# Jacobian

For a transformation, $LaTeX: y=f(x)$, such that $LaTeX: f$ is differentiable, the Jacobian is the determinant, $LaTeX: \mbox{det}\left( \nabla f(x) \right)$. Historically, the number of $LaTeX: y$-variables equals the number of $LaTeX: x$-variables, say $LaTeX: n$, so $LaTeX: \nabla f(x)$ is $LaTeX: n \times n$. The Jacobian matrix is defined as the matrix, $LaTeX: \nabla f(x)$, allowing the number of functions to be less than the number of variables.