eth-janishutz/eth-summaries
1
0
Fork 0
mirror of https://github.com/janishutz/eth-summaries.git synced 2026-03-14 10:50:05 +01:00
Code Issues Packages Projects Releases Wiki Activity

[Analysis] Update various sections

Browse Source 
Improve legibility, add some more remarks
This commit is contained in:
Janis Hutz
2026-01-29 18:25:56 +01:00
parent 05ac7b7568
commit 237e2720e4
6 changed files with 13 additions and 10 deletions
Download Patch File Download Diff File Expand all files Collapse all files

6
semester3/analysis-ii/cheat-sheet-jh/parts/vectors/differentiation/02_differential.tex
View File

@@ -47,12 +47,12 @@ The Jacobi matrix is $J_{g \circ f}(x_0) = J_g(f(x_0)) J_f(x_0)$ (RHS is a matri
% ────────────────────────────────────────────────────────────────────
\stepLabelNumber{all}
\compactdef{Directional derivative} $f$ has a directional derivative $w \in \R^m$ in the direction of $v \in \R^n$,
if the function $g$ defined on the set $I = \{ t \in \R : x_0 + tv \in X \}$ by $g(t) = f(x_0 + tv)$ has a derivative at $t = 0$ and is equal to $w$
if the function $g$ defined on the set $I = \{ t \in \R : x_0 + tv \in X \}$ by $g(t) = f(x_0 + tv)$ has a derivative at $t = 0$ and is equal to $w$\\
% ────────────────────────────────────────────────────────────────────
\shortremark Because $X$ is open, the set $I$ contains an open interval $]-\delta, \delta[$ for some $\delta > 0$.
\shortremark Because $X$ is open, the set $I$ contains an open interval $]-\delta, \delta[$ for some $\delta > 0$.\\
% ────────────────────────────────────────────────────────────────────
\shortproposition Let $f$ as previously be differentiable. Then for any $x \in X$ and non-zero $v \in \R^n$,
$f$ has a directional derivative at $x_0$ in the direction of $v$, given by$\dx f(x_0)(v)$
$f$ has a directional derivative at $x_0$ in the direction of $v$, given by$\dx f(x_0)(v)$\\
% ────────────────────────────────────────────────────────────────────
\shortremark The values of the above directional derivative are linear with respect to the vector $v$.
Suppose we know the dir. der. $w_1$ and $w_2$ in directions $v_1$ and $v_2$, then the directional derivative in direction $v_1 + v_2$ is $w_1 + w_2$