Explaining Coordinate Rotation in Arfken & Weber Chapter 1

[sams]

In Mathematical Methods for Physicists, 6^th Edition, by Arfken and Weber, Chapter 1 Vector Analysis, pages 8-9, the authors make the following statement:

"If A_x and A_y transform in the same way as x and y, the components of the general two-dimensional coordinate vector r, they are the components of a vector A. If A_x and A_y do not show this form invariance (also called covariance) when the coordinates are rotated, they do not form a vector."

I understand how to use equations (1.9) and their derivations, but could anyone please explain the above statement?

Thank you so much for your help...

 

[Stephen Tashi]

In Mathematical Methods for Physicists, 6^th Edition, by Arfken and Weber,

For those interested, this edition of the book is available online as a PDF - perhaps legally?

Chapter 1 Vector Analysis, pages 8-9, the authors make the following statement:

"If A_x and A_y transform in the same way as x and y, the components of the general two-dimensional coordinate vector r, they are the components of a vector A. If A_x and A_y do not show this form invariance (also called covariance) when the coordinates are rotated, they do not form a vector."

I find that passage to be confusing. Here's my guess at what it means: Suppose some physical phenomenon is assigned cartesian coordinates ##(A_x,A_y)## and has coordinates ##(A'_x, A'_y)## in a rotated coordinate system. Is the physical phenomenon a vector? The passage says that if ##(A'_x, A'_y)## can be computed from ##(A_x,A_y)## in the same way we would compute the new coordinates for a geometric point ##(A_x,A_y)## in a rotated coordinate system then the phenomenon is a vector.

For that passage to have significance, you must be able to imagine that there physical phenomenon described by two numbers ##(A_x,A_y)## whose coordinates in a rotated coordinate system cannot be computed by imagining ##(A_x,A_y)## to be the cartesian coordinates of a point and computing the new coordinates as we would compute the new coordinates for a geometric point.

Examples the authors give for such phenomena are "elastic constants" and "the index of refraction in isotropic crystals". Perhaps experts on those topics can elaborate.

If there is a phenomenon with a "magnitude and direction", it is tempting to think that it must be a vector and that it can be represented as an arrow from the origin of a cartesian coordinate system to some point in the coordinate system. The book says such a representation doesn't work for some phenomena.