Pages similar to: Introduction to a line integral of a scalar-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. - 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. - 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. - 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 integrals of multivariable calculus
A summary of the integrals of multivariable calculus, including calculation methods and their relationship to the fundamental theorems of vector calculus. - 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. - Introduction to double integrals
The definition of a double integral is motivated through a hair density example. - Double integrals as iterated integrals
A description of how to convert double integrals into two single integrals. - Double integral examples
Examples of integrating double integrals over rectangles and triangles. - Double integrals as volume
Explanation of how double integrals could be used to represent volume. - Examples of changing the order of integration in double integrals
Examples illustrating how to change the order of integration (or reverse the order of integration) in double integrals. - Double integrals as area
Explanation of how double integrals could be used to represent area. - Double integrals where one integration order is easier
Illustration of a region where integrating a double integral in one order is must easier than integrating in the other order.