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The gradient vector
Multivariable chain rule examples
Examples demonstrating the chain rule for 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.
Directional derivative and gradient examples
Examples of calculating the directional derivative and the gradient.
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.
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.
Special cases of the multivariable chain rule
Illustrations of different special cases of the multivariable chain rule and their relationship to the general case.
A method to approximate the roots to an equation.
Matrices and determinants for multivariable calculus
Brief description of matrices and determinants, at the level needed for multivariable calculus.
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.
The multivariable linear approximation
Introduction to the linear approximation in multivariable calculus and why it might be useful.
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.
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.
Introduction to the multivariable chain rule
Introduction to the multivariable chain rule. The basic concepts are illustrated through a simple example.
The derivative matrix
The derivative matrix is presented as a natural generalization of the single variable derivative to multivariable functions.
The gradient vector