Calculus problem No.2

Calculus Level 3

Given the function f ( x , y ) = x y e − ( x 2 + y 2 ) f(x,y) = xye^{-(x^{2}+y^{2})} . Calculate:

∇ 2 f ( 2 , ln ⁡ ( 3 ) ) \nabla^{2} f(2,\ln(3))

Notation:

∇ 2 \nabla^{2} denote the Laplace operator .


The answer is 0.106261888.

Anh Khoa Nguyễn Ngọc by Anh Khoa Nguyễn Ngọc , 18, Vietnam

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1 solution

Tom Engelsman
Aug 15, 2020

We are interested in computing:

f x x + f y y = [ 4 x y ( x 2 + y 2 ) − 12 x y ] ⋅ e − ( x 2 + y 2 ) = ( 4 x 2 + 4 y 2 − 12 ) ⋅ f ( x , y ) f_{xx} + f_{yy} = [4xy(x^2 + y^2) - 12xy] \cdot e^{-(x^2+y^2)} = (4x^2 + 4y^2 - 12) \cdot f(x,y)

which at the point ( x , y ) = ( 2 , ln ⁡ ( 3 ) ) (x,y) = (2, \ln(3)) equals 0.10626 . \boxed{0.10626}.

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