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.

  • $\begingroup$ I think Thomas' Calculus has a section on setting-up integrals from a level-surface perspective. It's not all they do since that would be very non-standard. The parametric view point is just easier for curves in $\mathbb{R}^3$. Personally, I think embracing both views and learning how they are intertwined makes a stronger course. That said, I am also interested to see more things from the perspective you desire. $\endgroup$ May 18, 2018 at 14:26
  • $\begingroup$ I wouldn't call anything involving derivatives of smooth functions "algebraic geometry". Viewing graphs as a special case of level sets is standard in most presentations of vector calculus. Certainly such a perspective can be found in standard textbooks such as Apostol (chapter 8 of volume 2). $\endgroup$
    – Dan Fox
    May 18, 2018 at 20:18
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    $\begingroup$ @DanFox I'm looking for the opposite though. I want a textbook or resource that doesn't talk about the graphs of functions at all, but instead talks about these topics from the perspective of a geometric shape being the vanishing set of a polynomial/rational/analytic/smooth function. I feel like having both creates confusion/clutter. Or at least I'm suspicious it does; I haven't tried to map out a vector-calculus course from just one perspective (I'm kinda asking if someone else already has). $\endgroup$ May 20, 2018 at 1:05
  • $\begingroup$ @DanFox But yeah, I don't know if characterizing these two perspectives as the differential geometry perspective and the algebraic geometry perspective is quite right, but I couldn't quite find better words to describe the two sides. The reason I'm thinking of these being the "two sides," was that I, an algebraist, was trying to discuss some of these topics with a differential geometer, and we were having quite a lot of trouble understanding each other. :) $\endgroup$ May 20, 2018 at 1:09
  • $\begingroup$ As far as I understand, you want to focus more on curves and surfaces (understood as general "subvarieties") instead of functions (which correspond to special subvarieties in a product space). But gradients, differentials or directional derivatives are things you apply to functions, not general subvarieties, so I can't imagine how such a book would treat these topics. $\endgroup$ Feb 1, 2021 at 12:18

1 Answer 1


I think this is only a partial answer to your question.

Ideals, Varieties, and Algorithms by Cox, Little & O'Shea presents affine varieties and their parametrizations in Sections 2 and 3, respectively, of Chapter 1. This material could complement or supplement the coverage of curves and surfaces in a multivariable calculus course. Then Section 4 of Chapter 3 (Elimination Theory) introduces singular points (and envelopes), including how to find tangent lines to algebraic curves by looking for multiple roots. Perhaps it could be useful.


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