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 12-20-2008, 08:02 AM #1 mEmmerrr journeyman   Join Date: Oct 2008 Location: The 19th Hole Posts: 290 Vector Divergence Quick question for any vector calculus gurus: How do these differ? (vector divergence calcs.) (grad . a)b where the . is dot product obv. with (a . grad)b Given a and b as standard i j k vectors and asked to solve at a given point. Surely just solve divergence inside both brackets, which I dont think can depend on the ordering? then get the outcome, times every part of b by the result and plug in the numbers. How can this be different?
 12-20-2008, 01:26 PM #3 thylacine Pooh-Bah   Join Date: Jul 2003 Posts: 4,029 Re: Vector Divergence consider the difference between (d/dx f(x)) g(x) and f(x) d/dx g(x)
 12-20-2008, 01:54 PM #4 Matt R. veteran   Join Date: Mar 2005 Posts: 3,032 Re: Vector Divergence thylacine, Is (a . del), a is a vector, defined? I've never seen it written like that. And if it is, I would think it would be defined such that (a . del) = (del . a), so the two operations in OP would be equal, right? I don't believe (a . del) is defined, but your example threw me off. I don't quite see how they are analogous since we have two operations (div and grad) in OP, and only one (differentiation) in your example.
 12-20-2008, 02:02 PM #5 thylacine Pooh-Bah   Join Date: Jul 2003 Posts: 4,029 Re: Vector Divergence hmm, OP would need to clarify what the exact notation is.
12-20-2008, 02:51 PM   #6
LongLiveYorke
grinder

Join Date: May 2008
Posts: 527
Re: Vector Divergence

Quote:
 Originally Posted by mEmmerrr (grad . a)b where the . is dot product obv. with (a . grad)b

To me, grad . a means dax/dx + day/dy + daz/dz, which is a scalar function that can then be multiplied by the vector b to form a vector

a . grad means ax * d/dx + ay * d/dy + az * d/dz which is a scalar operator which, when acting on a vector, returns a vector.

 12-20-2008, 04:46 PM #7 mEmmerrr journeyman   Join Date: Oct 2008 Location: The 19th Hole Posts: 290 Re: Vector Divergence sorry if this has been unclear. grad is the name we gave the upside down triangle denoting the gradient function, hence TRIANGLE THING dot producted with a vector field gives you its divergence. I thought grad was standard, apologies;. SO afaik: grad . a if a = Xi + Yj + Zk = d(x)/dx + d(y)/dy + d(z)/dz
 12-20-2008, 05:26 PM #8 Matt R. veteran   Join Date: Mar 2005 Posts: 3,032 Re: Vector Divergence I think you need to clarify what you mean by (a . grad)b then. LLY's definition makes sense to me, though, and if he's right the difference b/w your two statements would be: (grad . a)b = [d/dx (ax) + d/dy (ay) + d/dz (az)]b where a = ax i + ay j + az k So it's your divergence of a (a scalar) times your vector b. (a . grad)b = (ax * d/dx + ay * d/dy + az * d/dz)b In this would you would distribute (a . grad) into each component of b. (sorry I think it would be more clear if I could use better notation) But what it boils down to is in the former you are differentiating the components of a, adding, then multiplying by b, but in the latter you are going to end up differentiating the components of b because the ordering is different (which is what thylacine was getting at).
 12-20-2008, 08:21 PM #9 Borodog Enemy of the State     Join Date: Jan 2004 Location: I Didn't Vote, Bitch Posts: 25,491 Re: Vector Divergence Matt R is correct. Calling the nabla operator "grad" is going to be deliberately confusing; it's non-standard. "Grad a" is typically shorthand for "the gradient of a", but that does not imply the nabla operator means "grad". Call the nabla operator del (as is standard) and you won't have a problem. For example, "Del dot b" is the divergence of vector b.
12-21-2008, 02:59 AM   #10
blah_blah
old hand

Join Date: Feb 2007
Posts: 1,660
Re: Vector Divergence

Quote:
 Originally Posted by Borodog Call the nabla operator del (as is standard) and you won't have a problem.
let's not generalize too much, no one says 'del' in math (i personally say grad/div or 'D', the latter being the notation Evans uses for the gradient in his graduate pde text). that being said, the notation used by the OP is bad.

12-21-2008, 10:09 AM   #11
thylacine
Pooh-Bah

Join Date: Jul 2003
Posts: 4,029
Re: Vector Divergence

Quote:
 Originally Posted by blah_blah let's not generalize too much, no one says 'del' in math (i personally say grad/div or 'D', the latter being the notation Evans uses for the gradient in his graduate pde text). that being said, the notation used by the OP is bad.

12-21-2008, 03:17 PM   #12
blah_blah
old hand

Join Date: Feb 2007
Posts: 1,660
Re: Vector Divergence

Quote:
 Originally Posted by blah_blah let's not generalize too much,
Quote:
 Originally Posted by blah_blah no one says 'del' in math

12-21-2008, 03:28 PM   #13
Borodog
Enemy of the State

Join Date: Jan 2004
Location: I Didn't Vote, Bitch
Posts: 25,491
Re: Vector Divergence

Quote:
 Originally Posted by blah_blah let's not generalize too much, no one says 'del' in math
Really? I had no idea. I guess that's only standard in physics.

Anyone know what engineers call it?

12-21-2008, 03:29 PM   #14
Borodog
Enemy of the State

Join Date: Jan 2004
Location: I Didn't Vote, Bitch
Posts: 25,491
Re: Vector Divergence

Quote:
 Originally Posted by blah_blah i've made a huge mistake.
lol

12-21-2008, 06:17 PM   #15
PtMx

Join Date: Nov 2007
Posts: 823
Re: Vector Divergence

Quote:
 Originally Posted by blah_blah i've made a huge mistake.

 12-21-2008, 07:01 PM #16 blah_blah old hand   Join Date: Feb 2007 Posts: 1,660 Re: Vector Divergence thank you for explaining the joke to everyone
 01-05-2009, 06:20 PM #17 mEmmerrr journeyman   Join Date: Oct 2008 Location: The 19th Hole Posts: 290 Re: Vector Divergence Whos ready for round 2?!?! I have looked high and low through every textbook I have and the internet, and nothing turns up. Now remembering my amateur-ness of describing my last problem, I state this exactly. Find equations of the field line C of: a=yi+xj+xyk through (1,1,1) Im sure this is really simple, but I dont recall ever using the phrase "field line" of a vector and I cannot find a definition of it ANYWHERE. (I have used italics just to highlight the vectors, as I have already used bold!) Many thanks folks.
01-05-2009, 08:22 PM   #18
jason1990
old hand

Join Date: Sep 2004
Posts: 1,876
Re: Vector Divergence

Quote:
 Originally Posted by mEmmerrr Find equations of the field line C of: a=yi+xj+xyk through (1,1,1) Im sure this is really simple, but I dont recall ever using the phrase "field line" of a vector and I cannot find a definition of it ANYWHERE.
Quote:
 Originally Posted by Wikipedia A field line is a locus that is defined by a vector field and a starting location within the field. A vector field defines a direction at all points in space; a field line may be constructed by tracing a path in the direction of the vector field. More precisely, the tangent line to the path at each point is required to be parallel to the vector field at that point.
Try r(t) = e^t i + e^t j + (0.5e^{2t} + 0.5)k.

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