Pages similar to: Orienting curves
- 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.
- 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.
- 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.
- 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.
- Parametrized curve arc length examples
Examples of calculating the arc length of parametrized curves.
- 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.
- Orienting surfaces
How to orient a surface by choosing a normal vector.
- A Möbius strip is not orientable
Explanation why a Möbius strip cannot be oriented by choosing a normal vector to point to one side.
- Proper orientation for Stokes' theorem
The importance of orientating the surface and its boundary correctly when using Stokes' theorem.
- Determinants and linear transformations
A description of how a determinant describes the geometric properties of a linear transformation.
- Geometric properties of the determinant
Explanation of some of the basic geometric properties of the determinant.
- Length of curves
An integral to find the length of a curve.
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