let us describe how the euclidean duality is carried to the tangent bundle of a surface.
We know how euclidean duality is: For each euclidean space, , the coordinated (rectangular) functions: are linear, so their derivatives, , are equal themselves i.e the gradients obey , they are dubbed and they satisfy duality:
Now, if is a parameterization of the surface, , and is a “measure” then, doing calculus in the surface means do calculus to . Let .
Let us name the coordinated (rectangular) functions on .
So by the chain rule we have: that is, in terms of gradients:
So for the functions we get and by the same rule just above
where evaluating at the basis of give
but since it is known that the generate , then both: generate , the co-tangent space at ,… wanna see a picture?