Math Insight

Pages similar to: 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.
• Newton's Method
  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.

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