Pages similar to: Parametrized curve and derivative as location and velocity
- Derivatives of parameterized curves
The derivative of the vector-valued function parameterizing a curve is shown to be a vector tangent to the curve. - 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.