I have just started to (self) read Tensors, Differential forms, and Variational Principles by Lovelock and Rund.
Section 2 of chapter 1 deals with vector components in curvilinear coordinates. For spherical coordinates, the gist is that:
We have 3 surfaces: ρ = $c_1$ (or r), θ = $c_2$, and φ = $c_3$. [Sphere, cone, plane]
Their intersections form curves. [Straight line, great circle, circle]
- We define the basis vectors as those tangents to those curves:
- We have a different basis for each point in space... (Always mutually orthonormal, yet different with respect to rectangular coordinates).
My questions started coming from the fact that we can describe a vector starting from a given point $P$ with respect to the basis at $P$, using spherical components $\bar{A_i}$ rather than rectangular components $A_i$.
$\mathbf{A}=\bar{A_1}\epsilon _1 + \rho \bar{A_2}\epsilon _2+\rho \;Sin(\theta )\bar{A_3}\epsilon _3$
I went and programmed a visualization [Feel free to try it out!] using visual python, in which I can move the point $P$ and give the spherical components for a vector:
What I do not get my head around is how a different radius or angle $\theta $ can change the 2nd and 3rd basis vector and thus the vector $\mathbf{A}$ as a whole, if I set $\theta =0$:
It vanishes! (Having $\bar{A_1}=\bar{A_2}=0$ for this example). The same happens with increasing radius for the 2nd and 3rd basis vectors.
My take on this: It obviously has to do with the fact that basis vectors are tangent to those curves, so the bigger the curve, the bigger the effects on the basis is. Or something related to them being tangent to those curves.
I'm looking for any other explanation, or a proper explanation regarding tangency in my idea in case it is not properly worded.
Follow-up question: Is a vector with the same spherical components seen differently by different points in space?
Thanks for any ideas you share!
Good. You self discovered the hairy ball theorem. (*)
"If f is a continuous function that assigns a vector in R3 to every point p on a sphere such that f(p) is always tangent to the sphere at p, then there is at least one pole, a point where the field vanishes (a p such that f(p) = 0)."
In the way we view intrinsic geometry, given a point on the surface, we draw a tangent plane on this surface and use the tangent vectors at that point to span this basis. We can't really add or subtract vectors between two points on the surface unless we define a notion of connection (maybe I am wrong here, please comment if so)
*: 59:57 of this lecture by prof. schuller has a nice discussion