vector analysis
I was going through an Electrodynamics textbook, and as a prerequisite it requires elemenets of Vector calculus and Multivariable calculus. They discussed divergence, and gave examples of fields with positive and negative divergence. And they also gave a graphic example of a vector field where all the vectors are equal and parallel to each other as a field with 0 divergence.
However, to me it seems that this should have a positive divergence, not 0. Can someone explain this to me?
The complete answer is given on pages 22-27 of my 2011 vector calculus notes. I think many good calculus text include these heuristic arguments, I found them in Thomas' calculus a few editions back. Long story short, what you should really do to understand is to prove Greene's and Stokes' Theorems, this will give you deeper insight into the nature of your question. Let me summarize the method here:
All of this said, I really would rather give the less helpful answer that $d$ is the natural exterior derivative operation on the exterior algebra of $\mathbb{R}^3$ and it is just the case that: 1. $df = \omega_{\nabla f}$, 2. $d\omega_{\vec{F}} = \Phi_{\nabla \times \vec{F}}$ and 3. $d \Phi_{\vec{G}} = \nabla \cdot G dx \wedge dy \wedge dz$
where $\omega_{\langle a,b,c \rangle} = adx+bdy+cdz$ and $\Phi_{\langle a,b,c \rangle} = ady \wedge dz+bdz \wedge dz+cdx \wedge dy$. Therefore, gradient, curl and divergence are just different levels of the cohomological operator which ultimately reveals the deeper shape of space.
You've had some complex analysis, so you know what a harmonic function is. Take the gradient of any harmonic function. They also have harmonic functions in three dimensions, same example.
You said not to do that. Life is tough.
Two dimensional, we can take harmonic function $x^2-y^2,$ which is the real part of $(x+yi)^2,$ to get vector field $$ (2x, -2y). $$ This has divergence zero and "curl" (as used in Green's Theorem) zero. It really is the curl, we just write it as a scalar.
No more difficult in three dimensions, we may take function $x^2 + y^2 - 2 z^2,$ giving vector field $$ (2x, 2y,-4z). $$ Again, zero divergence and zero curl.
Best Answer
The divergence of a vector field at a point is the net flow generated by a vector field into (or out of) a small region around the point. If all the vectors of the field are parallel, then in any small region, there is just as much flow inwards as outwards, so the net flow is 0.