Pages similar to: Non-differentiable functions must have discontinuous partial derivatives
- A differentiable function with discontinuous partial derivatives
Illustration that discontinuous partial derivatives need not exclude a function from being differentiable. - The multidimensional differentiability theorem
Discussion of theorem that gives conditions which guarantee that a multivariable function is differentiable. - 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 derivative matrix
The derivative matrix is presented as a natural generalization of the single variable derivative to multivariable functions. - 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. - 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. - 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. - Multivariable chain rule examples
Examples demonstrating the chain rule for 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. - Special cases of the multivariable chain rule
Illustrations of different special cases of the multivariable chain rule and their relationship to the general case. - The gradient vector
The gradient vector is the matrix of partial derivatives of a scalar valued function viewed as a vector. - Newton's Method
A method to approximate the roots to an equation. - 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.