### Pages similar to: Parametrized curve arc length examples

- The arc length of a parametrized curve

Introduction to the arc length of a parametrized curve. The arc length definition is illustrated with interactive graphics. - Introduction to a line integral of a scalar-valued function

Introduction with interactive graphics illustrating the line integral of a scalar-valued function and informally deriving the formula for calculating the integral from the parametrization of the curve. - Introduction to a line integral of a vector field

The concepts behind the line integral of a vector field along a curve are illustrated by interactive graphics representing the work done on a magnetic particle. The graphics motivate the formula for the line integral. - 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. - Examples of scalar line integrals

Examples demonstrating how to calculate line integrals of scalar-valued functions. - 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. - 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. - 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 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. - 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. - Orienting curves

Orienting curves by choosing a tangent vector. - Length of curves

An integral to find the length of a curve. - Vector line integral examples

Example of calculating line integrals of vector fields. - A simple example of using the gradient theorem

An example of using the gradient theorem to calculate the line integral of a conservative, or path-independent, vector field. - Introduction to changing variables in double integrals

Introduction to the concepts behind a change of variables in double integrals. The transformation is illustrated with interactive graphics. - Surface area of parametrized surfaces

An introduction to surface area of parametrized surfaces, illustrated by interactive graphics. - 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. - An introduction to conservative vector fields

An introduction to the concept of path-independent or conservative vector fields, illustrated by interactive graphics. - Double integral change of variable examples

Examples of calculating double integrals through changing variables. - An introduction to ordinary differential equations

An introduction using simple examples explaining what an ordinary differential equation is and how one might solve them.