Vector Calculus: Gradient, Divergence, and Curl

Three operators, three different outputs

Gradient ( $\nabla f$ ) takes a scalar field and returns a vector field, pointing in the direction $f$ increases fastest.
Divergence ( $\nabla \cdot F$ ) takes a vector field and returns a scalar, measuring how much the field spreads outward from a point.
Curl ( $\nabla \times F$ ) takes a vector field and returns another vector, measuring the field's local rotation.

\nabla f = \left(\dfrac{\partial f}{\partial x}, \dfrac{\partial f}{\partial y}, \dfrac{\partial f}{\partial z}\right)

\nabla \cdot F = \dfrac{\partial F_1}{\partial x} + \dfrac{\partial F_2}{\partial y} + \dfrac{\partial F_3}{\partial z}

\nabla \times F = \left(\dfrac{\partial F_3}{\partial y}-\dfrac{\partial F_2}{\partial z},\; \dfrac{\partial F_1}{\partial z}-\dfrac{\partial F_3}{\partial x},\; \dfrac{\partial F_2}{\partial x}-\dfrac{\partial F_1}{\partial y}\right)

Find $\nabla f$ at the point $(1,2,1)$ for $f(x,y,z) = x^2yz^3$ .

Solution: Take each partial derivative in turn:

\nabla f = (2xyz^3,\; x^2z^3,\; 3x^2yz^2)

Evaluating at $(1,2,1)$ :

\nabla f(1,2,1) = (2(1)(2)(1),\; (1)(1),\; 3(1)(2)(1)) = (4,\,1,\,6)

Find the divergence and curl of the rotating field $F = (y,\, -x,\, 0)$ .

Divergence:

\nabla\cdot F = \dfrac{\partial}{\partial x}(y) + \dfrac{\partial}{\partial y}(-x) + \dfrac{\partial}{\partial z}(0) = 0+0+0 = 0

Zero divergence makes sense here: this field just rotates points around the origin without anything flowing in or out.

Curl:

\nabla\times F = \left(\dfrac{\partial(0)}{\partial y}-\dfrac{\partial(-x)}{\partial z},\; \dfrac{\partial(y)}{\partial z}-\dfrac{\partial(0)}{\partial x},\; \dfrac{\partial(-x)}{\partial x}-\dfrac{\partial(y)}{\partial y}\right)

= (0-0,\; 0-0,\; -1-1) = (0,\,0,\,-2)

A nonzero curl, entirely along the $z$ -axis, exactly what you'd expect from a field that rotates around that axis.

Gradient ( $\nabla f$ ) takes a scalar field and returns a vector field, pointing in the direction $f$ increases fastest.
Divergence ( $\nabla \cdot F$ ) takes a vector field and returns a scalar, measuring how much the field spreads outward from a point.
Curl ( $\nabla \times F$ ) takes a vector field and returns another vector, measuring the field's local rotation.

\nabla f = \left(\dfrac{\partial f}{\partial x}, \dfrac{\partial f}{\partial y}, \dfrac{\partial f}{\partial z}\right)

\nabla \cdot F = \dfrac{\partial F_1}{\partial x} + \dfrac{\partial F_2}{\partial y} + \dfrac{\partial F_3}{\partial z}

\nabla \times F = \left(\dfrac{\partial F_3}{\partial y}-\dfrac{\partial F_2}{\partial z},\; \dfrac{\partial F_1}{\partial z}-\dfrac{\partial F_3}{\partial x},\; \dfrac{\partial F_2}{\partial x}-\dfrac{\partial F_1}{\partial y}\right)