Derivative of function R^2 to R^2

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Derivative of function R^2 to R^2

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Multivariable Calculus

functional equation

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#1 h Saturday at 7:09 AM

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Give a function $f:\mathbb{R}^2 \to \mathbb{R}^2: f(x,y)=(x^2+xy,y^2+x)$ . Calculate the first and second derivative of the function at the point $(1,-1)$ .

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#2 h Saturday at 3:38 PM

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In the point $(x,y)$ we have $Df\in L(R^2,R^2)$ and $Df=\begin{pmatrix} 2x+y& x\\1&2y\end{pmatrix}$ . To simplify notation we identify $M_2(R)$ with $R^4$ and then $Df=(2x+y,x,1,2y)$ . Then $D^2f=\begin{pmatrix} 2&1\\1&0\\0&0\\0&2\end{pmatrix}$ . To be exact, it is the linear map with $(1,0)\rightarrow \begin{pmatrix} 2&1\\0&0\end{pmatrix}$ and $(0,1)\rightarrow \begin{pmatrix} 1&0\\0&2\end{pmatrix}$ .

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