Jacobian matrix
High school
The Jacobian of a function with several inputs and several outputs is the table of all of its partial derivatives, with one row per output and one column per input.
Undergraduate
The Jacobian of $f : \reals^n \to \reals^m$ is the $m \times n$ matrix of partial derivatives, whose entry in row $i$ and column $j$ is $\frac{\partial f_i}{\partial x_j}$.
Examples
Worked examples
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$f(x, y, z) = x^2 + y^2$
: $Df(x, y, z) = \begin{bmatrix} 2x & 2y & 0 \end{bmatrix}$
Undergraduate
The Laplacian of $f$ is the sum of its second partial derivatives, $2 + 2 + 0 = 4$, and its Jacobian is the row of its first partial derivatives.
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$f(x, y) = x^2 + 3y$
: $Df(x, y) = \begin{bmatrix} 2x & 3 \end{bmatrix}$
Undergraduate
A $1 \times 2$ matrix, the transpose of the gradient.
Properties
Notes
“Jacobian” sometimes means the Jacobian determinant (when $m = n$), as in the change-of-variables formula. See Jacobian: matrix vs. determinant.
Distinctions involving this term
- Types of multivariable derivatives: The directional derivative, the Jacobian, the gradient, the differential, and the Lie derivative all package the partial derivatives of a function differently: as a vector (gradient), a matrix (Jacobian), a linear map (differential), or a single number in a chosen direction (directional derivative) or along a vector field (Lie derivative).
- Types of derivatives in vector calculus: A function from $\reals^n$ to $\reals^m$ has several derivatives in vector calculus, each combining its partial derivatives differently. The Jacobian of a vector field contains all of its first partial derivatives. The divergence (a scalar) and curl (a vector) summarize parts of it. The Laplacian is a second-order operator on scalar functions.