I came across the dot product in polar, cylindrical, and spherical coordinates, today. After checking they were equivalent to the Cartesian versions, I started wondering how one would figure them out without resorting to conversion to Cartesian coordinates. Of course, one could use the fact that $\langle a,b\rangle =|a||b|\cos(\theta)$, IF one knew some convenient formula for the angle between two vectors in whatever coordinate system they were considering. But what if one were working in an unfamiliar coordinate system — say elliptical coordinates or bipolar cylindrical or something even more exotic — and didn't know a formula for that angle off the top of their head? Is there a general way to proceed in finding the formula for the dot product in curvilinear coordinates without converting them first to Cartesian coordinates?
[Math] Dot product in Curvilinear Coordinate Systems
coordinate systemsgeometryinner-products
Related Solutions
A couple points:
- the coordinate chart (or inversely a patch) does not approximately describe the manifold. It is exactly on the manifold. For example, at the top of the unit-sphere in $\mathbb{R}^3$ near the point $(0,0,1)$ it is true that $z=1$ locally approximates the sphere (it is the tangent plane), however, it is certainly not true that $\Phi(x,y) = (x,y,1)$ provides a patch of the sphere near $(0,0,1)$. We could use $\Psi ( x, y) = (x,y, \sqrt{1-x^2-y^2} )$ as the image of $\Psi$ is on the sphere.
- there are abstract examples of manifolds formed by sets of matrices, or projective spaces. Such examples have points which are not even in $\mathbb{R}^n$, thus, without some fine print, it is clearly impossible to use coordinates in $\mathbb{R}^n$ as coordinates for such manifolds. But...
- item 2. is not quite as imposing as it appears because it is usually possible to find a model of the abstract manifold which fits inside some copy of $\mathbb{R}^n$. Moreover, Whitney's Embedding Theorem and Nash's Embedding Theorem show that we can find a set $S$ inside $\mathbb{R}^k$ for $k$ sufficiently large to represent an abstract $n$-dimensional manifold $\mathcal{M}$ in such a way that $S$ has the same structure as $\mathcal{M}$. That structure could involve the metric, or just the topology, it depends on the type of manifold and theorem we wish to invoke. I point you to the links.
- what is distance on a manifold ? This would seem to be part of your current confusion. For a given point-set, there are multiple structures we can place. For example, the plane can be given a metric which gives it spherical or hyperbolic geometry (angles add up to more or less than 180 degrees in a triangle). Of course, those metrics are not induced from the ambient Euclidean metric in $\mathbb{R}^2$. Likewise, for manifolds, the metric need not be induced from the metric on the larger space on which it is embedded. We develop a theory of geometry for Riemannian manifolds which is completely based on the abstract structure of the manifold itself. The intrinsic geometry of a manifold is independent of the details of its embedding. This is a bit of a mind-bender when you first come across the idea. In the classical differential geometry we have some mixture of intrinsic and extrinsic quantities for surfaces. For example, the mean curvature is extrinsic whereas the Gaussian curvature is intrinsic.
After doing so more intense googling I found my answer by the addition of the phrase affine transformation.
The answer mathematically can be found here
Accordingly my first assumption was incorrect, it can be done with only two, therefore all my other assumptions are incorrect.
Best Answer
When the coordinates are orthogonal (as are all of your examples), the following works.
For each coordinate do the following: fix the other coordinates and consider the curve obtained by changing the chosen coordinate. Then find a unit vector tangent to this curve.
The chosen vectors will form an orthonormal basis for the space, adapted to the particular curvilinear coordinates. (Key words: moving frame.) Then you can find the dot product of two vectors by expanding them in this basis and using the standard formula $\sum a_i b_i$.
Example: for polar coordinates, the basis consists of unit vectors $\hat e_r$, pointing away from the origin, and $\hat e_\theta$, pointing at right angle to $\hat e_r$. Given two vectors $\vec u = u_1 \hat e_r+ u_2\hat e_\theta$ and $\vec v = v_1 \hat e_r+ v_2\hat e_\theta$, we can compute $\vec u\cdot \vec v = \sum_{i=1}^2 u_i v_i$.