The Nabla Vector

Poor maths tutor. By the way he's going, it seems that he would fall asleep before the rest of the class does.
The syllabus is actually quite ok. One just has to get around all that stuff about the del operator (∇) which only looks intimidating. The rest is zooming in and out of the infinitesimal, as usual.


Analogy. I hate it when the notes gives such dry-as-dust statements like "the curl/divergence of a vector field F is defined by [formula]". O, how we will revel in the rote memory! Here I'll try a more helpful description on the concepts of gradient, divergence and curl.

The Mysterious Nabla vector ∇ (and applications thereof)
∇ is basically (∂/∂x)i~ + (∂/∂y)j~ + (∂/∂z)k~. It is not a vector in totally good faith, but it behaves like one and we treat it like one anyway (to hell with details).

Gradient: product of ∇ and a scalar field
Metaphor: Liken the scalar field f to altitude in a hilly area (say in Temburong).
Imagine that are a 2LT navigating your way to Lakiun Camp with your mates. At some point of time (and some point of f(x,y)) you decide to torture them by leading them directly upslope. The gradient of the terrain is a vector that points to the direction of "directly upslope".

Note: A force field which is a gradient of a scalar is called a conservative force because it "conserves mechanical energy". But why does it have to be that (F~ = ∇f) ⇔ (F~ is conservative)? This needs some looking into.

Divergence: dot product of ∇ and a vector field
Metaphor: Liken the vector field to a crowd of people moving with various velocities.
Imagine a crowded city square in Xi'an. At this instant, someone unearths a timed explosive at A, and someone else unearths an ancient national treasure at B. The crowd near point A will scramble away from A out of fear, while those near point B will scramble towards B out of curiosity. We say that the divergence at A is positive while the divergence at B is negative.

Curl: cross product of ∇ and a vector field
Metaphor: Liken the vector field to the surface of a rough sea.
Imagine going to fish in your new yacht when suddenly a freak storm arrives and your boat is send tossing and turning. The curl of the surface of the sea at (x,y) is the force that sends your boat turning (rather than tossing).

Note that curl of a conservative field (∇f) is zero. This is because
curl(∇f) is ∇×∇f.
∇ and ∇ are "parallel"
so ∇×∇ is 0
curl(∇f) is ∇×∇f, which is zero times f, which is zero.
Weird! And questionable.