Proving the Equality of Indicial Notation Using Cross Product Properties

[Mugged]

Hello, I am supposed to prove that the below is true

[$$\delta_{}im$$$$\delta_{}jn$$(em x en)]$$\bullet$$[$$\delta_{}pr$$$$\delta_{}qs$$(er x es)] = $$\delta_{}ip$$$$\delta_{}jq$$ - $$\delta_{}iq$$$$\delta_{}jp$$

where em, en,... are random unit vectors and that bullet point is supposed to be the dot product. I am supposed to consider all possible combinations of m,n,r, and s to show this.

Thank you

 

[lanedance]

first you can write everything in one test line, is this what you mean, clik on it to see how
$$\delta_{im} \delta_{jn} (e_m \times e_n) \cdot \delta_{pr} \delta_{qs} (e_r \times e_s) = \delta_{ip}\delta_{jq} -\delta_{iq}\delta_{jp}$$

now have a try simplifying right hand side using the properties of the delta function

also note they're not "random" unit vectors, the cross product should be easy to evaluate if you assume they are an orthornormal basis set (with a handedness)

 

[Mugged]

thank you lanedance for showing me how to properly do that tex code stuff, i was wondering why mine looked so funky. and yeah those are orthonormal vectors.

as for the delta property, is there one you had in mind because i still haven't a clue as to which one to use. the one i have in front of me all deal with the levi-civita symbol and i don't want to use that in the proof

 

[lanedance]

start with the defnition
$$\delta_{ij}= 0, \ \ if i=j$$

for example
$$\delta_{im} e_m = e_i$$

 

[Mugged]

im sorry i still don't quite follow

 

[lanedance]

ok what don't you follow?

 

[lanedance]

PS, I'm happy to help you work it, but am not going to do the whole thing for you

 

[Mugged]

just the first step would be a gift, please I am still really confused how i go about using $$\delta_{im} e_m = e_i$$

 

[lanedance]

ok here's first step (same as post #4)
$$\delta_{im} (e_m \times e_n) = (e_i \times e_n)$$

now apply the rest of the delta functions

 

[Mugged]

ok so after doing that to the left hand side i end up with:
$$(e_i \times e_j) \cdot (e_p \times e_q)$$

and then going in reverse to produce the deltas i want, i get:

$$\delta_{ip} \delta_{jq} (e_p \times e_q) \cdot \delta_{jp} \delta_{iq} (e_j \times e_i)$$

but i don't know how i would produce the right hand side...

 

[Dick]

Ok, start with:
$$(e_i \times e_j) \cdot (e_p \times e_q)$$
Can you give me an argument using the properties of the cross product that that is zero unless the set {i,j} is equal to the set {p,q}? And that the same is true of the right side?