"I have made a great discovery in mathematics; I have suppressed the summation sign every time that the summation must be made over an index which occurs twice..." Albert Einstein (Kollros 1956; Pais 1982, p. 216).

Now that we have a good dual basis for linear functionals on our tangent space, we can use it to express an important functional - the gradient of a function defined on our surface.

The tangent space and the metric lets us do basic calculus at a point on our surface: vector displacements with the metric telling us distances and angles.

The next step is to "do calculus" with scalar functions on our surface. At a point on our surface, we can approximate a scalar function with a Linear Functional These are what we are going to need: