Gradient

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The gradient of f(x,y) = −(cos²x + cos²y)² as a projected vector field on the bottom plane. (source: commons.wikimedia.org)

The gradient vector (or gradient) of a differentiable function $f : R^{2} \to R$ is the vector-valued function $\nabla f : R^{2} \to R^{2}$ defined as $\nabla f = (\frac{\partial f}{\partial x}, \frac{\partial f}{\partial y})$ , where $x, y$ are the variables of the function $f$ .
- $\nabla f (p_{0}) = (\frac{\partial f}{\partial x} (p_{0}), \frac{\partial f}{\partial y} (p_{0}))$ is the gradient vector at the point $p_{0}$
- $\frac{\partial f}{\partial v} (p_{0}) = \nabla f (p_{0}) \cdot v$
- $\frac{\partial f}{\partial u ^} (p_{0}) = \nabla f (p_{0}) \cdot \hat{u} = ∥\nabla f (p_{0}) ∥ \cdot cos θ$ , where $\hat{u}$ is a unit vector and $θ$ is the angle between $\nabla f (p_{0})$ and $\hat{u}$ .
  - $\frac{\partial f}{\partial u ^} (p_{0})$ is maximized when $\hat{u}$ is in the direction of $\nabla f (p_{0})$ , minimized when $\hat{u}$ is in the opposite direction, and zero when $\hat{u} ⊥ \nabla f (p_{0})$ .

Divergence

The divergence of a continuously differentiable vector field $F = F_{x} i + F_{y} j + F_{z} k$ is defined as the scalar-valued function $div F = \nabla \cdot F = (\frac{\partial}{\partial x}, \frac{\partial}{\partial y}, \frac{\partial}{\partial z}) \cdot (F_{x}, F_{y}, F_{z}) = \frac{\partial F _{x}}{\partial x} + \frac{\partial F _{y}}{\partial y} + \frac{\partial F _{z}}{\partial z}$

The curl of a vector field $F = F_{x} i + F_{y} j$ is defined as the vector-valued function $curl (F) = \nabla \times F = (\frac{\partial F _{y}}{\partial x} - \frac{\partial F _{x}}{\partial y}) k$ .
The curl of a vector field $F = F_{x} i + F_{y} j + F_{z} k$ is defined as the vector-valued function $curl (F) = \nabla \times F = (\frac{\partial F _{z}}{\partial y} - \frac{\partial F _{y}}{\partial z}) i + (\frac{\partial F _{x}}{\partial z} - \frac{\partial F _{z}}{\partial x}) j + (\frac{\partial F _{y}}{\partial x} - \frac{\partial F _{x}}{\partial y}) k$ .