Change of variables

A substitution that rewrites a mathematical problem using a different set of variables, often simplifying its structure or computation.

Substitution: A replacement of one expression or variable with another to simplify a mathematical problem. A change of variables is a substitution applied to the variables of a problem.
U-substitution: An integration technique that replaces a composite expression with a new variable. It is the standard one-variable change of variables for simplifying integrals.
Function composition: The construction formed by applying one function to the output of another. A transformed expression is often a composition whose derivatives require careful tracking.
Linear transformation: A function between vector spaces that preserves addition and scalar multiplication. Linear substitutions have constant Jacobians, unlike general nonlinear changes of variables.
Jacobian matrix: The matrix of first derivatives of a vector-valued function. Its entries describe how a differentiable coordinate transformation changes local directions.
Integration by substitution: A method for evaluating integrals by replacing the integration variable and transforming the differential. It applies variable changes directly to antiderivatives and definite integrals.
Bijective function: A function that is both one-to-one and onto its codomain. A global change of variables typically needs a one-to-one correspondence between domains.
Affine transformation: A linear transformation followed by a translation. Affine substitutions shift or rescale coordinates without the curvature of nonlinear maps.
Jacobian determinant: The determinant of a Jacobian matrix, measuring local signed volume scaling. Its absolute value adjusts area or volume when integrating in new coordinates.
Change of variables formula: A theorem relating integrals under a differentiable transformation, including its Jacobian determinant. It gives the precise correction needed when transforming a multiple integral.

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