From Tensor Analysis
5.9 The Principal Curvatures of a Surface
Let us discuss in more detail the properties of a surface connected with its second fundamental form. We now denote the coordinates without indices, using u=u 1, v=u 2. We also use subscripts u, v to indicate partial differentiation:
Hence the second fundamental form is
where
Consider the first differential of n at a point P:
Since n is a unit vector its differential d n is orthogonal to n and thus lies in the osculating plane to the surface at P. We know that the differential d r= r u du+ r v dv also lies in the plane osculating at P. Let us consider the relation between the differentials d r and d n with respect to the variables du, dv now considered as independent variables. It is clearly a linear correspondence d r 
d n. Thus it defines a tensor A in two-dimensional space such that
This tensor is completely defined by its values n u= A r u and n v= A r v.
Lemma 5.1
The tensor A is symmetric.
Proof. It is enough to establish the equality
for a pair of linearly independent vectors ( x 1, x 2). To show symmetry of A consider
Similarly
The symmetry of A follows from the identity
this is derived by...
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