Pages similar to: Tangent line to parametrized curve examples
- 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. - Derivatives of parameterized curves
The derivative of the vector-valued function parameterizing a curve is shown to be a vector 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. - 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. - 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 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. - 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. - 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. - An introduction to parametrized surfaces
An introduction to how a vector-valued function of two variables can be viewed as parametrizing a surface. Interactive graphics illustrate the way in which the function maps a planar region onto a surface. - Parametrized surface examples
Examples showing how to parametrize surfaces as vector-valued functions of two variables. - Parametrization of a plane
Introduction to how one can parametrize a plane. Interactive graphics illustrate basic concepts. - Plane parametrization example
Example showing how to parametrize a plane. - Surface area of parametrized surfaces
An introduction to surface area of parametrized surfaces, illustrated by interactive graphics. - Calculation of the surface area of a parametrized surface
A calculation deriving the expression for the surface area of a parametrized surface.