Analysis II (Lecture)

Hold by Prof. Reinhard Racke at theUniKonstanz during the2nd_Semester. Here follows a recap of all chapters relevant for the exam.

Recap

9

9.3 Taylorreihen

• $f:[a,b]\to \mathbb{R}$, there is a $c\in (a,b)$ s.t. $f(b)={\sum }_{j=0}^n\frac{f^{(j)}(a)}{j!}(b-a{)}^j+\frac{f^{(n+1)}(c)}{(n+1)!}(b-a{)}^{n+1}$ .12
• .13
• $f\in C^{\infty }([a,b],\mathbb{R})$ s.t. ${\exists }_{c>0}{\forall }_{n\in \mathbb{N}}{\forall }_{c\in [a,b]}|f^{(n)}(x)|\le c^n$ $\Rightarrow {\sup }_{x\in [a,b)}{\sup }_{|h|\le h_0,x\pm h\in [a,b]}|R_n(x,h)|\to 0\text{ for }n\to \infty$.
• for $f\in C^{\infty }(U(x),\mathbb{R})$: $f(x+h)={\sum }_{j=0}^{\infty }\frac{f^{(j)}(x)}{j!}(h{)}^j$ .16 such func. are called analytisch .18

9.4 Weierstraßsche Approximation

• for $f\in C([a,b],\mathbb{R})$: ex. $P_n$ s.t. $|f(x)-P_n(x)|<ϵ$, coeff. $a_j^{(n)}$ dep. on $n$ which dep. on $ϵ$ .21
• .22
• $(P_{n}f)(x):={\sum }_{j=0}^n\left(\begin{array}{c}n\\ j\end{array}\right)x^j(1-x{)}^{n-j}f(\frac{j}{n})$ is the formula to find each $a_j^{(n)}$ given $n$ .23
• Cor: the system of Polynomials is dense in $C(J,\mathbb{R})$ respect to $||\cdot |{|}_{\infty }$ .23
• Also: $T_n(x)=a_0+{\sum }_{j=1}^n(a_j\cos (jx)+b_j\sin (jx))$, the trig. sum is dense in $C(J,\mathbb{R})$ res. $||\cdot |{|}_{\infty }$ .24

9.5 ONS

• Let ${\delta }_{mn}:=⟨v_n,v_m⟩$ s.t. $n\ne m\to {\delta }_{mn}=0$, exam. at the beginning, ONB of $C(\mathbb{R},\mathbb{R})$ if $2\pi$-periodic
• Note: in $C(\mathbb{R},\mathbb{R})$ is $⟨f,g⟩:={\int }_{-\pi }^{\pi }f(x)g(x)dx$. Q:? see script, p.119.
• def. OS if $⟨u_n,u_m⟩=c_n{\delta }_{nm}$, ONS if $c_n=1$ i.e. $||u_n|{|}_2:=\sqrt{⟨u_n,u_n⟩}=1$ .25
• Gram-Schm. want $(w_i{)}_{i\in I}$ $w_1=v_1$, $w_2=v_2-\frac{⟨v_2,w_1⟩}{|w_1^2|}{w}_1$, $w_3=v_3-\frac{⟨v_3,w_1⟩}{|w_1{|}^2}{w}_1-\frac{⟨v_3,w_2⟩}{|w_2{|}^2}{w}_2$.
• $L^2:=\{C(\bar{I},\mathbb{R}),||\cdot |{|}_2\}$ closed under convergence (vervollständigt)
• $\{u_n{\}}_{n\in \mathbb{N}}\subset L^2$ an ONS: $f_n:=⟨f,u_n⟩\in \mathbb{R}$ and $s_n[f](x):={\sum }_{j=i}^n{f}_j{u}_j(x)$, so: $s_n\to f$ .26
• ${\sum }_{j=1}^m|{\int }_a^{b}f(x)u_j(x)dx|={\sum }_{j=1}^n|f_j{|}^2\le ||f|{|}_2^2={\int }_a^b{f}^2(x)dx$
• $||s_n[f]|{|}_2\le ||f|{|}_2$, we are approaching from below since we sum pos elements.
• ${\sum }_{j=1}^{\infty }|f_j{|}^2=||f|{|}_2^2\iff ||f-s_n|{|}_2\to 0$ , boh & $(s_n{)}_n$ conv.
• $\{u_m{\}}_{n\in \mathbb{N}}\subset L^2(I,\mathbb{C})$ ONVS $\iff {\forall }_{f\in L^2(I\mathbb{C})}{\sum }_{j=1}^{\infty }=||f|{|}_2^2$ (.28) $\iff {\forall }_{f\in L(I,\mathbb{C})}{\forall }_{n\in \mathbb{N}}{f}_n=0\to f=0$ .30
• but .29?
• The Fourier Serie is the best .32

9.6

• for $\mathbb{R}$: $f(x)=a_0+{\sum }_{j=1}^{\infty }(a_j\cos (jx)+b_j\sin (jx))$: $a_0=\frac{1}{2\pi }{\int }_{-\pi }^{\pi }f(x)dx$; $a_j=\frac{1}{\pi }{\int }_{-\pi }^{\pi }f(x)\cos (jx)dx$; $b_j=\frac{1}{\pi }{\int }_{-\pi }^{\pi }f(x)\sin (jx)dx$
• for $\mathbb{C}$: $f_n:=⟨f,u_n⟩=\frac{1}{\sqrt{2\pi }}{\int }_{-\pi }^{\pi }f(x)e^{-inx}dx$; $a_0=\frac{1}{\sqrt{2\pi }}{f}_0$; $a_n=\frac{1}{\sqrt{2\pi }}(f_n+f_{-n})$; $b_n=\frac{-i}{\sqrt{2\pi }}(f_n-f_{-n})$
• $||f-{\sum }_{j=-n}^n{f}_j{u}_j|{|}_{\infty }\le \frac{1}{\sqrt{\pi n}}||f^{\prime }|{|}_2$ .33 make Bsp. .34

10

10.1 Normed VS

• Normed spaces examples
  • $(C(\bar{I},\mathbb{R}),||\cdot |{|}_p)$, $||f|{|}_p:=({\int }_I|f{|}^p{)}^{\frac{1}{p}}$ is normed space
  • The space of sequences $(l_p,||\cdot |{|}_p)$ s.t. $||\cdot |{|}_p:x\mapsto ({\sum }_{j=1}^{\infty }|\xi {|}^p{)}^{\frac{1}{p}}$ & infinite vectors in $\mathbb{C}$, $l_p:=\{x=({\xi }_n{)}_{n\in \mathbb{N}}|{\forall }_{j\in \mathbb{N}}{\xi }_j\in \mathbb{C},{\sum }_{j=1}^{\infty }|{\xi }_j{|}^p<\infty \}$, bounded seq. in $\mathbb{C}$, similarly for $p\to \infty$
• $\frac{1}{p}+\frac{1}{q}=1$ then (watch)
  • Young: ${\forall }_{a,b>0(\in \mathbb{R})}a\cdot b\le \frac{a^p}{p}+\frac{b^q}{q}$
  • Hölder: ${\sum }_{i=1}^{\infty }|{\xi }_i\cdot {\eta }_i|\le ||x|{|}_p\cdot ||y|{|}_q$ ($\iff ||xy|{|}_1\le ||x|{|}_p\cdot ||y|{|}_q$)
  • Mikowski: $||x+y|{|}_p\le ||x|{|}_p+||y|{|}_p$

10.2 Stetigkeit und Kompaktheit

• Bolzano-Weierstraß: $(x_k{)}_{k\in \mathbb{N}}\subset {\mathbb{R}}^n$ bounded $\Rightarrow$ there is a converging sequence in $(x_k{)}_{k\in \mathbb{N}}$.
• $K\subset {\mathbb{R}}^n$ clos. and boun. $\iff K$ is compact $\iff$ every $(x_k{)}_{k\in \mathbb{N}}$ has $(x_{k_j}{)}_{j\in \mathbb{N}}$ s.t. ${\lim }_{j\to \infty }{x}_{k_j}\in K$ .5
• $f$ cont. $\iff f:$ op/cl comes from op/cl $\iff {\forall }_{(x_k{)}_{k\in \mathbb{N}}\subset X},x_k\to x,f(x_k)\to f(x)$ .6
• $f:X\to Y$ cont., $K\subset X$ compact $\Rightarrow f(K)$ compact & $f{|}_K$ glm. stet.
• Nichtzus. $\iff {\exists }_{A,B\ne \varnothing }A\cap B=\varnothing \wedge X=A\cup B$ ($A,B$ both open or both closed) .11
• Zwischenwe.: $X$ zusam., $f:X\to Y$ cont. $R(f)$ ($=Im(f)$ as a space) is zusam. .12
• a cont. fun $\gamma :[a,b]\to {\mathbb{R}}^n$ is a Weg; for all $m_1,m_2$ then $\gamma :[a,b]\to M$ s.t. $\gamma (a)=m_1\wedge \gamma (b)=m_2$
• wegzusamm $\Rightarrow$ zusamm. .15; a lin. func. is cont. iff it is bounded
• $L(X,Y)$ lin. bound. cont. func.; $||\cdot ||:L(X,Y)\to {\mathbb{R}}_0^{+},A\mapsto {\sup }_{||x||=1}||Ax||$($={\sup }_{x\ne 0}\frac{||Ax||}{||x||}={\sup }_{||x||\le 1}||Ax||$)
• For $A$ lin.: $A$ stet. $\iff A$ stet. in $0\iff {\sup }_{||x|{|}_X=1}||Ax|{|}_Y<\infty \iff {\exists }_{c\in \mathbb{K}}{\forall }_{x\in X}||Ax|{|}_Y\le c||x|{|}_X$ (Blatt 3.)

11

11.1 Diff. Func.

• $(f_k{)}_{k\in \mathbb{N}}$ glm con. iff ${\lim }_{k\to \infty }||f_k-f|{|}_{\infty }=0$
• Sequences of continuous functions glm. converge to continuous functions .2
• $f$ is diff. if there is such $A(x),r(x,\cdot )$ s.t. $f(x+h)=f(x)+A(x)h+r(x,h)|h|$ .3
• For $im(f)\subset \mathbb{R}$ but also $im(f)\subset {\mathbb{R}}^m$ if $f$ always part. diff $\Rightarrow$ f diff.
• Note: $A(x)$ is unique and $r(x,0)=0$ stet. in $0$ $\Rightarrow f^{-1}$ is defined
• For $im(f)\subset \mathbb{R}$ but also $im(f)\subset {\mathbb{R}}^m$ if $f$ always part. diff $\Rightarrow$ f diff.
• ext. all ${\partial }_i$ are stetig $\Rightarrow f$ diff. ($A(x)=\nabla f(x)\Rightarrow f^{\prime }(x,h)\ne ⟨\nabla f(x),h⟩$ why??)
• $f$ diff in $x\Rightarrow f$ cont. in $x$
• ext: ${\lim }_{t\to 0}\frac{f(x+t)-f(x)}{t}$ der. in $\mathbb{R}$ in $x$, in ${\mathbb{R}}^n$, $h\in {\mathbb{R}}^n$ s.t. $|h|=1$, $D_y(h):={\lim }_{t\to 0}\frac{f(y+th)-f(y)}{t}$, $f$ diff. $\Rightarrow {\forall }_{h\in {\mathbb{R}}^n}{D}_y(h)=⟨\nabla f(t),h⟩$
• $f:U(\subset \mathbb{R})\to \mathbb{R}$, $\frac{\partial f(x)}{\partial x_i}:={\lim }_{h\to 0}\frac{f(x+h{e}_i)-f(x)}{h}$ .6
• $\nabla f(x):=(\frac{\partial f(x)}{\partial x_1},...,\frac{\partial f(x)}{\partial x_n}{)}^T=:({\partial }_{1}f(x),...,{\partial }_{n}f(x){)}^T$
• Note: $f^{\prime }(x)=(\nabla f(x){)}^T$ if $f:U(\subset {\mathbb{R}}^n)\to \mathbb{R}$ so $f^{\prime }(x)$ is an orisontal vector.7
• $J_f(x):=\left(\begin{array}{ccc}{\partial }_1{f}_1(x)& ...& {\partial }_n{f}_1(x)\\ ⋮& & ⋮\\ {\partial }_1{f}_m(x)& ...& {\partial }_n{f}_m(x)\end{array}\right)=:({\partial }_{1}f,...,{\partial }_{n}f)(x)=:\frac{\partial f(x)}{\partial x}$for $f:{\mathbb{R}}^n\to {\mathbb{R}}^m$, it is a gen. of the $\nabla$-operator .8,
• also $f^{\prime }:U\to L({\mathbb{R}}^n,{\mathbb{R}}^m),x\mapsto f^{\prime }(x)$, or similarly $f^{\prime }:(x,h)\mapsto f^{\prime }(x)\cdot h$ (matr. mult) .10
• $f:U(\subset {\mathbb{R}}^n)\to \mathbb{R}:(fg{)}^{\prime }=g{f}^{\prime }+f{g}^{\prime }\wedge (g\circ f{)}^{\prime }=(g^{\prime }\circ f)\cdot f^{\prime }\wedge (f+g{)}^{\prime }=f^{\prime }+g^{\prime }\wedge (\alpha f{)}^{\prime }=\alpha f^{\prime }$ .11
• $\nabla f(x_0)\cdot h_0$ for $|h_0|$ is the Richtungsableitung von $f$ in Richtung $h_0$ an der Stelle $x_0$ .12
  • To Compute: $D_x(h)={\lim }_{t\to 0}\frac{f(x+th)-f(x)}{t}$, $f$ diff in $x$, ${\forall }_{h\in {\mathbb{R}}^n}{D}_x(h)=⟨\nabla f(x),h⟩$
• Höhelinien: $N_f(c):=\{x\in U|f(x)=c\}$ clear, $U$ is the domain of $f:{\mathbb{R}}^2\to \mathbb{R}$
• Falllinien: for $\beta :\mathbb{R}\to {\mathbb{R}}^2$ s.t. ${\beta }^{\prime }(t)=\nabla f(\beta (t))$ uhm
• $divf:{\mathbb{R}}^n\to \mathbb{R},x\mapsto divf(x):={\sum }_{i=1}^n{\partial }_i{f}_i(x)$; $rotf:{\mathbb{R}}^3\to {\mathbb{R}}^3,x\mapsto ({\partial }_2{f}_3(x)-{\partial }_3{f}_2(x),{\partial }_3{f}_1(x)-{\partial }_1{f}_3(x),{\partial }_1{f}_2(x)-{\partial }_2{f}_1(x){)}^T$; $\Delta :C^2({\mathbb{R}}^n,\mathbb{R})\to C({\mathbb{R}}^n,\mathbb{R}),f\mapsto {\sum }_{i=1}^n{\partial }_i^2{f}_i$, $\Delta (f)=div(\nabla (f))$, called Laplace-Operator .14
• $f:U(\subset {\mathbb{R}}^n)\to {\mathbb{R}}^n$ inj. diff in $a$ s.t. $\det f^{\prime }(a)\ne 0$ (so that $f^{\prime }(a)$ is an inv. matrix), for $b:=f(a)$ for $f(a)$ is an inner point of $f(U)$ (or you can’t derivate) def. $g:=f^{-1}:f(U)\to {\mathbb{R}}^n$. then $g^{\prime }(b)=(f^{\prime }(a){)}^{-1}$, see those as matrices. Maybe it is to say when ${\cdot }^{-1}$ and ${\cdot }^{\prime }$ commute .16
• Actually you need only $\det f^{\prime }(a)\ne 0$ .17
• Q! dom(g) wtf? .18
• $f\in C^1(U,{\mathbb{R}}^n)$, inv., $f(U)$ open, $f^{-1}\in C^1$ then $f$ is a Diffeomorphismus on $U$. .19

11.2 Mittelwertsatz

• for $a,b\in {\mathbb{R}}^n,t\in (0,1)$: ${\gamma }_{ab}(t):=a+t(b-a)$; ${\Gamma }_{ab}:=\{x\in {\mathbb{R}}^n|{\exists }_{t\in (0,1)}x={\gamma }_{ab}(t)\}$
• Mittelwertsatz: ${\exists }_{c\in {\Gamma }_{ab}}f(b)=f(a)+f^{\prime }(c)(b-a)$, ${\Gamma }_{ab}$ is a the Gerade from $a$ to $b$, for $Im(f)\subseteq \mathbb{R}$
• It’s ok $\gamma :[0,1]\to U$ s.t. $\gamma (0)=a\wedge \gamma (1)=b$ with $\gamma$ diff. so $f(b)=f(a)+f^{\prime }(\gamma (\theta )){\gamma }^{\prime }(\theta )$, $\theta \in (0,1)$
• for $f:U\to {\mathbb{R}}^n$, $f^{\prime }=0\iff f=k$ for $k\in \mathbb{R}$
• ${\partial }_{1}f,...,{\partial }_{n}f$ cont. around $a$ $\Rightarrow f$ diff. .23, comp both lim. ${\partial }_{i}f(a)$, for $Im(f)\subset {\mathbb{R}}^n$

11.3 Höhere Ableitungen

• $f^{\prime \prime \prime }:U\to L({\mathbb{R}}^n,L({\mathbb{R}}^n,L({\mathbb{R}}^n,{\mathbb{R}}^m)))$ or $f^{\prime \prime }:U\times {\mathbb{R}}^n\times {\mathbb{R}}^n\to {\mathbb{R}}^m,(x,h,k)\mapsto f^{\prime \prime }(x,h,k)=(f^{\prime \prime }(x)h)k$
• In general $f^{(p)}:U\times ({\mathbb{R}}^n{)}^p\to {\mathbb{R}}^m,(x,h_1,...,h_p)\mapsto f^{(p)}(x,h_1,...,h_p)$
• also remember: ${\partial }_j{\partial }_{i}f(x)=f^{\prime \prime }(x,e_i,e_j)$ .24
• Schwarz: if $f\in C^2({\mathbb{R}}^n,{\mathbb{R}}^m)\wedge U$ open $\Rightarrow {\forall }_{i,j\in \{1,...,n\}}{\partial }_i{\partial }_{j}f(x)={\partial }_j{\partial }_{i}f(x)$ .25
• check .26
• Taylor: $f(x+h)=f(x)+f^{(1)}(x,h)+...+\frac{1}{p!}{f}^{(p)}(x,h,...,h)+R_p(x,h)$, .27 where $R_p(x,h)=\frac{1}{(p+1)!}{f}^{(p+1)}(x+\theta h,h,...,h)$ for $\theta \in (0,1)$; $Im(f)\subset \mathbb{R}$ and $f^{(n)}=$ $f(x+h)={\sum }_{|\alpha |=0,\alpha \in {\mathbb{N}}_0^n}^p\frac{1}{\alpha !}{\partial }^{\alpha }f(x)h^{\alpha }+R_p(x,h)$ where: $h^{\alpha }:=h_1^{{\alpha }_1}\cdot ...\cdot h_n^{{\alpha }_n}$ same for the others
  • e.g. $f(x,y)=f(a,b)+f_x(a,b)(x-a)+f_y(a,b)(y-b)+\frac{1}{2}{f}_{xx}(a,b)(x-a{)}^2$ $+\frac{1}{2}{f}_{yy}(a,b)(y-b{)}^2+f_{xy}(a,b)(x-a)(y-b)+R_2(x,h)$ for $R_2(x,y)=\frac{1}{6}{f}^{\prime \prime \prime }(x+\theta h,h,h,h)$ for $\theta \in (0,1)$.
• $Im(f)\subset \mathbb{R}$: $a$ critical point of $f$ $\iff \nabla f(a)=0\iff {\forall }_{h\in {\mathbb{R}}^n}{f}^{\prime }(a,h)=0$
• If $a$ extremstelle of $f\Rightarrow a$ critical point of $f$ .29
  • for $f\in C^2({\mathbb{R}}^n,\mathbb{R})$ then $Hes{s}_f(x):=({\partial }_i{\partial }_{j}f(x){)}_{i,j\in \{1,...,n\}}$ .31, .32, .34
  • $pd(Hes{s}_f(a))\Rightarrow a$ minimum; $nd(Hes{s}_f(a))\Rightarrow a$ maximum; $Hes{s}_f(a)$ indef. $\Rightarrow a$ not an extr.
• $M$ convex if the line between any two $m_1,m_2$ is in $M$ .35
• $im(f)\subset \mathbb{R}$ op, conv: $f$ conv. $\iff {\forall }_{x_1,x_2,x_1\ne x_2}{\forall }_{t\in (0,1)}f(x_1+t(x_2-x_1))\le f(x_1)+t(f(x_2)-f(x_1))$
• if only $<$ it is streng convec, also $f$ conv. $\Rightarrow$ $f$ cont. .37
• $f$ convex $\iff {\forall }_{x_1,x_2\in U}f(x_1+t(x_2-x_1))\le f(x_1)+t(f(x_2)-f(x_1))$, with $>$ for streng convex .36
• .38 if $f$ diff.
• if $pd(Hes{s}_f)\Rightarrow f$ streng conv.; $spd(Hes{s}_f)\Rightarrow f$ conv.

11.4 Extrema mit Nebenbedingungen

• ext: Berechnung der Extrema von $f:{\mathbb{R}}^n\to \mathbb{R}$ ($n=2$ in the Klausur)
  1. $\nabla f(x){=}^{!}0$ and get the points, $P_i$
  2. Compute $Hes{s}_f(x)$ and insert $P_i$
     1. $pd(Hes{s}_f(P_i))\Rightarrow P_i$ is a min.
     2. $nd(Hes{s}_f(P_i))\Rightarrow P_i$ is a max.
     3. if $Hes{s}_f(P_i)$ is indef: $P_i$ is Sattelpunkt
     4. else: we can’t say (for $spd$ and $snd$)
• $im(f)\subset \mathbb{R}$, else there can’t be an exercise but to compute the Jacobi.
  1. Checks:
     1. Check if ${\wedge }_{i=1}^n(g_1=0,...,g_m=0)$ (for $m<n$) holds for the extrema of $f$.
     2. Check that $M=\{x\in {\mathbb{R}}^n|g(x)=0\}$ is compact (necess. to use this method).
        1. Remember $M=g^{-1}(\{0\})\wedge g$ stet $\wedge$ ($\{0\}$ abg.) $\Rightarrow M$ abg.
  2. Solve ${\sum }_{i=1}^n(\nabla f(x){)}_i+{\lambda }_i(\nabla g(x){)}_i{=}^{!}0\wedge g_1{=}^{!}0\wedge ...\wedge g_m{=}^{!}0$, and get $P_i$ for some ${\lambda }_i$
  3. Compute $f(P_i)$ and compare them, they are extrema (not rel.), check if minima or maxima
• If we have $n-m=1$ then $h:\mathbb{R}\to \mathbb{R}$
  1. notice $h=f(\varphi (t),\psi (t))$ for some $\varphi$ and $\psi$ then see simply .41 write alg.
• Define $F:{\mathbb{R}}^n\times {\mathbb{R}}^m\to \mathbb{R},(x,\lambda )\mapsto f(x)+{\sum }_{i=1}^m{\lambda }_i{g}_i(x)$ and then ($\lambda$ is Lagrange Moltiplikator) $\nabla F(x,\lambda )=(\nabla f(x)+{\sum }_{i=1}^m{\lambda }_i{g}_i(x),g_1(x),...,g_n(x){)}^T$ set all to $0$, .42 is an ez application. .43
• $spd(Hes{s}_F(x_0,{\lambda }_0))\vee snd(Hes{s}_F(x_0,{\lambda }_0))\Rightarrow x_0$ is extr. of $f{|}_M$ for $M:=\{x\in U|g_j(0){\forall }_{j\in \{1,...,m\}}\}$
  1. compute $\nabla F(x,\lambda )=0$ and get $x_0$ and ${\lambda }_0$, then compute $Hes{s}_F(x_0,{\lambda }_0)$ check if definit.

12

12.1 Weglängen

• $\gamma :[a,b]\to {\mathbb{R}}^n$, if $\gamma$ stetig diff. and ${\forall }_{s\in (a,b)}{\gamma }^{\prime }(s)\ne 0$ then $\gamma$ is glatt; def. $t=s\mapsto \frac{{\gamma }^{\prime }(s)}{|{\gamma }^{\prime }(s)|}$
• $L(\gamma ):={\int }_a^b|{\gamma }^{\prime }(s)|ds$; additiv, ${\lim }_{n\to \infty }L({\gamma }_n)=L(\gamma )$
• $dom(\gamma )=[a,b]\sim dom(\tilde{\gamma })=[c,d]$ if ex. $\varphi :[a,b]\to [c,d]$ s.t. stetig diff. and bij with ${\forall }_{s\in [c,d]}{\varphi }^{\prime }(s)>0$, $\tilde{\gamma }(\tilde{s})=\gamma (\varphi (\tilde{s}))$; the equivalence classes on $\sim$ are a orientierte stückwise glatte Kurve.
• Every class has rapresentatvie a $\gamma$ s.t. $|{\gamma }^{\prime }(s)|=1$. The Bogenlänge of $\gamma$ is ${\int }_0^s|{\gamma }^{\prime }(t)|dt$ .6 .7
• $k(s):=|{\gamma }^{\prime \prime }(s)|$ is called Krümmung, $⟨{\gamma }^{\prime }(s),{\gamma }^{\prime \prime }(s)⟩=0$ direction of $k$ and $t$ are $\perp$ .7

12.2 Kurve in der Ebene und im Raum

• for $|{\gamma }^{\prime }(s)|=1$, (set speed to 1) we have $t(s)=\left(\begin{array}{c}{x}^{\prime }(s)\\ y^{\prime }(s)\end{array}\right)$; $n(s)=\left(\begin{array}{c}-y^{\prime }(s)\\ x^{\prime }(s)\end{array}\right)$, now define $\alpha ,\beta ,\delta ,ϵ$ s.t. $t^{\prime }(s)=\alpha (s)t(s)+\beta (s)n(s)$; $n^{\prime }(s)=\delta (s)t(s)+ϵ(s)n(s)$ like if they were funcitons. But $\alpha =ϵ=0$ and $\beta =-\delta$ called Frenetsche Gleichungen
• $a\times b=\left(\begin{array}{c}{a}_2{b}_3-a_3{b}_2\\ a_3{b}_1-a_1{b}_3\\ a_1{b}_2-a_2{b}_1\end{array}\right)$, def. $b(s):=t(s)\times n(s)$, $b\perp t\wedge b\perp n\wedge det(t(s),n(s),b(s))=1\wedge |b(s)|=1$
• def. $\tau (s):=⟨n^{\prime }(s),b(s)⟩=⟨-b^{\prime }(s),n(s)⟩$ called Torsion.

12.3 $m$-dimensionale Flächen im $\mathbb{R}$

• $\gamma :U(\subset {\mathbb{R}}^m)\to {\mathbb{R}}^n$, $m<n$ so ${\gamma }^{\prime }:U\to L({\mathbb{R}}^m,{\mathbb{R}}^n)$. $\tau :=\gamma (u)+{\gamma }^{\prime }(u,h)$, so $\tau$ is aff. subsp. in ${\mathbb{R}}^n$
• $\gamma (u,v)=\left(\begin{array}{c}f(u)\cos (v)\\ f(u)\sin (u)\\ g(u)\end{array}\right)$ Rotationsfläche, check ${\gamma }^{\prime }(u,v)$ is full rank then $\gamma (u,v)$ is a $2$-Fläche
• $dom(\gamma )=U$ & $dom(\tilde{\gamma })=\tilde{U}$ then $\gamma \sim \tilde{\gamma }\iff \exists \Phi :\tilde{U}\to U$ s.t. $\tilde{\gamma }=\Phi (\gamma )$ and $\Phi$ is a diffeomorph. .12 then the class is called $m$-Fläche with the gleiche Orientierung if $det({\Phi }^{\prime })>0$. uhm what?
• .13

13

13.1 Measure Theory

• $\mathcal{A}\subseteq \mathcal{P}(X)$ is a $\sigma$-algebra if: (i) $\varnothing \in \mathcal{A}$, (ii) $A\in \mathcal{A}\to A^c\in \mathcal{A}$, (iii) $({\forall }_{n\in \mathbb{N}}{A}_n\in \mathcal{A})\to {\bigcup }_{n\in \mathbb{N}}{A}_n\in \mathcal{A}$ .1
• char. of $\sigma$-al., (i) $X\in \mathcal{A}$ (ii) $B\subseteq A\to A\setminus B\in \mathcal{A}$ (iii) $A\cap B\in \mathcal{A}$ (iv) ${\bigcup }_{n\in \mathbb{N}}{A}_n\in \mathcal{A}$ Blatt 9.4
• see blatt other rules & given a fixed $X$ ${\bigcap }_{i\in I}{\mathcal{A}}_i$ is a $\sigma$-algebra, so $\sigma (\mathcal{E}):=\bigcap \{\mathcal{A}|\mathcal{E}\subset \mathcal{A}\}$ is the smallest $\sigma$-algebra containing $\mathcal{E}$.
• $\mu :\mathcal{A}\to [0,\infty ]$ a measure on $\mathcal{A}$ if (i) $\mu (\varnothing )=0$, (ii) $\mu ({\bigcup }_{n\in \mathbb{N}}^{\cdot }{A}_n)={\sum }_{n\in \mathbb{N}}\mu (A_n)$
• $\mu$ is said $\sigma$-endlich if ex. $(A_n{)}_{n\in \mathbb{N}}\subset \mathcal{A}$ s.t. ${\bigcup }_{n\in \mathcal{N}}{A}_n=X$ and ${\forall }_{n\in \mathbb{N}}\mu (A_n)<\infty$ .3
• $\mu (\{x\in X|\neg M(x)\})=0$ : $M(x)$ is true fast überall.
• ${\delta }_x(A)={\chi }_A(x)=1$ if $x\in A$, $0$ if $x\notin A$; $\zeta (A):=|A|$ if $|A|<{\aleph }_0$ , $\infty$ if $|A|\ge {\aleph }_0$ .5
• ${\mathcal{A}}_E:=\{A\cap E|A\in \mathcal{A}\}$ is a $\sigma$-al. $\mu {|}_E:=\mu {|}_{{\mathcal{A}}_E}$ then $(E,{\mathcal{A}}_E,\mu {|}_E)$ is a measure space
• $\sigma (f):=\{f^{-1}(B)|B\in \mathcal{B}\}$ for $f:X\to Y$ and $(Y,\mathcal{B})$, $(X,\mathcal{A})$ is the $\sigma$-al. generated by $f$.
• 1. $\iff$ 2. $\Rightarrow$ 3. (3. $\Rightarrow$ 2. if $\mu$ endlich, i.e. $\mu (x)<\infty$)
  2. $\mu$ is $\sigma$-additiv
  3. $({\forall }_{n\in \mathbb{N}}{A}_n\subset A_{n+1})$, so ${\bigcup }_{n\in \mathbb{N}}{A}_n=:A(\in \mathcal{A})$: ${\lim }_{n\to \infty }\mu (A_n)=\mu (A)$
  4. $({\forall }_{n\in \mathbb{N}}{A}_{n+1}\subset A_n)$, ${\bigcap }_{n\in \mathbb{N}}{A}_n=\varnothing$ and $\mu (A_1)<\infty$ ${\lim }_{n\to \infty }\mu (A_n)=0$
• (Borel-$\sigma$-algebra) $\mathcal{B}({\mathbb{R}}^n):=\sigma (\{(a,b]|a,b\in {\mathbb{R}}^n\wedge a<b\})$, all open and closed sets are $\mathcal{B}$-mess. .9
• Lebesgue-Maß: ${\lambda }_n:\mathcal{B}({\mathbb{R}}^n)\to [0,\infty ]$ s.t. $\lambda ({\prod }_{j=1}^n(a_j,b_j])={\prod }_{j=1}^n(b_j-a_j)$
• (i) $\lambda (\{x\})=0$ (ii) if ${\exists }_{j\in \{1,..,n\}}{a}_j{b}_j\to \lambda ({\prod }_{j=1}^n[a_j,b_j])$ (iii) $\lambda ((a,b))=\lambda ([a,b))=...$ ez (iv) $\lambda (\mathbb{Q})=0$ (v) $\lambda$ is $\sigma$-end., since ${\mathbb{R}}^n={\bigcup }_{N\in \mathbb{N}}{\prod }_{j=1}^n[-N,N]$.
• $\lambda (C)=1-\frac{1}{3}\cdot {\sum }_{n=0}^{\infty }(\frac{2}{3}{)}^n=0$ but $|C|=|\mathbb{R}|\Rightarrow \lambda (C)\ne 0$
• $f:X\to S$, $(X,\mathcal{A})$, $(S,\mathcal{S})$ $f$ $\mathcal{A}$-$\mathcal{S}$-meas. $\iff f^{-1}(\mathcal{S})\subset \mathcal{A}$ i.e. for all $B\in \mathcal{S}$: $f^{-1}(B)\in \mathcal{A}$, $\mathcal{A}$-meas.: $(S,\mathcal{S})=({\mathbb{R}}^n,\mathcal{B}({\mathbb{R}}^n))$; Borel-Meas if also $(X,\mathcal{A})=({\mathbb{R}}^m,\mathcal{B}({\mathbb{R}}^m))$. Meas. comes from meas.
• ext. for $f$ meas. $f:{\mathbb{R}}^n\to {\mathbb{R}}^n$ and $M\in \mathcal{B}({\mathbb{R}}^n)\Rightarrow f^{-1}(M)\in \mathcal{B}({\mathbb{R}}^n)$
• contant func. are always meas.; $f,g$ meas $\Rightarrow$ $g\circ f$ meas.
• for $(X,\mathcal{A})$, $(S,\sigma (\mathcal{E}))$: $f:X\to S$ $\mathcal{A}$-$\sigma (\mathcal{E})$meas. $\iff f^{-1}(\mathcal{E})\subset \mathcal{A}$ .15
• How to find $\mathcal{E}$ of $\mathcal{B}$? Check if with the rules of a $\sigma$-al. you get $(a,b]$ like in .16.(ii)$\Rightarrow$(i)
• For $im(f)\subset \mathbb{R}$: $f$ Borel-meas. $\iff$ ${\forall }_{a\in \mathbb{R}}\{x\in X|f(x)>a\}\in \mathcal{B}({\mathbb{R}}^n)\iff ...$ .16
• $f$ stetig $\Rightarrow f$ Borel-meas.; $(f_k{)}_{k\in \mathbb{N}}$ Borel-meas $\Rightarrow f$ Borel meas.; $f,g$ meas. $\Rightarrow f\cdot g,f+g,max\{f,g\},$ for $r>0$: $|f{|}^r$ and for $k\in \mathbb{N}$: $f^k$ are all meas. .18

13.2 Lebesgue Integral

• $f={\sum }_{i=1}^k{c}_i{\chi }_{A_i}$ for ${\chi }_{A_i}(x):=$ $1$ if $x\in A_i$, $0$ if $x\notin A_i$, is a Stufenf., $B(X,\mathcal{A};\mathbb{R})$ meas. bound func.
• like in 1-dim: $(s_k{)}_{k\in \mathbb{N}}\to f$. if $f\in B(X,\mathcal{A};\mathbb{R})$ then a $(s_k{)}_{k\in \mathbb{N}}$ conv. glm. so the space of the $s_k$ is dense in $B(X,\mathcal{A};\mathbb{R})$ respect to the $||\cdot |{|}_{\infty }$; $f_{+}:=\max \{f,0\}$, $f_{-}:=\min \{0,f\}$; $f=f_{+}-f_{-}$ .20
• $\int sd\mu :={\sum }_{j=1}^k{c}_j\mu (A_j)$ $(\in [0,\infty ])$, $\int f_{+}d\mu :=\sup \{\int sd\mu |0\le s\le f_{+}\}$; $\int fd\mu :=\int f_{+}d\mu -\int f_{-}d\mu$
• all int. func: ${\mathcal{L}}^1$ or ${\mathcal{L}}_1$, in $\int fd\mu$ for $\mu =\lambda$ becomes $\int f(x)dx$, ${\int }_{A}fd\mu :=\int {\chi }_{A}f\mu$
• $\{f\in B(X,\mathcal{A};\mathbb{R})|\mu (dom(f))<\infty \}\subset {\mathcal{L}}^1$; $f,g\in {\mathcal{L}}^1\wedge f\le g\Rightarrow \int fd\mu \le \int gd\mu$
• $\mu (A)=0\Rightarrow {\int }_{A}fd\mu =0$; ${\int }_{A{\cup }^{\cdot }B}fd\mu ={\int }_{A}fd\mu ={\int }_{B}fd\mu$
• $f\ge 0\wedge \int fd\mu =0\to f=0$; $|\int fd\mu |\le \int |f|d\mu$; $g\in {\mathcal{L}}^1(\mu )\wedge |f|\le g$ respect to $\mu$ $\Rightarrow f\in {\mathcal{L}}^1(\mu )$ .23
• "$\int$" is linear, on limits also for sequences s.t. $0\le f_n\le f_{n+1}$, lin on ${\sum }_{n=1}^{\infty }$ too .24 .25. 26
• Fatou: linear on $\lim {\inf }_{n\to \infty }$ .27;
• $(f_n{)}_{n\in \mathbb{N}}$ meas. $f\in \bar{\mathbb{R}}$ $\mu$-f.ü., $g\in {\mathcal{L}}^1(\mu )$: $|f_n(x)|\le g(x)\mu$-f.ü. $\Rightarrow f\in {\mathcal{L}}^1(\mu )\wedge li{m}_{n\to \infty }\int f_{n}d\mu =\int fd\mu$
• normally $A_i$ macchie di mucca for Riemann $A_i=(a_i,b_i]$, call those $s^{(r)}$
• Maj. Krit.: $|f_n|$ s.t. f.ü. $f(x)=f_n(x)\in \overline{\mathbb{R}}$ and ex. $g$ s.t. f.ü. ${\forall }_{n\in \mathbb{N}}|g(x)|\ge |f_n(x)|$ then $\lim$ com. $\int$.
• for $supp(f):=\overline{\{x\in {\mathbb{R}}^n|f(x)\ne 0\}}$: call $\overline{\int }f(x)dx:=\inf \{\int s^{(r)}(x)dx|s^{(r)}\ge f\}$ & $\underline{\int }f(x)dx:=\sup \{\int s^{(r)}(x)dx|s^{(r)}\le f\}$; $f$ Riem. int. $\iff \overline{\int }f(x)dx=\underline{\int }f(x)dx:=\int f(x)dx$ .29
• $f$ stetig $\Rightarrow f$ Riem. int.; $f$ Riem. int $\Rightarrow f$ Les. int $\wedge \int f(x)dx=\int fd\lambda$ .30

13.3 Iterierte Integrale (The Italians)

• Tonelli: $f:{\mathbb{R}}^2\to [0,\infty ]$ meas.,${\int }_{{\mathbb{R}}^2}f(x,y)d(x,y)={\int }_{\mathbb{R}}({\int }_{\mathbb{R}}f(x,y)dx)dy={\int }_{\mathbb{R}}({\int }_{\mathbb{R}}f(x,y)dy)dx$ .32
• Fubini: $f:{\mathbb{R}}^2\to \mathbb{R}$ Les. int. then the same above holds, Fubini holds for ${\mathbb{R}}^n$.
• Cavalieri: $A\in \mathcal{B}(\mathbb{R})$, $A^t:=\{(x_1,...,x_{n-1})\in {\mathbb{R}}^{n-1}|(x_1,...,x_{n-1},t)\in A\}$ $\Rightarrow A_t\in \mathcal{B}({\mathbb{R}}^{n-1})\wedge {\lambda }_n(A)={\int }_{\mathbb{R}}{\lambda }_{n-1}(A_t)dt$.
  • compute: prove $A_t$ is meas., compute $\lambda (A_t)$ then integrate ${\int }_I{\lambda }_{n-1}(A_t)dt$ for $t\in I$

13.4 Transformationssatz

• $U\subset {\mathbb{R}}^n$, $\mathcal{B}(U):=\{B\cap U|B\in \mathcal{B}\}$, $\Phi :U\to {\mathbb{R}}^n$ meas. $\Rightarrow \mu \circ {\Phi }^{-1}:\mathcal{B}({\mathbb{R}}^n)\to [0,\infty ],A\mapsto \mu ({\Phi }^{-1}(A))$
• $\mu$ translationsinvariant if $\mu \circ {\Phi }^{-1}=\mu$ for all translation of the form $\Phi :x\mapsto x+c$
• $\mu$ bewegungsinvariant if $\mu \circ {\Phi }^{-1}=\mu$ for all bewegungen of the form $\Phi :x\mapsto Sx+c$ for $S$ orth.
• $\mu$ translat. on $({\mathbb{R}}^n,\mathcal{B}({\mathbb{R}}^n))$ s.t. $\mu (W)<\infty$ for $W$ an Einheitswürfel then $\mu =\mu (W)\lambda$.
• $\lambda$ is bewegungs.;
• $T$ sq. inv. matrix $\lambda (T(B))=|det(T)|\cdot \lambda (B)$ .42
• $U,V\subset {\mathbb{R}}^n$ op. $\Phi :U\to V$ a $C^1$-Diffeo, let $f:V\to \mathbb{R}$ meas:$f$ int. on $V=\Phi (U)\iff (f\circ \Phi )\cdot |det{\Phi }^{\prime }|$ integr. $\Rightarrow {\int }_{\Phi (U)}f(x)dx={\int }_{U}f(\Phi (z))\cdot |det{\Phi }^{\prime }(z)|dz$ .43
• .44

13.5 Kurven- und Flächenintegrale

• Vector Field: $v:D(\subset {\mathbb{R}}^n)\to {\mathbb{R}}^n$; $\Gamma \subset {\mathbb{R}}^n=im(\gamma )$ and $\omega :\Gamma \times {\mathbb{R}}^n\to {\mathbb{R}}^n$ stet. linear in the second argument, then $\omega$ is an $1$-Form in $\Gamma$. Similarly for any $U\subset {\mathbb{R}}^n$ instead of $\Gamma$ we call $\omega$ an $1$-Form.
• for $\omega (x,h)=⟨\omega (x),h⟩$, $\omega (x)=({\omega }_1(x),...,{\omega }_n(x){)}^T$ for ${\omega }_i:=\omega (x,e_i)$
• ext: example: for any $f$ diff. then $(x,h)\mapsto ⟨\nabla f(x),h⟩$ is an $1$-Form
• call ${\int }_{\Gamma }\omega :={\int }_b^a\omega (\gamma (t),{\gamma }^{\prime }(t))dt$ Kurvenintegral; then lin. and for $\Gamma ={\Gamma }_1+{\Gamma }_2$: ${\int }_{\Gamma }\omega ={\int }_{{\Gamma }_1}\omega +{\int }_{{\Gamma }_2}\omega$
  1. Compute: ${\int }_{\Gamma }\Omega ={\int }_{\Gamma }⟨\omega (\gamma (t)),{\gamma }^{\prime }(t)⟩dt$, like a vectorfield $\cdot$ the directon of $\gamma$. For $\Omega :\Gamma \times {\mathbb{R}}^2\to \mathbb{R}$ and $\omega :{\mathbb{R}}^2\to {\mathbb{R}}^2$, so $\omega$ is like $v$ and $\Omega$ must give us something in one dimension so that we are allowed to integrate.
• for ${\Gamma }_1,{\Gamma }_2$ have the same starting and ending point, and ${\int }_{{\Gamma }_1}\omega ={\int }_{{\Gamma }_2}\omega$ then $\omega$ is wegunabhängig
• $U$ sternförmig to $p$ $\iff {\forall }_{x\in U}\overline{px}\subset U$.
• Exakt, How to prove:
  1. if you have a $f$ s.t. $\omega (x)=\nabla f(x)$, call $f$ Stammfunktion and $\omega$ exakt. .49
     1. ext: e.g. for $U$ sternf. find$f$ s.t. $f(x)={\int }_{\Gamma (x_0,x)}\omega$, for $\gamma :t\mapsto x_0+t(x-x_0)$ uhm
     2. ext: e.g. for all $i$ find: $f=\int {\omega }_{i}d{t}_i+c(x_1,...,x_{i-1},x_{i+1},...,x_n)$ uhm
  2. $\omega$ has $U$ open and zusamm.: $\omega$ exakt $\iff \omega$ wegunabähngig .50
  3. for ${\omega }_i={\partial }_{i}f$: $\omega$ exakt $\Rightarrow {\partial }_j{\omega }_i={\partial }_j{\partial }_{i}f={\partial }_i{\partial }_{j}f={\partial }_i{\omega }_j$, conver. holds if $U$ sternfö. for one $p$
     • Then thanks to Schwarz if $f\in C^2(U,{\mathbb{R}}^n)$ and $U$ sternf. then $f$ is exakt.
• $\gamma$ Par. von $m$-Fl. $g:u\mapsto {\gamma }^{\prime }(u{)}^T{\gamma }^{\prime }(u)=(⟨{\partial }_i\gamma (u),{\partial }_j\gamma (u)⟩{)}_{i,j=1,...,m}$ .55.i
• $A(\Gamma ):={\int }_U\sqrt{det(g(u))}du={\int }_U\sqrt{det({\gamma }^{\prime }(u{)}^T{\gamma }^{\prime }(u))}du$ .55.ii
  • Compute: the volume of $\Gamma$, say $\lambda (\Gamma )$ is $A(\Gamma )={\int }_U\sqrt{|det(g(u))|}du$ .58.iv
  • ext: But also if ${\gamma }^{\prime }(u)$ is square, then just compute ${\int }_{U}det({\gamma }^{\prime }(u))du$, no need of $g(u)$.
• $A$ ist bewegungsinvariant; $A(\Gamma )$ independent form $\gamma$
• if $u\mapsto f(\gamma (u))\sqrt{det(g(u))}$ int., then:${\int }_{\Gamma }fdA:={\int }_{\Gamma }f(x)dA(x):={\int }_{U}f(\gamma (u))\sqrt{det(g(u))}du$, Surf. Int.
  • ext: But also if ${\gamma }^{\prime }(u)$ is square, then just compute ${\int }_{U}f(\gamma (u))det({\gamma }^{\prime }(u))du$, no need of $g(u)$.
• a set $M$ is int. if ${\chi }_M$ int., so $A(M):={\int }_{\Gamma }{\chi }_{M}dA$ (useful?)
• $\gamma :I\times (0,2\pi )\to {\mathbb{R}}^3,(t,\varphi )\mapsto \left(\begin{array}{c}f(t)\cos (\varphi )\\ f(t)\sin (\varphi )\\ t\end{array}\right)$ Rotationsfläche, $A(\Gamma )=2\pi {\int }_{I}f(t)\sqrt{f^{\prime }(t{)}^2+1}dt$ .59
• $A($graph$F)={\int }_U\sqrt{1+|\nabla F(u){|}^2}du$, gr. $F:=$ a $(n-)1$-dim Flä. par: $\gamma :(U)\to {\mathbb{R}}^n,u\mapsto (u,F(u){)}^T$
• Compute volume integrals on circles: ${\int }_{{\mathbb{R}}^n}f(x)dx={\int }_0^{\infty }({\int }_{|x|=r}f(x)dA(x))dr$ .61
• ${\int }_{{\mathbb{R}}^n}f(|x|)dx=A(S^{n-1}){\int }_0^{\infty }f(r)r^{n-1}dr$ practice

13.6 Gauß and Stokes

${\int }_{G}divV(x)dx={\int }_{\partial G}⟨V(x),v(x)⟩dA(x)$ for $V:\bar{G}\to {\mathbb{R}}^n$ a glatt VF., $v:\partial G\to {\mathbb{R}}^n$ die äuß. normale

1. For a ${\int }_{\partial G}f(x)dA(x)$ find a $\psi (x)$ s.t. $\{x\in {\mathbb{R}}^n|\psi (x)=0\}=\partial G$ and $\psi \in C^1({\mathbb{R}}^n,\mathbb{R})$
2. Compute $v(x):=\frac{\nabla \psi (x)}{||\nabla \psi (x)||}$
3. Show that $G^{\circ }:=G\setminus \partial G=\{x\in {\mathbb{R}}^n|\psi (x)<0\}$ is bounded and that $\partial G$ is glatt
4. Define $V:G\to {\mathbb{R}}^n$ s.t. , ${\int }_{\partial G}f(x)dA(x)={\int }_{\partial G}⟨V(x),v(x)⟩dA(x)$ now use Gauß and compute:
5. ${\int }_{G}divV(x)dx={\int }_G{\partial }_1{V}_1(x)+...+{\partial }_n{V}_n(x)dx={\int }_U({\partial }_1{V}_1(\gamma (u))+...+{\partial }_n{V}_n(\gamma (u)))\sqrt{|\det g(u)|}du$

• Stokes in ${\mathbb{R}}^2$: $G$ bounded, $\partial G$ glatt, $V\in C^1(\bar{G},{\mathbb{R}}^2)$ then: ${\int }_{G}rot(V(x))dx={\int }_{\partial G}⟨V(x),dx⟩$
  • $rot(V):={\partial }_1{V}_2-{\partial }_2{V}_1$ in ${\mathbb{R}}^2$
• In ${\mathbb{R}}^3$: $M\subset {\mathbb{R}}^3$ a $2$-Fl., $G$ b., $\partial G$ gl., $V\in C^1(\overline{G},{\mathbb{R}}^3)$: ${\int }_G⟨rot(v(x)),v(x)⟩dA(x)={\int }_{\partial G}⟨V(x),dx⟩ds$
  • $rot(V):=\nabla \times V:=\left(\begin{array}{c}{\partial }_2{V}_3-{\partial }_3{V}_2\\ {\partial }_3{V}_1-{\partial }_1{V}_3\\ {\partial }_1{V}_2-{\partial }_2{V}_1\end{array}\right)$

To memorise

• Taylor: $f(x+h)={\sum }_{j=0}^{\infty }\frac{f^{(j)}(x)}{j!}(h{)}^j$ 9.18
• Weier.: $(P_{n}f)(x):={\sum }_{j=0}^n\left(\begin{array}{c}n\\ j\end{array}\right)x^j(1-x{)}^{n-j}f(\frac{j}{n})$ 9.23
• Fourier: for $\mathbb{R}$: $f(x)=a_0+{\sum }_{j=1}^{\infty }(a_j\cos (jx)+b_j\sin (jx))$: $a_0=\frac{1}{2\pi }{\int }_{-\pi }^{\pi }f(x)dx$; $a_j=\frac{1}{\pi }{\int }_{-\pi }^{\pi }f(x)\cos (jx)dx$; $b_j=\frac{1}{\pi }{\int }_{-\pi }^{\pi }f(x)\sin (jx)dx$ 9.32
• Young: ${\forall }_{a,b>0(\in \mathbb{R})}a\cdot b\le \frac{a^p}{p}+\frac{b^q}{q}$
• Hölder: ${\sum }_{i=1}^{\infty }|{\xi }_i\cdot {\eta }_i|\le ||x|{|}_p\cdot ||y|{|}_q$ ($\iff ||xy|{|}_1\le ||x|{|}_p\cdot ||y|{|}_q$)
• Mikowski: $||x+y|{|}_p\le ||x|{|}_p+||y|{|}_p$
• Nichtzus. $\iff {\exists }_{A,B\ne \varnothing }A\cap B=\varnothing \wedge X=A\cup B$ ($A,B$ both open or both closed) 10.11
• Gram-Schm. want $(w_i{)}_{i\in I}$ $w_1=v_1$, $w_2=v_2-\frac{⟨v_2,w_1⟩}{|w_1^2|}{w}_1$, $w_3=v_3-\frac{⟨v_3,w_1⟩}{|w_1{|}^2}{w}_1-\frac{⟨v_3,w_2⟩}{|w_2{|}^2}{w}_2$.
• $divf:{\mathbb{R}}^n\to \mathbb{R},x\mapsto divf(x):={\sum }_{i=1}^n{\partial }_i{f}_i(x)$
• $rotf:{\mathbb{R}}^3\to {\mathbb{R}}^3,x\mapsto ({\partial }_2{f}_3(x)-{\partial }_3{f}_2(x),{\partial }_3{f}_1(x)-{\partial }_1{f}_3(x),{\partial }_1{f}_2(x)-{\partial }_2{f}_1(x){)}^T$
• $\Delta :C^2({\mathbb{R}}^n,\mathbb{R})\to C({\mathbb{R}}^n,\mathbb{R}),f\mapsto {\sum }_{i=1}^n{\partial }_i^{2}f$, $\Delta (f)=div(\nabla (f))$, called Laplace-Operator 11.14
• ext: $D_y(h):={\lim }_{t\to 0}\frac{f(y+th)-f(y)}{t}$,
• Mittelwertsatz: ${\exists }_{c\in {\Gamma }_{ab}}f(b)=f(a)+f^{\prime }(c)(b-a)$, ${\Gamma }_{ab}$ is a the Gerade from $a$ to $b$, for $Im(f)\subseteq \mathbb{R}$
• Taylor in ${\mathbb{R}}^n$: $f(x,y)=f(a,b)+f_x(a,b)(x-a)+f_y(a,b)(y-b)+\frac{1}{2}{f}_{xx}(a,b)(x-a{)}^2$ $+\frac{1}{2}{f}_{yy}(a,b)(y-b{)}^2+f_{xy}(a,b)(x-a)(y-b)+R_2(x,h)$ for $R_2(x,y)=\frac{1}{6}{f}^{\prime \prime \prime }(x+\theta h,h,h,h)$ for $\theta \in (0,1)$. 11.27
• $f$ convex $\iff {\forall }_{x_1,x_2\in U}f(x_1+t(x_2-x_1))\le f(x_1)+t(f(x_2)-f(x_1))$, with $<$ for streng conv. 11.36
• Solve ${\sum }_{i=1}^n(\nabla f(x){)}_i+{\lambda }_i(\nabla g(x){)}_i{=}^{!}0\wedge g_1{=}^{!}0\wedge ...\wedge g_m{=}^{!}0$, and get $P_i$ for some ${\lambda }_i$
• $t=s\mapsto \frac{{\gamma }^{\prime }(s)}{|{\gamma }^{\prime }(s)|}$, $k(s):=|{\gamma }^{\prime \prime }(s)|$, $⟨{\gamma }^{\prime }(s),{\gamma }^{\prime \prime }(s)⟩=0$
• For $|{\gamma }^{\prime }(s)|=1$, $t(s)=\left(\begin{array}{c}{x}^{\prime }(s)\\ y^{\prime }(s)\end{array}\right)$; $n(s)=\left(\begin{array}{c}-y^{\prime }(s)\\ x^{\prime }(s)\end{array}\right)$, Frenetsche Gleichungen, $a\times b=\left(\begin{array}{c}{a}_2{b}_3-a_3{b}_2\\ a_3{b}_1-a_1{b}_3\\ a_1{b}_2-a_2{b}_1\end{array}\right)$, def. torsion as $\tau (s):=⟨n^{\prime }(s),b(s)⟩=⟨-b^{\prime }(s),n(s)⟩$
• $\mathcal{A}\subseteq \mathcal{P}(X)$ is a $\sigma$-algebra if: (i) $\varnothing \in \mathcal{A}$, (ii) $A\in \mathcal{A}\to A^c\in \mathcal{A}$, (iii) $({\forall }_{n\in \mathbb{N}}{A}_n\in \mathcal{A})\to {\bigcup }_{n\in \mathbb{N}}{A}_n\in \mathcal{A}$ .1
• $\sigma (\mathcal{E}):=\bigcap \{\mathcal{A}|\mathcal{E}\subset \mathcal{A}\}$
• $\mu :\mathcal{A}\to [0,\infty ]$ a measure on $\mathcal{A}$ if (i) $\mu (\varnothing )=0$, (ii) $\mu ({\bigcup }_{n\in \mathbb{N}}^{\cdot }{A}_n)={\sum }_{n\in \mathbb{N}}\mu (A_n)$
• $\mathcal{B}({\mathbb{R}}^n):=\sigma (\{(a,b]|a,b\in {\mathbb{R}}^n\wedge a<b\})$
• $f:X\to S$, $(X,\mathcal{A})$, $(S,\mathcal{S})$ $f$ $\mathcal{A}$-$\mathcal{S}$-meas. $\iff f^{-1}(\mathcal{S})\subset \mathcal{A}$ i.e. for all $B\in \mathcal{S}$: $f^{-1}(B)\in \mathcal{A}$, $\mathcal{A}$-meas.: $(S,\mathcal{S})=({\mathbb{R}}^n,\mathcal{B}({\mathbb{R}}^n))$; Borel-Meas if also $(X,\mathcal{A})=({\mathbb{R}}^m,\mathcal{B}({\mathbb{R}}^m))$
• Tonelli if $f>0$ and meas., Fubini if $f$ Les. Int.
• $A(\Gamma ):={\int }_U\sqrt{det(g(u))}du$, and${\int }_{\Gamma }fdA:={\int }_{U}f(\gamma (u))\sqrt{det(g(u))}du$ Surf. Int.
• $\gamma :I\times (0,2\pi )\to {\mathbb{R}}^3,(t,\varphi )\mapsto \left(\begin{array}{c}f(t)\cos (\varphi )\\ f(t)\sin (\varphi )\\ t\end{array}\right)$ Rotationsfläche, $A(\Gamma )=2\pi {\int }_{I}f(t)\sqrt{f^{\prime }(t{)}^2+1}dt$ .59
• Gauß, 5 steps
  1. For a ${\int }_{\partial G}f(x)dA(x)$ find a $\psi (x)$ s.t. $\{x\in {\mathbb{R}}^n|\psi (x)=0\}=\partial G$ and $\psi \in C^1({\mathbb{R}}^n,\mathbb{R})$
  2. Compute $v(x):=\frac{\nabla \psi (x)}{||\nabla \psi (x)||}$
  3. Show that $G^{\circ }:=G\setminus \partial G=\{x\in {\mathbb{R}}^n|\psi (x)<0\}$ is bounded and that $\partial G$ is glatt
  4. Define $V:G\to {\mathbb{R}}^n$ s.t. , ${\int }_{\partial G}f(x)dA(x)={\int }_{\partial G}⟨V(x),v(x)⟩dA(x)$ now use Gauß and
  5. compute ${\int }_{G}divV(x)dx$.