notes / Math 55b / chapter 6: Analysis in Several Variables
27 Stokes’ Theorem; Introduction to Complex Analysis
· Denis Auroux
Recall: differential forms
Definition 27.1. A -form on an open subset is a function with values in :
Given and , this gives
The space of -forms on :.
Exterior product: with .
The exterior derivative is . On a convex domain in , such that (Poincaré lemma).
Pullback: if is a smooth map (,), then we have a map defined by
Concretely, denoting by coordinates on and on :, and
E.g. for and ,.
Integration of differential forms
Given , we can integrate over a -dimensional submanifold parametrized by a smooth map from a -cell to (or any other nice enough domain for integration), ,:
splitting into small grid parallelepipeds:
i.e.
Check:
for 1-forms this agrees with the path integral formula (similarly for , e.g. surface integrals in reduce to ordinary by parametrizing the surface);
for -forms on ,.
Theorem 27.2 (Pullback formula). Given a smooth map ,, and a -dimensional :
( the change of variables formula).
Example 27.3. on ; = the circle of radius , oriented counterclockwise (as a path ).
Show solution
Pulling back via (polar coordinates),
So
(independent of , or in fact the same along the image of any curve in the -plane).
Note: (by direct calculation, or using ), but such that on : if such an existed, then . This is the differential-topological way of seeing that isn’t simply connected (cohomology ).
Stokes’ theorem
For parametrized as , a -cell (or other nice domain), define = the -dimensional boundary (for , this has pieces!), with suitable orientation. (Most relevant to us:)
Theorem 27.4 (Stokes’ theorem).,
Application: if is a closed 1-form on a simply connected , the path integral is independent of the choice of the path from the base point to .
Indeed, path-independence comes from Stokes for the surface traced by a path homotopy:
So we can define for any path . Stokes again ( the fundamental theorem of calculus) gives
and we find that is exact. ( Poincaré lemma.)
Remark. Stokes’ theorem for differential forms in and specializes to all the theorems of multivariable calculus:
: the fundamental theorem of calculus for path integrals;
: Green’s theorem in , the curl theorem in ;
in : the Gauss/divergence theorem.
The most useful case for complex analysis is: for a region with boundary curve ,
Proof (Sketch proof (of Stokes’ theorem)). Both sides obey the pullback formula (using , and ), so we can do changes of coordinates / pullback by the parametrization .
We can decompose into pieces: either by writing as a sum of forms with support contained in subsets that have a single parametrization, or by observing that if with ,
then and contain with opposite orientations, and so
By cutting + change of variables, we can reduce to the case = a -cell in .
Over a -cell, consider each component of separately: e.g. and
So
using that vanishes on the other faces of ( the -plane) and the orientation convention for (which we didn’t state, but which is designed to make this work).
(Remark: the same calculation works for = the region between 2 graphs over , but this is not needed.) ∎
Complex analysis
Our next topic: complex analysis (in 1 complex variable).
We’ll study functions , open, . Writing , these are instances of functions , and the notion of continuity is the same, but we introduce a different (more restrictive) notion of differentiability.
Definition 27.5. The (complex) derivative of at (if it exists) is
(i.e. ).
The catch is: this limit has to hold for in …
Definition 27.6. We say is analytic (or holomorphic) if exists for all .
Example 27.7. Assume only takes real values, … Then in the definition the numerator is always real, so taking in we get , while taking imaginary we get . So: the complex derivative of a function which takes real values either doesn’t exist or is equal to …!
Complex vs. real differentiability
We can treat as a function of 2 real variables, . If exists then, taking real, resp. imaginary, we find:
Equivalently, writing for real-valued functions ,, this becomes
i.e. is of the form . This is the matrix of complex multiplication by , viewed as an -linear operator on .
In the language of differentials, (), a complex-valued 1-form on , can be written in terms of and as:
Then: if exists, then
Conversely! If is real differentiable at , then gives
where is linear ,. The complex derivative exists iff .
Proposition 27.8. is analytic
is differentiable and (rescale + rotate: conformal transformations)
(Cauchy–Riemann equation).
Remark. Geometrically, conformal transformations of the plane preserve angles between vectors (and orientation). So: analytic functions in 1 complex variable are conformal mappings (differentiable, in 2 real variables). If you draw a square grid in the plane and map it by , the resulting curves meet at right angles everywhere.
Example 27.9 ().
Notes not reached in this lecture, covered at the start of Lecture 28
The miracle
Even though analyticity only requires the existence of a complex derivative, it has many far-reaching consequences, which we’ll see and prove in the next few classes. Among these:
If is analytic, then it has derivatives to all orders! (Unlike the real case, where e.g. is only , not .)
The Taylor series expansion of at any point is convergent and equal to over a disc ; in particular can be expressed as a power series in ! (Unlike: has all derivatives zero at , so its Taylor series converges to , not .)
Local determination: if are analytic, is connected, and over any subset of that has a limit point (e.g. a small ball, or a small real interval, or…), then on all of !!!
…and more! But first let’s see examples and work out basic properties.
Example 27.10 (Polynomials). Polynomials : are analytic, and the complex derivative = the usual derivative (this follows from the usual rules of differentiation, which hold in the complex case too).
By contrast, a polynomial in 2 variables can be rewritten as a polynomial in (set ,):. Check:
so such a polynomial is analytic iff there are no ’s in the expression.
Example 27.11 (Rational functions). Rational functions :
(removing common factors, we assume ). This function has zeros at the , and poles at the . The order of a zero or pole is the multiplicity of the root or in or . Rational functions are analytic on their domain of definition .