differential of a function $f :\mathbb{R}^2 \rightarrow \mathbb{R}^2$

Question

I struggling in finding the linear application ( the differential ) of the function above $$ f\left(x,y\right)=\left(xe^{y},ye^{x}\right) $$

When i write $f\left(a+h\right)$ with $a=\left(x,y\right)$ and $h=\left(h_1,h_2\right)$ i have \begin{align*}\displaystyle f\left(a+h\right)=f\left(x+h_1,y+h_2\right)&=\left(\left(x+h_1\right)e^{y+h_2},\left(y+h_2\right)e^{x+h_1}\right)\\ &=\left(xe^{y},ye^{x}\right)+\left(xe^{y}\left(e^{h_2}-1\right)+h_1e^{y+h_2},ye^{x}\left(e^{h_1}-1\right)+h_2e^{x+h_1}\right)\end{align*}

Which doesnt make an obvious linear application appears... Any help please ? I'm quite rusty on this subject...

Answer 1

For a function $f:\mathbb{R}^2\to \mathbb{R}^2$, $f(x,y)=(f^1(x,y),f^2(x,y))$ the derivative is given by $$ Df(x,y)=\begin{bmatrix}f^1_x&f^1_y\\ f^2_x&f^2_y \end{bmatrix} $$ So yours is $$ Df(x,y)=\begin{bmatrix}e^y&xe^y\\ ye^x&e^x \end{bmatrix} $$ Then $$ f(x+h_1,y+h_2)\approx f(x,y)+Df(x,y)(h_1,h_2)\\ =(xe^y,ye^x)+(e^yh_1+xe^yh_2,ye^xh_1+e^xh_2) $$