Electromagnetic Fields and Waves: Notes, Solved Problems and Interactive Lab

2.2.8 Recap

Recap

Nabla $\nabla = a_{x} \partial / \partial x + a_{y} \partial / \partial y + a_{z} \partial / \partial z$ is a vector operator. It gives the gradient $\nabla f$ (scalar $\to$ vector), the divergence $\nabla \cdot A$ (vector $\to$ scalar), the curl $\nabla \times A$ (vector $\to$ vector) and the Laplacian $\nabla^{2} f = \nabla \cdot \nabla f$ (scalar $\to$ scalar).
Gradient: $\nabla f = \frac{\partial f}{\partial x} a_{x} + \frac{\partial f}{\partial y} a_{y} + \frac{\partial f}{\partial z} a_{z}$ gives the direction and size of the fastest increase of $f$ and is normal to the level surfaces, with $df = \nabla f \cdot d ℓ$ and $df / d ℓ = \nabla f \cdot a_{ℓ}$ .
Divergence: $\nabla \cdot A = \frac{\partial A _{x}}{\partial x} + \frac{\partial A _{y}}{\partial y} + \frac{\partial A _{z}}{\partial z} = lim \oint_{S} A \cdot d S /Δ v$ : net outward flux per unit volume (sources $> 0$ , sinks $< 0$ ).
Curl: the determinant (2.2.8), and $(\nabla \times A) \cdot a_{n} = lim \oint_{C} A \cdot d ℓ /Δ s$ : circulation per unit area (the paddle wheel), right-hand rule.
Laplacian: $\nabla^{2} f = f_{xx} + f_{y y} + f_{z z}$ , and $\overset{ˉ}{f} = f + \frac{a ^{2}}{6} \nabla^{2} f$ . It is positive in a dip, negative on a bump, zero where $f$ equals the average of its neighbours.
Null identities: $\nabla \times \nabla f = 0$ , $\nabla \cdot (\nabla \times A) = 0$ .