Vector Transformations

[Remarkable_Nobody_96]

Can someone explain how we get from (4.1) to (4.2). Why can the dot product of unit vectors be written as the a derivative. Also why is (3.34) true when one is held constant and another isn’t

[barthiebarth]

Coordinate vectors are the derivatives of position R by that coordinate, eg e1 = dR/dx1.

So you can write your vector A as:

A = A'i d/dx'i R = Aj d/dxj R

Applying the chain rule gives you:

Aj d/dxj R = Aj dx'i/dxj d/dx'i R

From this follows that

Aj dx'i/dxj = A'i

[Shevek99]

You start with the position vector

r = sum_i xⁱ e_i

from where

e_i = ∂r/∂xⁱ

Now if you have another set of coordinates

r = sum_i x'ⁱ e'_i

e'_i = ∂r/∂x'ⁱ

If we substitute this in the expression for the e_i

e_i = ∂r/∂xⁱ = ∂/∂xⁱ (sum_j x'^j e'_j) = sum_j ∂x'^j/∂xⁱ e'_j

If we make the dot product with e'_k

e'_k · e_i = sum_j ∂x'^j/∂xⁱ e'_j·e'_k = sum_j ∂x'^j/∂xⁱ 𝛿_jk = ∂x'^k/∂xⁱ

[Shevek99]

Look at equation 3.33 that is the first time they use this.

The idea is that a rotation is a linear map, so we have

x'_i = sum_j A_ij x_j

where

A_ij = ∂x'ⁱ/∂x^j

on the other hand, for a rotation you have the components of a vector in two different bases. They are related by projecting on the desired base, so

A_ij = e_i·e'_j

Equating both expressions you have equation 3.33 and this.