion that she spearheaded. The second sentence in that Charter states that, “Compassion impels us to work tirelessly to alleviate the suffering of our fellow creatures, to dethrone ourselves from the centre of our world and put another there.”

These are ideas that impact me deeply:

“Regard one another as more important than yourselves; do not merely look out for your own personal interests, but also for the interests of others.”

“Dethrone ourselves from the centre of our world and put another there.”

“What do we owe to each other?”