Jacobian

In vector calculus, the Jacobian is shorthand for either the Jacobian matrix or its determinant, the Jacobian determinant.

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Also, in algebraic geometry the Jacobian of a curve means the Jacobian variety: a group structure, which can be imposed on the curve.

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They are all named after the mathematician Carl Gustav Jacobi; the term "Jacobian" may be pronounced as "yah-KO-bee-un".

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The Jacobian matrix is the matrix of all first-order partial derivatives of a vector-valued function. Its importance lies in the fact that it represents the best linear approximation to a differentiable function near a given point. In this sense, the Jacobian is akin to a derivative of a multivariate function.

Vector calculus: Vector calculus is a field of mathematics concerned with multivariate real analysis of vectors in two or more dimensions. It consists of a suite of formulas and problem solving techniques very useful for engineering and physics....

Determinant: In linear algebra, a determinant is a function depending on n that associates a scalar det(A) to every n×n square matrix A. The fundamental geometric meaning of a determinant is as the scale factor for volume when A is regarded as a linear transformation. Determinants are important both in cal...

Algebraic geometry: Algebraic geometry is a branch of mathematics which, as the name suggests, combines abstract algebra, especially commutative algebra, with geometry. It can be seen as the study of solution sets of systems of algebraic equations. When there is more than one variable, geometric considerations enter, a...