syllabus.org

Functions

R^n as a vector space

functions from R^n to R^m

linear functions from R^n to R^m

the matrix of a linear function

composition of functions and matrix multiplication

applications

formula for angle sum of sine and cosine

basic image manipulation

Inner product, length, and angles

Limits and the derivative

look at graphs

define limits

**(Which deltas work? Is this limit right, if not give an epsilon which has no delta.)

compute limits

very rapid single variable calculus

definition of total derivative

a bunch of approximation questions

partial derivatives

low dimensional examples

points versus vectors

Find the tangent plane for a surface at a point.

gradient vector for maps to R, and geometry (level curves, “steepist increase”)

Chain rule and review

chain rule

computations

reprove product rule and rule for f(x)^g(x)

Review everything up til now by generalizing:

Vector spaces and linear maps (in general) Examples: spaces of polynomials, space of linear maps

basis, dimension

Inner products and norms in general, orthogonal basis (gram-schmidt)

The derivative between normed spaces

Second Derivative and Optimization

Operator norm on L(V,W)

Second derivative as a bilinear form, Hessian “Matrix”

Second order taylor expansion

Compactness (maybe just “closed and bounded”)

Find critical points of a function.

Define local maximum and local minimum.

Positive definite bilinear forms, second derivative test

Lagrange multipliers

Eigenvectors

Spectral theorem for real symmetric matricies (Use lagrange multipliers proof)

(Show connections to completing the square)

Using spectral theorem to test for positive definiteness (use computer algebra software to compute eigenvalues)

Multivariable Talyor’s theorem

nth derivatives as symmetric n-tensors

multivariable taylors theorem with remainder

analyze critical points with higher level taylor polynomials if second derivative test failed

Integration of “functions” (Really top level forms)

definition of integral

determinants

chain of variables

Integration of 1-form on a 1-chain

Observe that total derivative of a function is a one form: use this to motivate definition of general 1-form.

Prove FTC for 1-forms. Observe that “dθ” cannot be exact on the annulus.

When can we find f so that df=w? Natural question because then we can use FTC. At least need mixed partials to commute. (secretly this is just saying dw=0, but they do not know that yet.). Answer: Poincare’s lemma says that at least on simply connected domains this is always true.

Talk about how Poincare lemma gives LOCAL solution to df=w for any closed form. Global obstruction comes from going around loops.

See how dθ is “essentially” only closed 1-form on annulus up to an exact form

At least see in principle how there should be “essentially” n closed 1-forms on a region in R^2 with n holes.

(ive explicit formula for generators of the 1st de Rham cohomology group)

Note that one form level down if df = 0, f is locally constant. Number of constants needed gives number of connected componenets.

higher dimensional forms

To be able to integrate over a parameterized k-chain, we will need something that eats k tangent vectors and spits out numbers.

Motivative multilinearity and the fact that they are alternating.

Show that the n choose k wedges of dx_i’s generate all k-forms.

Define pullback of forms.

Define wedge product.

Define exterior derivative.

Observe that dw=0 exactly says w is closed using our previous definition for 1-forms.

Define hodge star operator.

(Maybe?) write laws of electromagnetism. Hodge star needs to be modified for the lorentzian metric though, so I am not sure…

Integration and Stoke’s theorem

Definition of integration

Stoke’s theorem

(Show that Stoke’s theorem gives us a more intuitive definition of exterior derivative as integral around “infinitesmal parallopipeds”)

Topological consequences

*Focus on R^3 (Anticlimax)

See how to interpret 0,1,2,3 forms as functions or vector fields in R^3

Work out the definitions of grad, curl, and div from this and defintion of exterior derivative

Work out what Stoke’s theorem says in this notation