CalcBLUE 2 : Ch. 12.2 : Multivariate Taylor Expansion

What does Taylor expansion mean when you have multiple inputs? What's the formula? Ow! My eyes!

[@RahulMadhavan]

powerful visualization!

[@bridge5189]

Prof. Ghrist, I have always been curious about knowing the 'proof' of multivariable Taylor Series, can you please let me know how it is derived?
Thank you

[@prof-g]

there are several ways to prove it; i think the most standard uses the fundamental theorem of integral calculus, iterated.

[@bridge5189]

@@prof-g In fact, I made one video ua-cam.com/video/BfBaHtS68TE/v-deo.html for single variable which starts from the fundamental theorem of calculus itself. But I couldn't translate this concept beyond single variable. Any link for multivariable Taylor series' derivation would be kindly appreciated.