Pages similar to: How linear transformations map parallelograms and parallelepipeds
- Determinants and linear transformations
A description of how a determinant describes the geometric properties of a linear transformation. - Matrices and linear transformations
A description of how every matrix can be associated with a linear transformation. - Linear transformations
Definition of a linear transformation of Euclidean spaces. - Geometric properties of the determinant
Explanation of some of the basic geometric properties of the determinant. - Polar coordinates mapping
How polar coordinates can be viewed as mapping from the polar plane onto the Cartesian plane. - 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. - Matrices and determinants for multivariable calculus
Brief description of matrices and determinants, at the level needed for multivariable calculus. - The definition of differentiability in higher dimensions
The definition of differentiability for multivariable functions. Informal derivation designed to give intuition behind the condition for a function to be differentiable. - 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. - The derivative matrix
The derivative matrix is presented as a natural generalization of the single variable derivative to multivariable functions. - Area calculation for changing variables in double integrals
A derivation of how a mapping that changes variables in double integrals transforms area. - Spherical coordinates
Illustration of spherical coordinates with interactive graphics. - Cartesian coordinates
Illustration of Cartesian coordinates in two and three dimensions. - Lines (and other items in Analytic Geometry)
How to describe the equation for a line in terms of horizontal and vertical coordinates. - Polar coordinates
Illustration of polar coordinates with interactive graphics. - Cylindrical coordinates
Illustration of cylindrical coordinates with interactive graphics. - The elliptic paraboloid
Description of the elliptic paraboloid with interactive graphics that illustrate cross sections and the effect of changing parameters. - The hyperbolic paraboloid
Description of the hyperbolic paraboloid with interactive graphics that illustrate cross sections and the effect of changing parameters. - The ellipsoid
Description of the ellipsoid with interactive graphics that illustrate cross sections and the effect of changing parameters.