I'm working through the proof on Lagrange multipliers in Multivariate Calculus and Geometry by Sean Dineen. For, reference, these are the pages which contain the proof. Page 1 Page 2
My confusion arises in the second page where the author specifies the constraint $f_i(X_1,\theta(X_1))=c_i$
A few sentences down below, he says: "Applying the chain rule to 3.3 ... " and proceeds to partially differentiate the above identity. Now, I know the matrix $A$ in the proof is the tangent vector space of the constraint manifold, and that is specifically, because partial differentiation of a parametric surface yields tangential velocity vectors. However, I'm still not able to understand how the proof hand waves a $ \nabla f_i(P)$ in there.
If I write the function $ \sigma(X_1) → (X_1, \theta(X_1)) $ From my understanding, applying the chain rule for partial differentiation with respect to $X_1$ yields:
$ D_{\sigma (X_1)}f_i(P)\cdot \frac{d\sigma(X_1)} {dX_1}=O $
How is the author deducing that the first differential becomes $\nabla f_i(P)$? More importantly, how am I expected to symbolically differentiate $ f_i $ with respect to $\sigma(X_1)$ ? Especially, since differentiation of a function in its parametric form gives a tangential vector, and not a normal vector, which is what $\nabla f_i $ implies.
From what I understand, and have checked after playing around with examples, partial differentiation of a set of parametric equations yield tangential Vectors, and partial differentiation of an equation expressed implicitly as x,y, z, etc, yields the normal vector,
Most of my issue with the last part of this proof is based on this understanding. I'd appreciate any insight or corrections to my thinking. A long form elaboration of how the chain rule is eexactly applied, and how the results fall out of its would also be enormously appreciated.
The same line of reasoning occurs in Vector Calculus, Linear Algebra, and Differential Forms by Hubbard, but the stepspresented there are even more brief, with mostly text, and almost no rigour.
Please help!