Both Green's theorem and Stokes' theorem are higher-dimensional versions of the fundamental theorem of calculus, see how!
What we're building to
- Both Green's theorem and Stokes' theorem, as well as several other multivariable calculus results, are really just higher dimensional analogs of the fundamental theorem of calculus.
Quick review of the fundamental theorem of calculus
Remember the fundamental theorem of calculus?
Here's what it says:
In other words, when you integrate the derivative of a function on a region
of the number line, it's the same as evaluating the function itself on the boundary of that region, meaning the numbers and , and taking the difference.
Green's theorem can be seen as completely analogous to the fundamental theorem, but for two dimensions.
- Instead of taking the derivative of a single-variable function
, it involves the of a two-variable vector-valued function .
- Instead of integrating this over a region
of the number line, take its double integral over a region of the -plane.
- The boundary of the one-dimensional range
is just the pair of points and . But because is two-dimensional, its boundary is a curve .
- Instead of evaluating
at the two boundary points and and taking the difference, take the line integral of around the boundary oriented counterclockwise.
The underlying idea here is that when you integrate the "derivative" of a thing over a region, the value only depends on the value of that thing on the boundary of the region. It's just that in two dimensions, the relevant notion of a derivative is
, and the boundary of a region involves an entire curve rather than a pair of points.
Stokes' theorem takes this to three dimensions. Instead of just thinking of a flat region
on the -plane, you think of a surface living in space. This time, let represent the boundary to this surface.
- Instead of a single variable function
, or a two-dimensional vector field, is a three-dimensional vector field.
- Instead of taking the derivative
, or the , take the full-blown three-dimensional curl of .
- Instead of taking the single integral over an interval
, or a double integral in a two-dimensional region, take the surface integral over in three-dimensions. Taking the surface integral of a vector field involves dotting that vector field with unit normal vectors.
- On the right-hand side, instead of writing
, which involves evaluating on the boundary of the interval and taking the difference, we have the line integral of our function around the boundary of the surface , just as we did for Green's theorem.
The divergence theorem, covered in just a bit, is yet another version of this phenomenon. It relates the triple integral of the divergence of a three-dimensional vector field in a three-dimensional volume to the surface integral of that vector field on the boundary of that volume.
The fundamental theorem of line integrals also falls under the same overarching principle, relating the line integral of the gradient of a function to the values of that function on the bounds of the line.
In general, it seems that the universe is trying to tell us that when you integrate the "derivative" of a function within a region, where the type of integration/derivative/region/function involved might be multidimensional, you get something that just depends on the value of that function on the boundary of that region. I think this is just one of the most beautiful things in math.
The generalized Stokes' theorem
In case you are curious, pure mathematics does have a deeper theorem which captures all these theorems (and more) in a very compact formula. It is called the generalized Stokes' theorem. The language to describe it is a bit technical, involving the ideas of "differential forms" and "manifolds", so I won't go into it here. But if you understand all the examples above, you already understand the underlying intuition and beauty of this unifying theorem.
Want to join the conversation?
- I am not sure I get it or not.
if all the curl(s) in the 2d region will cancel each other out, then the curls in the 3d region will also cancel each other out. That is why taking the "line integral of the gradient of a function to the values of that function on the bounds of the line" works.(5 votes)
- Just because you mentioned "Differential Forms" and "Manifolds", I am curious to know if you have any plans to make courses on topics like Multilinear Algebra, Differential Geometry, Tensor Analysis, etc.? I would love to learn them from you (either Sal or Grant).(2 votes)