CalcBLUE 2 : Ch. 12.2 : Multivariate Taylor Expansion

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What does Taylor expansion mean when you have multiple inputs? What's the formula? Ow! My eyes!
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How do you not have more subs>? Damn the quality~!!!!

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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

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