Is there a resource or textbook that presents the basics of vector calculus, specifically the gradient, directional derivatives, curves and surfaces, and extrema, from a more algebraic geometry flavored perspective? I suppose I mean algebraic geometry flavored as a opposed to differential geometry flavored, which I am less familiar with.
For an example of what I mean by the two flavors, you can present some surfaces in $\mathbf{R}^3$ as the graph of a function $f(x,y)$. The tangent plane to this surface at a point $(x_0,y_0)$ is given by
$$ z = f(x_0,y_0) + \left(\frac{\partial f}{\partial x}(x_0,y_0)\right)(x-x_0) + \left(\frac{\partial f}{\partial y}(x_0,y_0)\right)(y-y_0) $$
This is how the textbook I'm using now presents it initially, but it's strikingly asymmetric because we're thinking of the surface as a graph. Instead you could think of your surface as the vanishing set of the function $p(x,y,z) = f(x,y) - z$ and the equation of our tangent plane at $(x_0,y_0,z_0)$ is much more symmetric and easy to write down in terms of the gradient:
$$ 0 = \nabla p(x_0,y_0,z_0) \cdot \langle x-x_0, y-y_0, z-z_0 \rangle $$
I'm looking for such a resource because it feels like the textbook I'm teaching from talks about things from a mix a both perspectives, and I'm suspicious that doing this leads to unnecessary confusion. I want to know if there is a complete treatment from a single perspective (preferably algebraic geometry), and to see if restricting to that one perspective leads to any problems within the scope of an undergraduate course is vector calculus.
