Analysis · Siblings

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.

Definitions

Jacobian

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
  • $f(x, y) = x^2 + 3y$ : $Df(x, y) = \begin{bmatrix} 2x & 3 \end{bmatrix}$
    A $1 \times 2$ matrix, the transpose of the gradient.
Properties
  • The Jacobian is the matrix of the derivative $Df(x)$ in the standard bases.
  • For $m = 1$, the Jacobian is a row vector, the transpose of the gradient.
  • Chain rule: the Jacobian of $g \circ f$ at $x$ is the product of the Jacobian of $g$ at $f(x)$ and the Jacobian of $f$ at $x$.

divergence

High school

Consider a flow, like air moving in a room. The divergence of the flow measures how much it spreads out from each point (positive) or gathers in toward it (negative).

Undergraduate

Consider a vector field $F = (F_1, \dots, F_n) : \reals^n \to \reals^n$. The divergence of $F$ is the scalar function

\[\nabla \cdot F = \frac{\partial F_1}{\partial x_1} + \dots + \frac{\partial F_n}{\partial x_n}.\]

It measures how much $F$ spreads out from a point.

Properties
  • The divergence is the trace of the Jacobian matrix of $F$.
  • By the divergence theorem, the integral of $\nabla \cdot F$ over a region equals the flux of $F$ out through its boundary.

curl

High school

Consider a flow, like water moving in a pond, and imagine placing a tiny paddle wheel at a point in it. The curl of the flow at that point measures how much the flow swirls around the point: how fast the paddle wheel would spin.

Undergraduate

Consider a vector field $F = (F_1, F_2, F_3) : \reals^3 \to \reals^3$. The curl of $F$ is the vector field

\[\nabla \times F = \left( \frac{\partial F_3}{\partial y} - \frac{\partial F_2}{\partial z}, \frac{\partial F_1}{\partial z} - \frac{\partial F_3}{\partial x}, \frac{\partial F_2}{\partial x} - \frac{\partial F_1}{\partial y} \right).\]

It measures the rotation of $F$ around each point.

Missing definition: laplacian

Examples

Each cell is the ___ of the vector field or function in that row.

Example Jacobian divergence curl laplacian Why
Missing example: rotation_field_3d
Missing example: paraboloid_xyz

Notes

  • The divergence is the trace of the Jacobian, and the curl comes from its antisymmetric part.
  • The divergence of a vector field is unrelated to a divergent sequence.

See also

All distinctions · All definitions