Question

Correct me if I'm wrong--more multivariable review, because I realized that I'm progressively getting worse and worse at things I should be good at.

For f(x,y,z) being a scalar field, ∇f(x,y,z)=(∂f/∂x)i+(∂f/∂y)j+(∂f/∂z)k

The above being the gradient of a scalar field, which gives us the vector of rate of change in the direction of greatest incremental change

For r=f(x)i+f(y)j+f(z)k, ∇⋅r=(∂f(x)/∂x)+(∂f(y)/∂y)+(∂f(z)/∂z)

Determines magnitude of... uh... flux? Does this mean that a plane with a constant slope will have a divergence of 0?

Answer 1

"Everyone, including me, is just sitting here staring at the question" ~ Bahrom

Answer 2

"Sometimes, Bahrom accidentally posts twice." ~ Buddah

Answer 3

you need more physical interpretations I think
Have you studied electromagnetism at all?

Answer 4

Yes, but honestly that stuff was a while back. I took introductory physics in reverse order, and now I'm taking modern physics in normal order. And my introductory physics didn't include calculus.

Answer 5

So, in short, I'd need a refresher of electromagnetics. XD

Answer 6

Fluid dynamics might work better. :o

Answer 7

Sorry I don't know fluid mechanics as well, so first electromagnetism:
if you remember that voltage is composed of equipotential surfaces, that will help you get a concept of gradient.\[\vec E=-\nabla V(x,y,z)\]where the voltage function V is a scalar function, and the gradient is the electric field, which is a vector

Answer 8

Oh my God, that makes sense. But why is it negative?

Answer 9

because electric field goes points from high voltage to low, now the other way around
the divergence\[\text{ div } \vec F=\nabla\cdot\vec F\]rep[resents a source or a sink

Answer 10

Oh riiight. A source is where the field is "fast" then proceeds to slow? And a sink is the reverse?

Answer 11

... in fluid mechanics divergence of the vector field is proportional to how much water is flowing in or out of the system
divergence zero means the fluid has no sources and is incompressible
that means its volume doesn't change with pressure

Answer 12

it has to do with the tendency of the water to diverge from a given point, as the name suggests
water is incompressible, so the flow is assumed to be unidirectional
if the liquid were expanding it would have a positive divergence, and negative if it were contracting

Answer 13

tendency of a fluid to diverge*

Answer 14

in Maxwell's equations we have\[\nabla\cdot\vec E=\frac\rho{\epsilon_0}\]which means the flux through any closed surface is proportional to the enclosed charge, and\[\nabla\cdot\vec B=0\]which means the magnetic flux through any closed surface is zero.

Answer 15

Give me a few moments to absorb this. XD I'm having trouble reconciling 1D and 2D divergence with 4D.

Answer 16

what does 1D divergence mean, i don't know ?
I only ever see it in 3D

Answer 17

Well, I mean it in the sense of, for instance, r=(1/2)xi, where its div(r)=∇⋅r=(1/2). Because the vector is only extended in 1D, I think of it as "1D".

In this case, this means that, as one gets farther, the amount of "flow" is greater...?

Answer 18

Ugh, I'm going to get dinner. XD

Answer 19

Thanks for the help!

Answer 20

anytime
relearn gauss's law and I can help more

Answer 21

"Sometimes Badreferences thanks people for the people's help." ~ Henry Ford

Answer 22

I'll ask you when I do.