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The Taylor Expansion of Multivariate Function

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Given two position vectors x and h, Taylor's formula up to order two can be written as:

(See attachment)

a) Write down Taylor's formula in two variables (x,y) with h = (h,k)^T, using Di to denote partial derivatives.
b) State the conditions that partial derivatives commute, namely, D1D2f = D2D1f (See attachment)
c) Given f(x,y) = cos(x^2 + y) show by direct calculation that D1D2f = D2D1f

See attachment for complete question.

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Solution Summary

The solution shows how to write the multivariate Taylor series expansion in a concise manner and then explains the steps to expand the function cos(x^2+y) up to the third order.