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The derivative matrix
Subtleties of differentiability in higher dimensions
A description of some of the tricky ways where a function of multiple variables can fail to be differentiable. Example two variable functions are illustrated with interactive graphics.
The definition of differentiability in higher dimensions
The definition of differentiability for multivariable functions. Informal derivation designed to give intuition behind the condition for a function to be differentiable.
An introduction to the directional derivative and the gradient
Interactive graphics about a mountain range illustrate the concepts behind the directional derivative and the gradient of scalar-valued functions of two variables.
The multidimensional differentiability theorem
Discussion of theorem that gives conditions which guarantee that a multivariable function is differentiable.
Non-differentiable functions must have discontinuous partial derivatives
A visual tour demonstrating discontinuous partial derivatives of a non-differentiable function, as required by the differentiability theorem.
A differentiable function with discontinuous partial derivatives
Illustration that discontinuous partial derivatives need not exclude a function from being differentiable.
Matrices and linear transformations
A description of how every matrix can be associated with a linear transformation.
Introduction to partial derivatives
The concept of partial derivatives is introduced with an illustration of heating costs. Interactive graphics demonstrate the properties of partial derivatives.
Partial derivative examples
Examples of how to calculate partial derivatives.
Partial derivative by limit definition
Description with example of how to calculate the partial derivative from its limit definition.
Introduction to differentiability in higher dimensions
An introduction to the basic concept of the differentiability of a function of multiple variables. Discussion centers around the existence of a tangent plane to a function of two variables.
Examples of calculating the derivative
Examples showing how to calculate the derivative and linear approximation of multivariable functions.
Derivation of the directional derivative and the gradient
Derivation of the directional derivative and the gradient from the definition of differentiability of scalar-valued multivariable functions.
Introduction to Taylor's theorem for multivariable functions
Development of Taylor's polynomial for functions of many variables.
Elementary partial derivative problems
Sample problems illustrating the partial derivative.
Solutions to elementary partial derivative problems
Solutions to sample problems illustrating the partial derivative.
Matrices and determinants for multivariable calculus
Brief description of matrices and determinants, at the level needed for multivariable calculus.
Multiplying matrices and vectors
How to multiply matrices with vectors and other matrices.
Matrix and vector multiplication examples
Examples demonstrating how to multiply matrices and vectors.
Introduction to matrices
A brief introduction to matrices.
The derivative matrix