> **The Jacobian Matrix and the Jacobian** > The *Jacobian Matrix* of a $C^1$ ([[Differentiable Function|differentiable]]) [[Map]] is the matrix of partial derivatives $\partial f_i/\partial x_j$. The determinant of the matrix is called the *Jacobian* and is often denoted by: > $J = \partial(f_1, \ldots f_n)/\partial(x_1, \ldots x_n)$ Consider the mapping from the function $g$ to $g_*$ as follows: $g_*(f_1(x_1 , \ldots x_n), \ldots f_n(x_1, \ldots x_n)) = g(x_1, \ldots x_n)$ Then $ \int_M g(x_1, \ldots x_n) dx_1 \ldots dx_n = \int_N g_*(y_1, \ldots y_n) J dy_1 \ldots dy_n $ Since $g$ and $g^*$ have the same value at appropriate points, it is often said that the volume element $dx_1 \ldots dx_n$ has changed to $J dy_1 \ldots dy_n$, a useful point of view if we view the mapping $f$ as a coordinate change.