Transformation of vector components

[kent davidge]

The components of a vector $v$ are related in two coordinate systems via $v'^\mu = \frac{\partial x'^\mu}{\partial x^\sigma}v^\sigma$. When evaluating this at a specific $x'(x_0) \equiv x'_0$, how should we proceed? $v'^\mu(x'_0) = \frac{\partial x'^\mu}{\partial x^\sigma}(x_0)v^\sigma(x_0)$ or $v'^\mu(x'_0) = (\frac{\partial x'^\mu}{\partial x^\sigma}v^\sigma)(x_0)$?

That is, should we first work out the sum of the functions and then evaluate the product? Or can we evaluate each separately?

 

[anuttarasammyak]

I do not see difference. Do they differ ?

 

[kent davidge]

I do not see difference. Do they differ ?

Not in general, I think. However you might get into trouble if the Jacobian $\frac{\partial x'^\mu}{\partial x^\nu}$ or the components $v^\sigma$ have some weird behaviour at $x_0$, but so that yet their combination is a function that is fine there.

 

[anuttarasammyak]

I interpret the two,
$$F(x_0)G(x_o)=[F(x)G(x)]_{x=x_0}$$
The both share weirdness if any.