Expansion of the Position Vector in Curvilinear coordinates

Expansion of the Position Vector in Curvilinear coordinates PhL 9.6.12 The error: "Comment Appendix. In tensor doc, I do mention polars as an example on page 166, but I never make a connection between vθ and θ. I was there just showing the various notations of a vector, use of italics, and all that stuff. So let's examine the connection right here. We are using primes-removed notation. r = rnen = rer + θeθ + zez = r + θ r + z " Correction of this error: The above is wrong because in cylindrical coordinates we really have r = r + z so what was I thinking? I was just making a parallel between vector r and vector v. So here is the v stuff v = vnen = vθeθ + vrer v = vnn = vθθ + vrr = vθ + vr v = vn = vx + vy The corresponding r stuff would be r = rnen = rθeθ + rrer = rrer r = rnn = rθθ + rrr = rθ + rr = rr r = rn = rx + ry = x + y So the main point is that rθ = rθ = 0 for polar coordinates. rθ = rθ = 0 // correct rθ = rθ = θ // totally wrong Negative Theorem: The components of the position vector r in curvilinear coordinates are NOT simply the curvilinear coordinates. We write things like r = r,θ,φ in spherical coordinates, but that does not mean that r = r + θ + φ . You can write r = rr + rθ + rφ if you want, but rθ = 0 and rφ = 0. The confusion arises because for a general vector v = vr, vθ, vφ you do have v = vr + vθ + vφ and in general all three components are non-zero. The position vector is always a special case. Where is this in tensor doc? Maybe I have failed to bring out this fact in tensor doc and that is why I fell into this hole. My first mention that relates I guess would be in the expansion section 6 (f) or 7 (s). In my expansion sections, I show the expansion of a vector V, but in general r = x is not a vector as I state many times. That is, the position "vector" x does not transform as a vector under F as a general rule. Does this preclude writing an expansion for x ? I never address this topic! Well, it gets some mention on page 65 of tensor doc in the Comments. My implication is that V can be any N-tuple in the expansion and the expansion is valid! I do say that directly. I later show that this is valid for expansions on en, not just on un. So the point is that you have to actually compute the dot products in the expansion of the position "vector" in any given curvilinear system. V = V'1 e1 + V'2 e2 +... = Σn V'n en where En V = V'n En = g'ni ei A few lines later I do the unit vector stuff V = V'11 + V'22 +... = Σn V'n n where En V = V'n = V'n Later in Section 7 I state that V = V'1e1 + V'2e2 +... = Σn V'n en where en V = V'n en = g'ni ei so this seems to say that V'n = en V = g'ni ei V Eventually you have to go compute ei V for your vector V and if V = x, for vector x. I am looking for a general formula here valid for any curvilinear system. How about this V'n = RnmVm and even though x is not a real vector, according to my notes we still define x'n = Rnmxm x'n = hnx'n x'n = hn Rnmxm // these are the unit vector components of the position vector in any system! In polar coordinates we have 1,2 = θ,r R = S-1 = hθ = r hr = 1 = = = But, -sinθ x + cosθ y = -sinθ rcosθ + cosθ rsinθ = 0 cosθ x + sinθ y = cosθrcosθ + sinθ rsinθ = r so we find that = = = so this is now we discover that values for rr and rθ. So the general formula is this: x'n = hn Rnmxm but more likely you would know all this from the way the curvilinear coordinates are defined! That is to say, for sphericals, you would know that r = r without doing all this work.