Pages similar to: Matrices and determinants for multivariable calculus
- Determinants and linear transformations
A description of how a determinant describes the geometric properties of a linear transformation. - Geometric properties of the determinant
Explanation of some of the basic geometric properties of the determinant. - 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. - Multiplying matrices and vectors
How to multiply matrices with vectors and other matrices. - Matrix and vector multiplication examples
Examples demonstrating how to multiply matrices and vectors. - Introduction to matrices
A brief introduction to matrices. - The transpose of a matrix
Definition of the transpose of a matrix or a vector. - Dot product in matrix notation
How to view the dot product between two vectors as a product of matrices. - Matrices and linear transformations
A description of how every matrix can be associated with a linear transformation. - 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. - The multidimensional differentiability theorem
Discussion of theorem that gives conditions which guarantee that a multivariable function is differentiable. - Non-differentiable functions must have discontinuous partial derivatives
A visual tour demonstrating discontinuous partial derivatives of a non-differentiable function, as required by the differentiability theorem. - A differentiable function with discontinuous partial derivatives
Illustration that discontinuous partial derivatives need not exclude a function from being differentiable. - Special cases of the multivariable chain rule
Illustrations of different special cases of the multivariable chain rule and their relationship to the general case. - The gradient vector
The gradient vector is the matrix of partial derivatives of a scalar valued function viewed as a vector.