Pages similar to: Geometric properties of the determinant
- Determinants and linear transformations
A description of how a determinant describes the geometric properties of a linear transformation. - Matrices and determinants for multivariable calculus
Brief description of matrices and determinants, at the level needed for multivariable calculus. - How linear transformations map parallelograms and parallelepipeds
Why linear transformations map parallelograms onto parallelograms and parallelepipeds onto parallelepipeds. - 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. - Matrices and linear transformations
A description of how every matrix can be associated with a linear transformation. - Translation, rescaling, and reflection
An illustration using interactive graphics how to translate, rescale, and reflect graphs of functions of two variables. - Linear transformations
Definition of a linear transformation of Euclidean spaces. - 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. - 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. - Area calculation for changing variables in double integrals
A derivation of how a mapping that changes variables in double integrals transforms area. - Triple integral change of variables story
Story illustrating the process of changing variables in triple integrals. - 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. - Orienting curves
Orienting curves by choosing a tangent vector. - The multidimensional differentiability theorem
Discussion of theorem that gives conditions which guarantee that a multivariable function is differentiable.