0

I know how to compute the differential, for example, for $f(x,y)=(x^2,y^2)$, we obtain:

$$f'=\left( \begin{array}{cc} 2 x & 0 \\ 0 & 2 y \\ \end{array} \right)$$

Now I want to apply it to points in $\Bbb{R}^2$. What do I do? First choose values for $x,y$ and then apply to points in $\Bbb{R}^2$? Just looking at the books I have it's not too clear how to do it.

Gary
  • 31,845
Red Banana
  • 23,956
  • 20
  • 91
  • 192
  • 2
    It might be helpful to think about it as something like a directional derivative. Choosing the point $x=1, y=2$ and applying $f'$ to the vector $[1,0]$ and will give you the derivative of the function $f$ at the point $(1,2)$ along the vector $[1,0]$. – Volk Aug 29 '22 at 05:58

1 Answers1

1

I don't exactly understand the question, because I think you have already applied it. Take for example the Gateaux derivative. (In direction $v=(v_{1},v_{2})$)

$\displaystyle \lim_{h \to 0}\dfrac{f(x+hv_{1},y+hv_{2})-f(x,y)}{h}$.

In your example it is $(2xv_{1},2yv_{2})^{T}$ which is $\begin{bmatrix} 2x&0 \\ 0& 2y \\ \end{bmatrix}$$\begin{bmatrix} v_{1} \\v_{2} \end{bmatrix}$.

But I think you have already done that!