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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.
Polar coordinates mapping
How polar coordinates can be viewed as mapping from the polar plane onto the Cartesian plane.
How linear transformations map parallelograms and parallelepipeds
Why linear transformations map parallelograms onto parallelograms and parallelepipeds onto parallelepipeds.
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.
Geometric properties of the determinant
Explanation of some of the basic geometric properties of the determinant.
Volume calculation for changing variables in triple integrals
A derivation of how a mapping that changes variables in triple integrals transforms volume.
The linear function
Overview of the linear function of one variable and a few of its properties.