# Multivariable analysis

Lecturer Roland van der Veen, assistants: Ruben Ijpma and Oscar Koster Please send all your email to multivariableanalysis19@gmail.com

## Course description

The goal of this course is to explore the notions of differentiation and integration in arbitrarily many variables. The material is focused on answering two basic questions:
1) How to solve an equation? How many solutions can one expect?
2) Is there a higher dimensional analogue the fundamental theorem of calculus? Can one find a primitive?
The equations we will address are systems of non-linear equations in finitely many variables and also ordinary differential equations. The approach will be mostly theoretical, schetching a framework in which one can predict how many solutions there will be without necessarily solving the equation. The key assumption is that everything we do can locally be approximated by linear functions. In other words, everything will be differentiable. One of the main results is that the linearization of the equation predicts the number of solutions and approximates them well locally. This is known as the implicit function theorem. For ordinary differential equations we will prove a similar result on the existence and uniqueness of solutions.
To introduce the second question, recall what the fundamental theorem of calculus says that the integral of the derivative is the same as the 'integral' of the function on the boundary. What if our function depends on two or more variables? In two and three dimensions, vector calculus gives some partial answers involving div, grad, curl and the theorems of Gauss, Green and Stokes. How can one make sense of these and are there any more such theorems perhaps in higher dimensions? The key to understanding this question is to pass from functions to differential forms. In the example above this means passing from f(x) to the differential form f(x)dx. Taking the \$dx\$ part of our integrands seriously clarifies all formulas and shows the way to a general fundamental theorem of calculus that works in any dimension, known as the (generalized) Stokes theorem. All the results mentioned in this paragraph are special cases of this powerful theorem.

There are lectures on Mondays 5115.0317 and Fridays 5118.-156 and problem sessions on Tuesdays 5114.0004 and Wednesdays 5115.0317. The Tuesday problem session is for students whose surname starts with A-M and the Wednesday problem session is for students whose name starts with L-Z.

## Schedule

Date Material covered Exercises
11-11 Introduction, Linear algebra, derivative. Homework (due Monday 18-11): Problems sec 2.1: 9 and sec 2.2: 2 Solutions
12/13-11 Problem session Problems Sec 1.0: 0. Sec 2.0: 0,1,3. Sec.2.1: 2,5,8. Sec. 2.2: 0,1.
15-11 Properties of derivative, partial derivative. -
18-11 Integration. Homework (due Monday 25-11): Sec 2.3: 1 and 2. Solutions
19/20-11 Problem session Problems Sec. 2.1: 10, Sec. 2.2: 4,5,6, Sec. 2.3: 3,4.
22-11 Fubini theorem sec. 2.3, definition of Exterior algebra sec. 3.1 -
25-11 Banach lemma, mean value theorem sec. 2.4, Inverse function theorem Homework: (due Monday 2-12): Sec. 2.4: 1, Sec 2.5: 0. Sec 3.1: 4. Solutions
26/27-11 Problem session Problems Sec 3.1: 1,2,5. Sec. 2.4: 2,3,4. Sec.2.6:0
29-11 Finish proof of inverse function theorem. sec. 2.5 -
2-12 Implicit function theorem sec. 2.5 Homework: (due Monday 9-12): Sec. 2.5: 7,10.Solutions
3/4-12 Problem session Problems Sec. 2.5:1,3,4,5,6,8,11,12.
6-12 Existence of solutions to ODE, Review of course so far. -
9-12 Wedge products, differential forms and their integrals Homework (due Monday 16-12): Sec. 2.6: 9 Sec. 3.3: 4 Solutions
11-12: Extra (optional) review/catch up session. Time 11-13, location Bernoulliborg 5161.0105, Exercises
10/11-12 Problem session Problems Sec 2.6. 8,5 Sec. 3.3: 0,3.Solutions to the tutorial exercises
13-12 Integration of differential forms -
16-12 Chains and exterior derivative, statement of Stokes theorem. Homework: Participate in the discussion forum Scottish Cafe on the nestor page of this class. The more you do the more points you get. To pass the homework you should at least ask two valid questions about the Lecture notes. Answering someone else's question also gives you points.
17/18-12 Problem session Problems Sec. 2.1: 11,12. Sec. 3.2: 4, 3.4:0, Sec 3.5: 0,1.
20-12 NO CLASS -
6-1 Recap on exterior derivative Homework (due Tuesday 14-1 at the problem session): Sec. 3.4: 1, Sec. 3.5: 5,6. Solutions
7/8-1 Problem session Problems: 3.5:2,3, 3.6:0,1
10-1 Proof of Stokes theorem -
13-1 Recap and Poincare lemma Homework: No homework.
14/15-1 Problem session Problems TBA.
17-1 Questions Mock exam
17-1 Catch-up session 13:00 to 15:00 in room NB 5114.0043
24-1, 8:30 - 11:30 Written Exam (solutions now available) Location BB 5161.0165
9-4, due 23-4 Resit Exam, Tex file available here at home

## Assessment

There will be seven homework sets and a final written exam. The final grade is computed as max((H+3T)/4,T), where H is the average grade of the best 6 homework sets. Homework will be assigned every Monday and should be handed in the next Monday before class. It may also be emailed to multivariableanalysis19@gmail.com. The final exam will contain at least one exercise that was discussed in the problem sessions.

## Literature

We closely follow the lecture notes available here. Please note that the contents may be improved over time so it is a good idea to download a new copy regularly. Tex also available here.
Note that these notes are SHORT, which means you should read single every word. This is not meant to be easy and there may be things that are unclear, if so please ask and never assume it is just your problem.
For more background material the books 'calculus on manifolds' by M. Spivak and real mathematical analysis by C. Pugh are recommended.