Let $$F:\Bbb R^n\times\Bbb R^n\to\Bbb R$$ be the function $F(x,y)=\langle Ax,y\rangle$ where $\langle , \rangle$ denotes the standard inner product on $\Bbb R^n$ and $A$ be an $n\times n$ real matrix. If $D$ denotes the total derivative. which of the fallowing is correct?
1). $(DF(x,y))(u,v)=\langle Au,y\rangle+\langle Ax,v\rangle$
2). $(DF(x,y))(0,0)=(0,0)$
3). $DF(x,y)$ may not exist for some $(x,y)\in\Bbb R^n\times\Bbb R^n$
4). $DF(x,y)$ doesnot exist at $(x,y)=(0,0)$
I don't know, how the total derivative is defined for Inner product. So, tell me the definition for total derivative on inner product space, also tell, How to proceed further to solve this problem? Thank you
The inner product is bilinear, so you would expect that the derivative exists. The derivative exists if we can find a linear map $B:\mathbb{R}^{n} \times \mathbb{R}^n \to \mathbb{R}$ such that $$ \lvert \langle A(x+h),(y+k) \rangle - \langle Ax,y \rangle - B(h,k) \rvert = o(\lvert h\rvert+\lvert k\rvert) $$ as $h,k \to 0$. Expanding out the inner product, we find $$ \langle A(x+h),(y+k) \rangle - \langle Ax,y \rangle = \langle Ah,y \rangle + \langle Ax,k \rangle + \langle Ah,k \rangle $$ Then the absolute value in the definition can only be $o(\lvert h\rvert+\lvert k\rvert)$ when the first two terms are cancelled out (notice that $h=|h| (h/|h|)$ and so on). Therefore take $$ B(h,k) = \langle Ah,y \rangle + \langle Ax,k \rangle. $$ Now all you have to do is show whether the remainder of the left-hand side satisfies $$ \lvert \langle Ah,k \rangle \rvert = o(\lvert h\rvert+\lvert k\rvert), $$ which you can do in the same manner as showing that $\langle Ah,y \rangle=O(h)$.