I'm taking multivariable calc this fall. I began self-studying on my own a couple months ago, using Salas's calc text. Then I stumbled on Ted Shrifin's MTH3500/10 incredible lecture series on Youtube. His text, Multivariable Mathematics, arrived in the mail yesterday evening! It's a freakin' gorgeous book, and I'm super excited to start. I have three questions.
(1) Question for anyone that's worked or taught from this book: In terms of coverage, how does this book compare with something like Munkres, Calculus on Manifolds? Is there significant overlap? Will I be prepared for Munkres after reading Shifrin?
(2) I decided I had to have the book after bing-watching the lectures. The ideas there are just so lovely, and so nicely explained. The idea of linear maps is a beautiful one, and I'm amazed at how it generalizes the results of single-variable calc. Matrices, matrix multiplication, and the like can seem so unmotivated and pointless, until one sees that matrix multiplication is the algebra behind the composition of linear maps. Historically, was it the need to put multivariable calc on a sound footing that motivated the development of linear algebra?
(3) I'll be taking linear algebra in the fall, too. I wonder: Why (or how!) would anyone successfully teach linear algebra without using multivariable calculus and the geometry of linear maps to reify matrices and their symbol-shunting? I mean, Shifrin strives to show the connection between linear algebra and multivariable calc, and this is an unusual approach, right? But then how else would linear algebra be taught?
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$\begingroup$In Ted's book he teaches you about the topology on $\mathbb{R}^n$, so you will have this as a reference when looking at more abstract examples of topological spaces. The space $\mathbb{R}^n$ becomes a metric space with the euclidean metric and metric spaces are topological spaces. So yes, you will be prepared for Munkres. Generally, the only hard part out point-set topology is the set theory and extracting useful information from pictures.
The concept of merging calculus and geometry has been around for sometime. In differential geometry they study non-linear objects i.e manifolds by linear ones i.e the differential (and Hessian) which you've seen is a linear map. This is beautiful because the theory of linear maps i.e linear algebra is very developed. It was after I seen linear algebra used in other fields that I thought, "my god, this should never be taught by itself." However, it was the exploration of this subject independent of these other fields that allow us to use it's theory at great lengths.
$\textbf{PS}$: I have his book and love it as well. It also has more advanced material on differential forms which will prepare you for differential topology and differential geometry.
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