Transformation Properties of Vector and Tensor Components

Abstract

In this Chapter we wish to investigate how the components of vectors and tensors change when another coordinate system is chosen. We want to find out what happens to the components under a general transformation of the form u^{i ′} ≡ uⁱ (u¹, u², u³) for i = 1, 2, 3. We find it convenient to use primes “on” the indices for quantities related to the new system; here, u^i′, rather than putting a prime or a bar on the u symbol, such as u′ⁱ or $\bar u^{i'}$. This is for notational purposes, since the indices then will match on the RHS and LHS of the equations. This will become clear below.