 Solution

ID: past-exam-of-the-mathematics-course-of-the-year-1-of-the-natural-sciences-course-of-the-university-of-cambridge/2016/paper-2/10/a/solution

In a multivariable Taylor series, order means total degree. The only nonzero monomial of total degree at most three is

x y

; there are no constant, linear, or cubic terms. Thus

ϕ (x, y) = x y + O ((x^{2} + y^{2})^{2})

near the origin.

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