Given a parametrized smooth embedded surface , where is an open subset of with coordinates , and a point , define what is meant by the tangent space at , the unit normal at , and the first fundamental form
[You need not show that your definitions are independent of the parametrization.]
The second fundamental form is defined to be
where and . Prove that the partial derivatives of (considered as a vector-valued function of ) are of the form , , where
Explain briefly the significance of the determinant .