Derivative of the cross and dot product

[bluelagoon]

Homework Statement

If you have two functions dependent on t, A(t) and B(t). Prove their derivatives are as follows:

d(A (dot) B) / dt = [A (dot) (dB)/(d(t)] + [d(A)/d(t) (dot) B]

{Where "(dot)" acts as the dot product}




d(A x B) / dt = [A x (dB)/(d(t)] + [d(A)/d(t) x B]

{Where "x" acts as the cross product}




I have little experience with differentiating cross products and dot products and appreciate any help in starting this one!

Thanks,
Janet

 

[nicksauce]

Well A . B = axbx + ayby + azbz (in 3 dimensions)
Differentiate that using the product rule, then regroup and show that its the expression you need it to be equal to. The cross product is similar.

 

[Biest]

I think the easy way around of using \delta_{ij} and \epsilon_{ijk} is a bit too advanced.

For the cross-product I would just say from experience that working backward may be a bit less confusing, e.g. show what two parts are and simplify the output expression, and then showing that it is equal to the derivative of the initial expression

 

[Hurkyl]

You've already proven the derivative rule for one kind of product; doesn't the same method work for these products?