+- Southperry.net (https://www.southperry.net)
+-- Forum: Social (https://www.southperry.net/forumdisplay.php?fid=14)
+--- Forum: Rubik's Cube (https://www.southperry.net/forumdisplay.php?fid=58)
+--- Thread: Anyone good with index notation and vector calculus? (/showthread.php?tid=1806)
Anyone good with index notation and vector calculus? - Wani - 2008-07-30
Okay, I'm studying the kronecker delta and the alternating tensor. (As in e[123] = e[231] = e[312] = 1, e[321] = e[132] = e[213] = -1, e[ijk] = 0 otherwise).
I have to prove that e[ijk]e[rsk] = δ[ir]δ[js] - δ[is]δ[jr] using the identity (This is gonna be hard to write
e[ijk]e[rst] =
| δ[ir] δ[is] δ[it] |
| δ[jr] δ[js] δ[jt] |
| δ[kr] δ[ks] δ[kt] |
But every time I go through it, I wind up with the signs backwards, ie e[ijk]e[rsk] = δ[is]δ[jr] - δ[ir]δ[js]
Anyone have any idea what I might be doing wrong?
Anyone good with index notation and vector calculus? - Stereo - 2008-07-30
| δ[ir] δ[is] δ[ik] |
| δ[jr] δ[js] δ[jk] |
| δ[kr] δ[ks] δ[kk] |
+δ[ir].
| δ[js] δ[jk] |
| δ[ks] 1.... |
- δ[is].
| δ[jr] δ[jk] |
| δ[kr] 1.... |
+ δ[ik].
| δ[jr] δ[js] |
| δ[kr] δ[ks] |
δ[ir].δ[js] - δ[ir].δ[jk].δ[ks]
- δ[is].δ[jr] + δ[is].δ[jk].δ[kr]
+ δ[ik].δ[jr].δ[ks] - δ[ik].δ[js].δ[kr]
I guess you wnt wrong just expanding the determinant? I'm pretty sure this is right.
edit: no code tags?
Anyone good with index notation and vector calculus? - Wani - 2008-07-31
Umm.... no, if you simplify what you got down, you got the exact same thing as me.
Everything but the last line cancels out:
δ[ik].δ[jr].δ[ks] - δ[ik].δ[js].δ[kr]
=+ δ[is].δ[jr] - δ[ir].δ[js]
Anyone good with index notation and vector calculus? - Worthyness - 2008-07-31
Holy Crap that looks ridiculous 0_o
Looks like a bunch of random symbols...
Mind telling me what they're used for so that i can read about it? It all looks so... Interesting.
Anyone good with index notation and vector calculus? - Wani - 2008-07-31
lol
The alternating tensor, e, is basically used because it's perfectly anti-symmetric, as in e[ijk] = -e[jik], e[ijk] = -e[kji], e[ijk] = -e[ikj], which has a lot of useful applications in things like cross products of vectors, probably more that I haven't learn yet.
The kronecker delta, δ is basically the identity matrix, if you've studied matrices before. Basically, it acts like 1. δ[ij] is 1 if i=j, and 0 anywhere else.
Anyone good with index notation and vector calculus? - Kalovale - 2008-07-31
Worthyness Wrote:Holy Crap that looks ridiculous 0_o
Looks like a bunch of random symbols...
Mind telling me what they're used for so that i can read about it? It all looks so... Interesting.
He introduced it already >.>
Kronecker's Delta, antisymmetric tensor
Hope you have fun, I didn't
Anyone good with index notation and vector calculus? - Wani - 2008-07-31
Aha! I figured what I did wrong!
Stereo, δ[kk] isn't 1, it's the trace of I. Since this is in 3 dimensions, it's 1 + 1 + 1 = 3.
| δ[ir] δ[is] δ[ik] |
| δ[jr] δ[js] δ[jk] |
| δ[kr] δ[ks] δ[kk] |
+δ[ir].
| δ[js] δ[jk] |
| δ[ks] 3.... |
- δ[is].
| δ[jr] δ[jk] |
| δ[kr] 3.... |
+ δ[ik].
| δ[jr] δ[js] |
| δ[kr] δ[ks] |
3.δ[ir].δ[js] - δ[ir].δ[jk].δ[ks]
- 3.δ[is].δ[jr] + δ[is].δ[jk].δ[kr]
+ δ[ik].δ[jr].δ[ks] - δ[ik].δ[js].δ[kr]
= δ[ir]δ[js] - δ[is]δ[jr]
As required.