Pages similar to: The arc length of a parametrized curve
- 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.
- 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.
- 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.
- Length of curves
An integral to find the length of a 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.
- 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.
- 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.
- 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.
- Parametrized curve arc length examples
Examples of calculating the arc length of parametrized curves.
- Alternate notation for vector line integrals
An alternative notation for the line integral of a vector field is introduced.
- Line integrals as circulation
Definition of circulation as the line integral of a vector field around a closed curve.
- Vector line integral examples
Example of calculating line integrals of vector fields.
- The idea behind Green's theorem
Introduction to Green's theorem, based on the intuition of microscopic and macroscopic circulation of a vector field.
- Using Green's theorem to find area
A trick to use Green's theorem to calculate the area of a region
- The definition of curl from line integrals
How the curl of a vector field is defined by line integrals representing circulation.
- Calculating the formula for circulation per unit area
A sketch of the proof for the formula for the component of the curl of a vector field.
- The gradient theorem for line integrals
A introduction to the gradient theorem for conservative or path-independent line integrals.
- The idea behind Stokes' theorem
Introduction to Stokes' theorem, based on the intuition of microscopic and macroscopic circulation of a vector field and illustrated by interactive graphics.