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.