Pages similar to: Determinants and linear transformations
- Geometric properties of the determinant
Explanation of some of the basic geometric properties of the determinant. - Matrices and determinants for multivariable calculus
Brief description of matrices and determinants, at the level needed for multivariable calculus. - Linear transformations
Definition of a linear transformation of Euclidean spaces. - Area calculation for changing variables in double integrals
A derivation of how a mapping that changes variables in double integrals transforms area. - Length, area, and volume factors
A summary of the expansion factors that arise from mappings in integrals of multivariable calculus. - Polar coordinates mapping
How polar coordinates can be viewed as mapping from the polar plane onto the Cartesian plane. - Volume calculation for changing variables in triple integrals
A derivation of how a mapping that changes variables in triple integrals transforms volume. - How linear transformations map parallelograms and parallelepipeds
Why linear transformations map parallelograms onto parallelograms and parallelepipeds onto parallelepipeds. - Matrices and linear transformations
A description of how every matrix can be associated with a linear transformation. - The relationship between determinants and area or volume
The properties of the cross product and the scalar triple product give links between determinants and area or volume. - Translation, rescaling, and reflection
An illustration using interactive graphics how to translate, rescale, and reflect graphs of functions of two variables. - 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. - Multivariable chain rule examples
Examples demonstrating the chain rule for multivariable functions. - Double integrals as volume
Explanation of how double integrals could be used to represent volume. - Double integrals as area
Explanation of how double integrals could be used to represent area. - Introduction to triple integrals
An introduction to the definition of triple integrals as well as their formulation as iterated integrals. - Triple integral examples
Examples showing how to calculate triple integrals, including setting up the region of integration and changing the order of integration. - Using Green's theorem to find area
A trick to use Green's theorem to calculate the area of a region