Differences
This shows you the differences between two versions of the page.
| Both sides previous revision Previous revision Next revision | Previous revision Last revision Both sides next revision | ||
|
linear_approximation [2014/02/20 18:29] nikolaj |
linear_approximation [2014/03/21 11:11] 127.0.0.1 external edit |
||
|---|---|---|---|
| Line 1: | Line 1: | ||
| ===== Linear approximation ===== | ===== Linear approximation ===== | ||
| ==== Set ==== | ==== Set ==== | ||
| - | | @#88DDEE: $X,Y$ ... Banach spaces with topology | | + | | @#55CCEE: context | @#55CCEE: $X,Y$ ... Banach spaces with topology | |
| - | | @#88DDEE: $\mathcal O$ ... open in $X$ | | + | | @#55CCEE: context | @#55CCEE: $\mathcal O$ ... open in $X$ | |
| - | | @#88DDEE: $x\in\mathcal O$ | | + | | @#55CCEE: context | @#55CCEE: $x\in\mathcal O$ | |
| - | | @#88DDEE: $f:\mathcal O\to Y$ | | + | | @#55CCEE: context | @#55CCEE: $f:\mathcal O\to Y$ | |
| - | | @#FFBB00: $J_x^f$ | | + | | @#FFBB00: definiendum | @#FFBB00: $J_x^f$ | |
| - | | @#55EE55: $J_x^f$ ... bounded linear operator from $X$ to $Y$ | | + | | @#55EE55: postulate | @#55EE55: $J_x^f$ ... bounded linear operator from $X$ to $Y$ | |
| - | | @#55EE55: $\mathrm{lim}_{h\to 0}\ \Vert f(x+h)-f(x)-J_x^f(h)\Vert / \Vert h\Vert\ =\ 0$ | | + | | @#55EE55: postulate | @#55EE55: $\mathrm{lim}_{h\to 0}\ \Vert f(x+h)-f(x)-J_x^f(h)\Vert / \Vert h\Vert\ =\ 0$ | |
| ==== Discussion ==== | ==== Discussion ==== | ||
| Line 45: | Line 45: | ||
| * The field $\mathbb C$ is $\mathbb R^2$ as a vector space. So to compute the linear approximation of a function $u(x,y)+i v(x,y)$, where $x,y$ and the functions $u,v$ are real, we must only identify it with the vector field $\langle u(x,y),v(x,y)\rangle$. | * The field $\mathbb C$ is $\mathbb R^2$ as a vector space. So to compute the linear approximation of a function $u(x,y)+i v(x,y)$, where $x,y$ and the functions $u,v$ are real, we must only identify it with the vector field $\langle u(x,y),v(x,y)\rangle$. | ||
| - | There is extensive discussion on this in [[holomorphic function]]. | + | Much more on this in the article [[holomorphic function]]. |
| === Reference === | === Reference === | ||
