Why must the form of v_i v_j be independent of coordinate system?

[member 428835]

Hey PF!

I am trying to understand what is meant when we say a vector is invariant, which I believe is independent of a coordinate system. I have already read a PF post here: https://www.physicsforums.com/showthread.php?t=651863.

I'm looking at DH's post, and this makes a lot of sense!

However, I have read the following, which I am trying to interpret. Please read this and help me out, if you can:

Consider the single point velocity stress tensor, $v_i v_j$ where $v_i$ is the $i$th component of velocity. First rotate the coordinate system 90 degrees around the $x_1$ axis so the old $x_3$ axis becomes the new $x′_2$ axis and the old negative $x_2$ axis becomes the new $x′_3$ axis. It is easy to see $v′_2 v′_3$ in the new coordinate system must be equal to $-v_2 v_3$ in the old. But isotropy [don't worry about interpreting this] requires that the form of $v_i v_j$ be independent of coordinate system. This clearly is possible only if $v_2 v_3 = 0$.

Thanks!

 

[member 428835]

Sorry to post again, but I received an email that someone replied but I cannot view the response. Could you post again, whoever it was?

 

[electricspit]

The easiest way, cutting all algebra out of the way, is to view a vector as a geometric object. Think of it like a rigid arrow in any coordinate frame you like.

Better yet, think of a vector as... a cube. Just a geometric object. You can make infinite coordinate systems surround and envelope the cube, but the cube will never change based on the coordinate system.

Vectors, like cubes, are just geometric objects. They do not change with a choice of coordinate system. The numbers which represent the components may change, but the form won't change. Does this make sense?

 

[member 428835]

totally, this confirms my intuition. thanks!

 

[HallsofIvy]

All vectors are independent of the coordinate system- that's the whole point of vectors! A vector can be "invariant" under a given linear transformation: if for linear transformation A, Av= v the v is "invariant under A".