Consider a point in cartesian and polar coordinates, P={x,y}={r,θ}, such that x=rcosθ and y=rsinθ. I have an arbitrary vector (e.g. a velocity vector) at point P, and I know it's cooridnates in cartesian coordinates, →v=aiei=a0e0+a1e1, where ei={ˆx,ˆy}, and I want to find the components in a bases that is the polar tangent vectors, e′i={ˆr,ˆθ}, i.e. →v=a′ie′i=a′0e′0+a′1e1.
(Ultimately I need to do this exercise in 3D spherical coordinates, and do a couple of coordinate rotations in between, but I think this case encompasses my [current] confusion).
Attempt:
I can define a transformation between coordinates based on, e′j=eiSij=ei∂e′j∂ei,
for example, e′0=ˆr=∂∂r=∂∂x∂x∂r+∂∂y∂y∂r.
Thus the tensor that transforms the bases {ˆx,ˆy}→{ˆr,ˆθ}, can be calculated as
Sij=(∂e′0∂e0∂e′1∂e0∂e′0∂e1∂e′1∂e1)=(∂r∂x∂θ∂x∂r∂y∂θ∂y)=(cosθ−1rsinθsinθ1rcosθ),
and the values of r,θ here are, again, fixed based on the point P. Using this, we immediately get, e′i={ˆr,ˆθ}={ˆxcosθ+ˆysinθ,−(ˆx/r)sinθ+(ˆy/r)cosθ}, which matches what I get from geometry. So this seems good.
The components of the vectors transform contravariantly, and so the transformation should be given by, Tij=∂ej/∂e′i. Going through the same procedure, I find the components of the tensor that transforms the coordinates {a0,a1}→{a′0,a′1} as,
Tij=(∂e0∂e′0∂e1∂e′0∂e0∂e′1∂e1∂e′1)=(∂x∂r∂y∂r∂x∂θ∂y∂θ)=(cosθsinθ−rsinθrcosθ).
Now if we try this, we immediately get a problem: a′i={xcosθ+ysinθ,−rxsinθ+rycosθ}. The 0th component seems fine, but the 1st has the wrong dimensions: length squared instead of dimensionless!
I also notice that while TS=I (the identity matrix), and thus T=S−1, also ST≠I, i.e. S≠T−1...
Am I missing a transpose somewhere? Or am I assuming an orthonormality somewhere where it doesn't exist? Any help or pointers (even on improper terminology / syntax) is much appreciated!
Edit: Rows-vs-columns, and left-vs-right matrix multiplication has always confused me, so I'm primarily following the indices and their positions, but I've still tried to be consistent with rows and columns. One "reason" to add a transpose, however, could be that while the inverse matrix should transform the components (as apposed to bases), the components are 'columns' while the bases 'rows', so perhaps that adds a transpose? But where does this come from in index notation?
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