# Lectures in TMA4110 - Calculus 3, fall 2012

Below is a description of what I intend to cover in each lecture and slides from the lectures. I will try to make slides without transitions and without the examples written out, available before the lectures so you can print them and follow along during the lectures. Slides with the examples written out will be made available after the lectures. If I find any mistakes on the slides during the lectures, I will fix these mistakes before I upload the slides with the examples written out, but I will not changes the slide I already have made available on the web page.

## Lecture 1

I will give a short introduction to the course and to the course web page. I will then introduce the complex numbers and tell how they can be represented as points in the plane, how we add and multiply complex numbers and tell what complex conjugation is. Pages ix-xiv in the book.

## Lecture 2

I will give a review of what we looked at in lecture 1 and then tell about polar representation of complex numbers, de Moivre's Theorem, root of complex numbers, roots of unity and solutions to second degree equations. Pages ix-xviii and xxiv-xxvi in the book.

## Lecture 3

I will start by giving a short review of what we did in the first week. We will then look at solutions of second degree equation. After that, we will look at complex functions, in particular the complex exponential function. Pages xix-xx and xxiii-xxvi in the book.

## Lecture 4

We now start on the part about second-order linear differential equations. I will go through the definitions of second-order differential equations and second-order linear differential equations, define when a second-order linear differential equation is homogeneous or inhomogeneous, and present the existence and uniqueness results for second-order linear differential equations. We will then look at linear combinations of solutions, the Wronskian of two solutions and fundamental set of solutions. We will skip the part about the vibrating spring for now. Pages xxxv-xxxvi and xxxix-xliv in the book.

## Lecture 5

I will start with a review of what we did last time. We will then look at the general solution for second-order homogeneous linear differential equations with constant coefficients. Pages xlix-lv in the book.

## Lecture 6

I will start with a review of what we did last time. We will then first look at harmonic motions, and then inhomogeneous second-order linear differential equations and the method of underdetermined coefficients. Pages xxxvi-xxxix and lv-lxxi in the book.

## Lecture 7

I will start with a review of what we did last time. We will then first look at forced harmonic motions, and then the method of variation of parameters. Pages lxxii-lxxxiii in the book.

## Lecture 8

We now begin on the linear algebra part. In this lecture we will study systems of linear equations and how to solve them, and introduce matrices, row reduction and echelon forms (Sections 1.1-1.2).

## Lecture 9

In this lecture we shall study vector equations and matrix equations (Sections 1.3-1.4).

## Lecture 10

In this lecture I shall introduce homogeneous and nonhomegeneous equations, write solution sets in parametric vector form and look at applications of linear systems (Sections 1.5-1.6).

## Lecture 11

In this lecture we shall define linear independence and introduce linear transformations (Sections 1.7-1.8).

## Lecture 12

In this lecture we shall look at the standard matrix of a linear transformation and some applications of linear models (Sections 1.9-1.10).

## Lecture 13

In this lecture we shall see how we can add and multiply matrices, and then we shall define and study invertible matrices (Sections 2.1-2.2).

## Lecture 14

In this lecture we shall study the inverse of a matrix, look at methods for determining when a matrix is invertible, and for finding the inverse, and finally at the inverse matrix theorem (Sections 2.2-2.3).

## Lecture 15

In this lecture we shall introduce and study determinants (Sections 3.1-3.2).

## Lecture 16

In this lecture we shall look at Cramer's rule, give a formula for the inverse of an invertible matrix, and look at the relationship between areas, volumes and determinants (Section 3.3).

## Lecture 17

In this lecture we shall shall introduce and study vector spaces, subspaces, null spaces and column spaces (Sections 4.1-4.2).

## Lecture 18

In this lecture we shall define linear independence and linear dependence in arbitrary vector spaces, and introduce and study bases for subspaces of vector spaces (Section 4.3).

## Lecture 19

In this lecture we shall further study bases of vector spaces, and we shall look at coordinate systems and the dimension of a vector space (Sections 4.4-4.5).

## Lecture 20

In this lecture we shall further study the dimension of a vector space, and we shall introduce and study the row space and the rank of a matrix, and briefly look at Markov chains and steady-state vectors (Sections 4.5-4.6 and 4.9).

## Lecture 21

In this lecture we shall introduce and study eigenvectors, eigenvalues, eigenspaces and the characteristic polynomial of a square matrix (Sections 5.1-5.2).

## Lecture 22

In this lecture we shall look at diagonalizable matrices and real matrices with complex eigenvalues (Sections 5.3 and 5.5).

## Lecture 23

In this lecture we shall look at systems of first order differential equations (Section 5.7 and Section 4.2 in Polking).

## Lecture 24

In this lecture we shall look at the inner product, the length of a vector, orthogonality and orthogonal sets in \(\mathbb{R}^n\) (Sections 6.1-6.2).

## Lecture 25

In this lecture we shall look at orthogonal projections, the Gram-Schmidt process and QR factorization (Sections 6.3-6.4).

## Lecture 26

In this lecture we shall look at least-squares problems and applications to linear models (Sections 6.5-6.6).

## Lecture 27

In this lecture we shall look at symmetric matrices and quadratic forms (Sections 7.1-7.2).