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Introduction to matrices
Multiplying matrices and vectors
How to multiply matrices with vectors and other matrices.
The derivative matrix
The derivative matrix is presented as a natural generalization of the single variable derivative to multivariable functions.
Matrices and determinants for multivariable calculus
Brief description of matrices and determinants, at the level needed for multivariable calculus.
Matrices and linear transformations
A description of how every matrix can be associated with a linear transformation.
The transpose of a matrix
Definition of the transpose of a matrix or a vector.
Dot product in matrix notation
How to view the dot product between two vectors as a product of matrices.
Matrix and vector multiplication examples
Examples demonstrating how to multiply matrices and vectors.
Determinants and linear transformations
A description of how a determinant describes the geometric properties of a linear transformation.
An introduction to ordinary differential equations
An introduction using simple examples explaining what an ordinary differential equation is and how one might solve them.
An introduction to networks
An overview of a network as a collection of connected elements. Different types of networks are illustrated as well as a way to represent them mathematically.
The absurd high dimensionality of random graphs
The space of random networks is so ridiculously large that one needs to take drastic measures to make it manageable.
How linear transformations map parallelograms and parallelepipeds
Why linear transformations map parallelograms onto parallelograms and parallelepipeds onto parallelepipeds.
Discrete dynamical systems as function iteration
The simplest versions of discrete dynamical systems can be viewed as evolving due to iterating a function repeatedly.
Solving linear ordinary differential equations using an integrating factor
Illustration of the procedure to find an integrating factor that allows integration of a first order linear ordinary differential equation.
Special cases of the multivariable chain rule
Illustrations of different special cases of the multivariable chain rule and their relationship to the general case.
Initial dynamical systems exploration
Exploring models of anteater population growth and lead decay in the bloodstream using interactive applets.
Introduction to the multivariable chain rule
Introduction to the multivariable chain rule. The basic concepts are illustrated through a simple example.
Introduction to changing variables in double integrals
Introduction to the concepts behind a change of variables in double integrals. The transformation is illustrated with interactive graphics.
Vectors in two- and three-dimensional Cartesian coordinates
A introduction to representing vectors using the standard Cartesian coordinate systems in the plane and in three-dimensional space.
The idea of a dynamical system
The basic concept of a dynamical system as the evolution of something over time. The fundamental ideas of the state space and temporal evolution rules are illustrated with examples featuring interactive graphics.
Introduction to matrices