Determinant of the Hessian

Calculus Level 2

What is the determinant of the Hessian matrix corresponding to the function

f ( x , y ) = 1 2 ( x 2 + y 2 ) 2 ? f(x,y) = \frac12 \big(x^2+y^2\big)^2 ?

4 4 12 ( x 2 + y 2 ) 2 12\big(x^2+y^2\big)^2 x 2 + y 2 x^2+y^2 x 4 + y 4 − 4 x y x^4+y^4 - 4xy

Matt DeCross by Matt DeCross , 27, USA

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1 solution

Matt DeCross
May 10, 2016

The vector of first derivatives is:

D f = ( 2 x ( x 2 + y 2 ) 2 y ( x 2 + y 2 ) ) . Df = \begin{pmatrix} 2x(x^2 + y^2) & 2y (x^2 + y^2) \end{pmatrix}.

The Hessian matrix is thus:

H = ( 2 y 2 + 6 x 2 4 x y 4 x y 2 x 2 + 6 y 2 ) . H = \begin{pmatrix} 2y^2 + 6x^2 & 4xy \\ 4xy & 2x^2 + 6y^2 \end{pmatrix}.

and so its determinant is:

det ⁡ H = 12 x 4 + 24 x 2 y 2 + 12 y 4 = 12 ( x 2 + y 2 ) 2 \begin{aligned} \det H &= 12x^4 + 24 x^2 y^2 + 12 y^4 \\ &= 12(x^2 + y^2)^2 \end{aligned}

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