Unifying concepts in vector calculus

[Gza]

I'm a week or two away fromt the end of my vector calculus class and we are covering topics like surface integration, green's theorem and such. The problem I'm having so far is that everything just seems so disjointed and ad hoc, with all these theorems I feel I have to memorize instead of understanding how they fit into the bigger scheme of mathematics. Can anyone explain what are some of the unifying principals of this field that will help me to remember what it's all about a few months from now? I haven't covered stokes theorem or divergence theorem, so a general idea of what they are about would be useful too.

 

[mathwonk]

Green's , Stokes, and Divergence theorems are all essentially the same idea, just in different dimensions. for that matter so is the fundamental theorem of calculus, of which these are just higher dimensional versions.

Namely you are relating what goes on in the interior of a region to what goes on on the boundary of that region. In the fundamnetal theorem of calculus, you relate the integral of df over an interval, to the values of f at the endpoints of that interval.

Lets go up one dimension. Say you are monitoring the situation inside a country but you are not allowed to actually enter that country. You could still count how many people cross the border of that country, positively if they leave, and negatively iff they enter the country. That would measure the total outflow from that country. Presumably in normal times the same number roughly would enter as leave. If lots more people leave than enter in a given year, you would deduce that somehting in that country is forcing them out, that there is some sort of explosion inside that country.

Or say you were watching ants cross a circle drawn on the ground. If the ants are just crossing that portion of ground, the same number would enter the circle on one side as leave on the other. This could be measured by just looking at the circle. If there is an anthill inside the circle, with ants pouring out of it, there could be a total outflow of ants at every point of the circle.

If you are monitoring the amount of pollution crossing the boundaries of a town, and there is a positive amount flowing out at every border of the town, or if just the average amount leaving is positive overall, you could conclude there is a pollution source inside the town.

So these theorems just say there is a precise relationship between the flow of a vector field (or covector field) across a loop bordering a two dimensional region, (or across a surface bordering a three dimensional region, or across an n-1 dimensional manifold bordering an n dimensional region), and the behavior of some other covector field summed over the interior of the region.

The precise relationship is that the integral of the covector field taken over the boundary of a region, equals the integral of the exterior derivative of that covector field, taken over the interior of the region. This only works if the covector field can actually be defined in the interior of the region, so you can compute the derivative there.

Purely mathematically, this is very powerful, as it gives you a form of induction, the most powerful proof technique we have. I.e. something happening in n dimensions can sometimes be measured by a related quantity occurring only in n-1 dimensions.

Anytime you can reduce the number of dimensions you are dealing with, you have made your problem easier.


Here is an example of using Green's theorem to prove the fundamental theorem of algebra. Consider the covector field measuring change in angle with respect to the origin, "dtheta". Integrating this around a closed path measures the number of times the path encircles the origin counterclockwise.

When written out in x,y, coordinates dtheta becomes

-y/(x^2+y^2) dx + x/(x^2+y^2) dy.

This shows that it is not defined at the origin, (0,0), since the denominator would vanish there. Thus we can apply Greens theorem to this form if and only if the origin is not inside the loop we are intergrating over.

If the origin is not inside the loop, then Greens theorem says the integral of dtheta over the loop equals the exterior derivative of this form over the plane region interior to the loop.

Now a slightly tedious calculation, ultimately reducing to the equality of mixed partials, shows that the exterior derivative of this "form" (covector field) equals zero. Thus its integral over the interior of the loop is also zero. Hence by Greens the integral of dtheta over the loop is also zero.

We have proved that if a loop does not have the origin in its interior, then the loop does not wind around the origin. , i.e. then the winding number of the loop about the origin is zero. This is analogous to saying that if a circle on the ground does not enclose the opening of an anthill, then the number of ants walking out of the circle minus the number walking into the circle should average out to zero.

Now it is not too hard to show that if a complex polynomial of positive degree n, is applied to a large enough circle, then the image circle does in fact wind around the origin exactly n times counterclockwise, (counted properly).

Hence by greens theorem, the interior of the image circle must contain the origin. But the interior of the image circle is the image under the polynomial of the interior of the source circle. hence some point is mapped onto the origin by the polynomial. I.e. there is a point z with f(z) = 0, which proves the fundamental theorem of algebra.


As to proving these stokes type theorems, they are all proved in the same way, induction plus the fundamental theorem of calculus in one variable.

I.e. the usual FTC says that the integral of f' over an interval equals the difference in values of f at the end points.

Now consider a rectangle, and sweep across it by a moving interval. Then by the "fubini" theorem, or repeated integration, the integral of f' over the rectangle can be computed from the family of integrals of f' over the moving intervals. But each of these by the usual FTC is computed from f at the end points. Now these endpoints sweep out boundary edges of the rectangle.

hence the integral of f' over the interior of the rectangle equals the integral of f over the edges of the rectangle.

Now you will see that only two opposite edges of the rectangle occur in this description, but a covector field has the form pdx + qdy, and the pdx part works over the edges parallel to the x-axis and the qdy part works over the twp edges parallel to the y axis.

In three dimensions, you integrate f' over the interior of a cube, via integrals over the faces of the cube. Again this is done by repated integration to reduce to the lower dimension and then FTC.

One can use an argument analogous to that above to show that on an even dimensional sphere, such as the usual 2 sphere, every vector field tangent to the sphere must have the zero vector somewhere, by integrating the covector field "domega", which measures solid angle change.

The argument is a little subtler since it does not reduce only to whether something is zero or non zero, but the harder question of whether an integral is positive or negative.

These theorems are ultimately the reason for the connection between topologya nd calculus, i.e. differential topology. if you have been told you should learn topology, a very important first step is to study greens and stokes theorems, as they are ways of using calculus to measure the holes in a space, i.e. topology of the space.

does this help?

a good book with this point of view is the recent text on advanced calculus by Theodore Shifrin.

 

[Gza]

does this help?

wow, most definitely. The idea that Green's Stokes' and Divergence theorems are all essentially the same idea in different dimensions is a beautiful idea. Thanks so much for the input mathwonk!

 

[mathwonk]

you are very welcome. it is wonderful to know that an answer has been useful.