Jacobian determinant

Definition

Let $U \subseteq \mathbb{R}^n$ be open, let $f \colon U \to \mathbb{R}^n$, and let $a \in U$ be a point at which all first partial derivatives of $f$ exist. The Jacobian determinant of $f$ at $a$ is the determinant $$\det J_f(a) = \det \left( \frac{\partial f_i}{\partial x_j}(a) \right)_{1 \le i, j \le n}$$ of the Jacobian matrix of $f$ at $a$. It is also written $\dfrac{\partial (f_1, \dots, f_n)}{\partial (x_1, \dots, x_n)}(a)$, and it is often called the Jacobian of $f$.