How do I generalize the dot product (bilinear form) in spherical coordinates?

In cartesian coordinates, the unit vectors $$\{u_x, u_y, u_z\}$$ are universal. That is, $$u_x(x, y, z)$$ is constant and so on for the rest of them. Because of that, the dot product $$\langle v | w \rangle$$ is defined everywhere in $$\mathbb{R}^n$$.

But in spherical, $$u_\phi(r, \phi, \theta)$$ is not constant. It varies. So the dot product $$\langle v | w \rangle$$ is only defined when $$u$$ and $$v$$ share an origin.

But still, I kind of miss that old dot product. Maybe I can get a bilinear form that is defined everywhere in spherical coordinates, by doing something different. What do I need to do to get my universal bilinear form in spherical coordinates?

• Even In Euclidean space, you should be taking the dot product of tangent vectors at different points. What shoukd this mean? – Ted Shifrin Apr 24 at 17:14
• @TedShifrin I think I wrote my question incorrectly. I mean, at point P <v, w> is going to have one formula and at point Q != P <v, w> is going to have a differen onet. Contrast the case in cartesian coordinates where <v, w> is the same formula at all points. – ErotemeObelus Apr 24 at 17:19
• Aha. Ultimately, the fact that the sphere has nonzero curvature prevents your having a coordinate system in which the inner product $g_{ij}$ is given by a constant matrix. That can happen only on a flat (locally Euclidean) space. – Ted Shifrin Apr 25 at 5:52
• Can you talk more about this $g_{ij}$ thingie? – ErotemeObelus Apr 25 at 21:48