Math Insight

Pages similar to: Derivatives of parameterized curves

• Parametrized curve and derivative as location and velocity
  Description of a parametrization of a curve as the position of a particle and the derivative as the particle's velocity. Illustrated with animated graphics.
• Tangent lines to parametrized curves
  The tangent vector given by the derivative of a parametrized curve forms the basis for the equation of a line tangent to the curve.
• An introduction to parametrized curves
  An introduction to how a vector-valued function of a single variable can be viewed as parametrizing a curve. Interactive graphics illustrate the way in which the function maps an interval onto a curve.
• Tangent line to parametrized curve examples
  Examples showing how to calculate the tangent line to a parameterized curve from the derivative of the underlying vector-valued function.
• Line integrals are independent of parametrization
  Description of how the value of a line integral over a curve is independent of the parametrization of the curve.
• Orienting curves
  Orienting curves by choosing a tangent vector.
• Parametrization of a line
  Introduction to how one can parametrize a line. Interactive graphics illustrate basic concepts.
• Parametrization of a line examples
  Examples demonstrating how to calculate parametrizations of a line.
• 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.
• 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.

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