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I'm looking for a literature reference for Taylor polynomials of two variables with Peano's form of remainder. More specifically, I need one that uses the little-o notation in two variables. For instance, the Taylor-MacLaurin polynomial of degree $2$ of a generic two-variable real function is expressed as follows: $$ f(x,y) = f(0, 0) + f_x (0, 0) x + f_y (0, 0) y + \frac{1}{2}\left[ f_{xx} (0,0)x^2 + 2f_{xy} (0, 0)xy + f_{yy} (0, 0) y^2 \right] + o \left(x^2 +y^2 \right) $$

This topic was covered in my real analysis course (our textbook doesn't even mention it, and our lecturer is not interested in helping me), but any other resource outside of it is welcomed. I therefore ask for your help in finding a possible source that uses this notation.

I thank you in advance for your attention.

J. W. Tanner
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ncant
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    https://math.stackexchange.com/questions/4651790/looking-for-a-book-that-teaches-high-dimensional-advanced-calculus-but-not-real/4651812#4651812 – zkutch Apr 29 '25 at 00:34
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    Or go to Cartan: https://math.stackexchange.com/questions/3271948/proving-limit-of-fx-tnfx-taylor-is-zero-in-multivariable-calculus/3272515#3272515 You might also find a reference in the literature of "jets", "$k$-jets", and "jet spaces" – Eric Towers Apr 29 '25 at 01:29
  • @EricTowers Thank you – ncant Apr 30 '25 at 08:30

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