# Tangent plane and Jacobian for a surface?

1. Jul 9, 2010

### 7thSon

S(u,v) ---> (x(u,v), y(u,v), z(u,v))

I could define a non-square Jacobian based on the derivatives along the coordinate curves, which will in general be non-orthogonal on the image of this mapping. Those tangent vectors will be the basis for a tangent space at some point on the surface.

I would like to construct an inner product or metric which obeys the kronecker delta in this "coordinate system" of the tangent plane. Someone on another forum said that I can get such a metric tensor by taking J(transpose)J, where J is defined by the cartesian coordinates of those tangent vectors in the columns of J. Then, I take the inverse of that.

My question is, what the heck is the basis for doing the steps above?

Also, I can multiply the 2 vectors defining the tangent plane by scalar constants and add them and span all of that tangent space. Are the scalar multipliers that allow me to span that tangent space "coordinates" with respect to basis of those tangent vectors? Or is the basis of such a vector still the standard basis, since the coordinates of the tangent vectors themselves are still being referred back to the standard R3 basis?