I was paging through my old electrodynamics text and stumbled on the following in a chapter on potentials and fields.
Rho is a function of r and t, or rho(r, t)
Del rho = [d(rho)/dt] (del t)
I am sadly at a loss to understand where this last equation came from, and help would be appreciated.
When using Del, you’re talking about a 3-D surface. The purpose of Del, as I recall, is to find the gradient - or, the extrema of the surface…to the best of my knowledge.
You’ll recall Del is a calculus function involving partial derivatives. As I recall, the Del finds what is called the gradient.
If you’d like more info, I’ll crack open my Calculus book, and see if I can help you. However, this part of Calc I just accepted without trying to understand. While you are using it to define an electromagnetic field, I know it is used in topography and weather maps, for example.
I was a ME (mechanical engineer), and this has little application for my purposes - so I’m a bit rutsy here.
It looks like a vector version of the chain rule. My vector analysis background is slightly non-existent, so consider this an educated guess.
On preview, I see what Jinx said, and I think (s)he’s right on, although I don’t have my calc book. Try looking up the definition of del; if it is the gradient, then I’m pretty sure the original equation follows from the chain rule.
Like Jinx said, Del doesn’t have much use for a mechanical engineer (not the mechanical engineer making this post, anyway. I’ve long forgotten this since my emag class). I just want to step in and restate the OP’s formula taking advantage of the new Symbol font capability.
[sym]Ñr[/sym]=(d[sym]r[/sym]/dt)[sym]Ñ[/sym]t
Is that pretty much how it looked? Or should it be a partial derivative, i.e.
[sym]Ñr[/sym]=([sym]¶r[/sym]/[sym]¶[/sym]t)[sym]Ñ[/sym]t
It should definitely be a partial derivative, as [sym]r[/sym] is a function of two variables.
That’s what I figgered. Also, I missed bolding the first [sym]r[/sym], which indicates that I’m also missing a dot or cross.
I found this link while doing a search on vectors and gradients. Does it help?
On a related note, I can’t believe how much of this stuff I’ve forgotten.
Yep, that’s how it should look… how did you do that?
What I can’t figure out is where the d(rho)/dt came from. The Del operator that I’m familiar with is strictly spatial.
[d/dx x + d/dy y + d/dz z]
d should be the symbol for a partial Derivative, but I don’t know how to do that.
Actually since I don’t know how to do the hat symbol for a unit vector I should have used i, j, k
It’s me again.
There’s no dot or cross involved del(scaler) is the gradient and it’s a vector.
Thanks so far, and keep trying guys this is driving me crazy.
There may be a couple of typos. If t is time, del(t) would normally be zero unless time is a function of position. Also, the scalar function, rho, is more likely to be a function of the scalar, r, instead of a vector, r. If you meant rho(r,t), then it is true that:
del(rho)= d(rho)/dr del®
Which is similar to what your book says with r instead of t.
Well, it looks to me like you’ve just got d[sub]i[/sub][sym]r[/sym] = d/dt [sym]r[/sym] * d[sub]i[/sub]t. Isn’t that just the chain rule, with [sym]r/sym = [sym]r[/sym][t(r)]?
The symbol font is now available. See the ATMB thread http://boards.straightdope.com/sdmb/showthread.php?threadid=90639
Thanks g8rguy it was a composite function, and Thanks to bibliophage, I’ll see if I can figure out how to use that stuff.