We define the partial derivatives of the various components of \boldsymbol{y} as functions of x_i in terms of the so-called Jacobian matrix
\boldsymbol{J}=\frac{\partial \boldsymbol{y}}{\partial \boldsymbol{x}}=\begin{bmatrix} \frac{\partial y_0}{\partial x_0} & \frac{\partial y_0}{\partial x_1} & \frac{\partial y_0}{\partial x_2} & \dots & \dots & \frac{\partial y_0}{\partial x_{n-1}} \\ \frac{\partial y_1}{\partial x_0} & \frac{\partial y_1}{\partial x_1} & \frac{\partial y_1}{\partial x_2} & \dots & \dots & \frac{\partial y_1}{\partial x_{n-1}} \\ \frac{\partial y_2}{\partial x_0} & \frac{\partial y_2}{\partial x_1} & \frac{\partial y_2}{\partial x_2} & \dots & \dots & \frac{\partial y_2}{\partial x_{n-1}} \\ \dots & \dots & \dots & \dots & \dots & \dots \\ \dots & \dots & \dots & \dots & \dots & \dots \\ \frac{\partial y_{m-1}}{\partial x_0} & \frac{\partial y_{m-1}}{\partial x_1} & \frac{\partial y_{m-1}}{\partial x_2} & \dots & \dots & \frac{\partial y_{m-1}}{\partial x_{n-1}} \end{bmatrix},which is an m\times n matrix. If \boldsymbol{x} is a scalar, then the Jacobian is only a single-column vector, or an m\times 1 matrix. If on the other hand \boldsymbol{y} is a scalar, the Jacobian becomes a 1\times n matrix.
When this matrix is a square matrix m=n , its determinant is often referred to as the Jacobian determinant. Both the matrix and (if m=n ) the determinant are often referred to simply as the Jacobian. The Jacobian matrix represents the differential of \boldsymbol{y} at every point where the vector is differentiable.