Intbar

intbar in native mode $⨍⨎$

intbar in svg mode $\intbar \intBar$

Integrale

MathML

Mittelwertintegrale

• Integral mit zwei Strichen: $\intBar _{A}f(x)\mathrm {d} x$
• Integral mit einem Strich: $\intbar _{A}f(x)\mathrm {d} x$
• Integral mit einem Strich in Textstyle: $\textstyle \intbar _{A}f(x)\mathrm {d} x$
• Integral mit einem Strich und mit Limits: $\intBar \limits _{A}f(x)\mathrm {d} x$

Normale Integrale

1. Normales Integral über eine Menge mit displaystyle: $\displaystyle \int _{A}f(x)\mathrm {d} x$
2. Normales Integral über ein Intervall mit Limits-Parameter mit displaystyle: $\displaystyle \int \limits _{0}^{\infty }f(x)dx$
3. Normales Integral über ein Intervall (ohne Limits-Parameter) mit displaystyle: $\displaystyle \int _{0}^{\infty }f(x)dx$
4. Normales Integral über ein Intervall mit NoLimits-Parameter mit displaystyle$\displaystyle \int \nolimits _{0}^{\infty }f(x)dx$
5. Normales Integral über ein Intervall (ohne Limits-Parameter) mit textstyle: $\textstyle \int _{0}^{\infty }f(x)dx$
6. $\oint _{\partial \Sigma }\langle F,\tau \rangle \,\mathrm {d} s=\iint _{\Sigma }\langle \operatorname {rot} \,F,\nu \rangle \,\mathrm {d} S=\iint _{\Sigma }\left({\frac {\partial v_{2}}{\partial x}}-{\frac {\partial v_{1}}{\partial y}}\right)\mathrm {d} x\mathrm {d} y$
7. $\iiint _{V_{\mathrm {Kart} }}f(x,y,z)\,\mathrm {d} x\mathrm {d} y\mathrm {d} z=\iiint _{V_{\mathrm {Kug} }}{\tilde {f}}(r,\theta ,\varphi )\cdot r^{2}\sin \theta \,\mathrm {d} r\mathrm {d} \theta \mathrm {d} \varphi$

Native

Mittelwertintegrale

• intbar in native mode mit textstyle $⨍⨎$
• intbar in native mode mit displaystyle $⨍⨎$
• intbar in native mode mit displaystyle und limits $\underset{0}{\overset{\mathrm{\infty }}{⨍}}⨎$

Normale Integrale

1. Normales Integral über eine Menge mit displaystyle: ${\int }_{A}f(x)\mathrm{d}x$
2. Normales Integral über ein Intervall mit Limits-Parameter mit displaystyle: $\underset{0}{\overset{\mathrm{\infty }}{\int }}f(x)dx$
3. Normales Integral über ein Intervall (ohne Limits-Parameter) mit displaystyle: $\underset{0}{\overset{\mathrm{\infty }}{\int }}f(x)dx$
4. Normales Integral über ein Intervall mit NoLimits-Parameter mit displaystyle$\underset{0}{\overset{\mathrm{\infty }}{\int }}f(x)dx$
5. Normales Integral über ein Intervall (ohne Limits-Parameter) mit textstyle: $\underset{0}{\overset{\mathrm{\infty }}{\int }}f(x)dx$
6. ${\oint }_{\partial \mathrm{\Sigma }}⟨F,\tau ⟩\mathrm{d}s={\iint }_{\mathrm{\Sigma }}⟨\mathrm{r}\mathrm{o}\mathrm{t}F,\nu ⟩\mathrm{d}S={\iint }_{\mathrm{\Sigma }}(\frac{\partial v_2}{\partial x}-\frac{\partial v_1}{\partial y})\mathrm{d}x\mathrm{d}y$
7. ${\iiint }_{V_{\mathrm{K}\mathrm{a}\mathrm{r}\mathrm{t}}}f(x,y,z)\mathrm{d}x\mathrm{d}y\mathrm{d}z={\iiint }_{V_{\mathrm{K}\mathrm{u}\mathrm{g}}}\tilde{f}(r,\theta ,\phi )\cdot r^2\sin \theta \mathrm{d}r\mathrm{d}\theta \mathrm{d}\phi$

Andere stix macros

• $\oiiint _{A}f(x)\mathrm {d} x$
• $\oiint _{A}f(x)\mathrm {d} x$
• $\ointctrclockwise _{A}f(x)\mathrm {d} x$
• $\ointctrclockwise _{A}f(x)\mathrm {d} x$
• $\varointclockwise _{A}f(x)\mathrm {d} x$
• $\varointclockwise _{A}f(x)\mathrm {d} x$

Summen

MathML

1. $\mathrm {T} ^{(n)}(V):=\bigoplus _{i=0}^{n}V^{\otimes i}$
2. ${\frac {1}{\pi }}\sum _{n=1}^{\infty }{\frac {\sin(2\pi nx)}{n}}={\frac {1}{2}}-x,\;0<x<1$
3. ${\frac {1}{\pi }}\sum \limits _{n=1}^{\infty }{\frac {\sin(2\pi nx)}{n}}={\frac {1}{2}}-x,\;0<x<1$
4. ${\frac {1}{\pi }}\sum \nolimits _{n=1}^{\infty }{\frac {\sin(2\pi nx)}{n}}={\frac {1}{2}}-x,\;0<x<1$
5. $\textstyle {\frac {1}{\pi }}\sum _{n=1}^{\infty }{\frac {\sin(2\pi nx)}{n}}={\frac {1}{2}}-x,\;0<x<1$
6. $F_{N}(s)=\prod _{p\leq N \atop p\ {\text{Primzahl}}}\sum _{k=0}^{\infty }{\frac {f(p^{k})}{p^{ks}}}$
7. $F_{N}(s)=\prod \limits _{p\leq N \atop p\ {\text{Primzahl}}}\sum _{k=0}^{\infty }{\frac {f(p^{k})}{p^{ks}}}$
8. $\sum _{I,J}f_{I,J}\mathrm {d} z_{I}\wedge \mathrm {d} {\overline {z}}_{J}:=\sum _{\stackrel {1\leq i_{1}<\ldots <i_{p}\leq p}{1\leq j_{1}<\ldots <j_{q}\leq q}}^{}f_{i_{1},\ldots i_{p},j_{1},\ldots j_{q}}\mathrm {d} z_{i_{1}}\wedge \cdots \wedge \mathrm {d} z_{i_{p}}\wedge \mathrm {d} {\overline {z}}_{j_{1}}\wedge \cdots \wedge \mathrm {d} {\overline {z}}_{j_{q}}$

Native

1. ${\mathrm{T}}^{(n)}(V):=\underset{i=0}{\overset{n}{⨁}}{V}^{\otimes i}$
2. $\frac{1}{\pi }\sum _{n=1}^{\mathrm{\infty }}\frac{\sin (2\pi nx)}{n}=\frac{1}{2}-x,0<x<1$
3. $\frac{1}{\pi }\sum _{n=1}^{\mathrm{\infty }}\frac{\sin (2\pi nx)}{n}=\frac{1}{2}-x,0<x<1$
4. $\frac{1}{\pi }\sum _{n=1}^{\mathrm{\infty }}\frac{\sin (2\pi nx)}{n}=\frac{1}{2}-x,0<x<1$
5. $\frac{1}{\pi }\sum _{n=1}^{\mathrm{\infty }}\frac{\sin (2\pi nx)}{n}=\frac{1}{2}-x,0<x<1$
6. $F_N(s)={\prod }_{\genfrac{}{}{0ex}{}{p\le N}{p\text{Primzahl}}}\sum _{k=0}^{\mathrm{\infty }}\frac{f(p^k)}{p^{ks}}$
7. $F_N(s)=\prod _{\genfrac{}{}{0ex}{}{p\le N}{p\text{Primzahl}}}\sum _{k=0}^{\mathrm{\infty }}\frac{f(p^k)}{p^{ks}}$
8. ${\sum }_{I,J}{f}_{I,J}\mathrm{d}{z}_I\wedge \mathrm{d}{\bar{z}}_J:=\sum _{\stackrel{1\le i_1<\dots <i_p\le p}{1\le j_1<\dots <j_q\le q}}^{}{f}_{i_1,\dots i_p,j_1,\dots j_q}\mathrm{d}{z}_{i_1}\wedge \cdots \wedge \mathrm{d}{z}_{i_p}\wedge \mathrm{d}{\bar{z}}_{j_1}\wedge \cdots \wedge \mathrm{d}{\bar{z}}_{j_q}$

Klammern

MathML

1. $\gamma \,x+\ln {\bigl [}\Gamma (x+1){\bigr ]}=\sum _{n=1}^{\infty }{\biggl [}{\frac {x}{n}}-\ln {\biggl (}1+{\frac {x}{n}}{\biggr )}{\biggr ]}$

Native

1. $\gamma x+\ln [\mathrm{\Gamma }(x+1)]=\sum _{n=1}^{\mathrm{\infty }}[\frac{x}{n}-\ln (1+\frac{x}{n}\left)\right]$

Weiteres

MathML

1. $\operatorname {Hom}$
3. ${\overline {\partial }}f:=\sum _{j=1}^{n}{\frac {\partial }{\partial {\overline {z}}_{j}}}f{\rm {d}}{\overline {z}}_{j}$
4. $\bigvee \limits _{i\in I}{\mathcal {A}}_{i}=\sigma \left(\bigcup \limits _{i\in I}{\mathcal {A}}_{i}\right)$
5. $\bigcap _{n\in \mathbb {N} }A_{n}={\biggl (}\bigcup _{n\in \mathbb {N} }A_{n}^{\mathsf {c}}{\biggr )}^{\!\!{\mathsf {c}}}$
6. $\left\{{\begin{array}{rl}{\frac {1}{2\pi }}\ln {|x|}\ ,&n=2,\\-{\frac {1}{(n-2)\,\omega _{n}}}{\frac {1}{|x|^{n-2}}}\ ,&n>2\\\end{array}}\right.$

Native

1. $\mathrm{H}\mathrm{o}\mathrm{m}$
2. $0⟶{\mathrm{\Gamma }}^{\mathrm{\infty }}(E_0)\stackrel{D_0}{⟶}{\mathrm{\Gamma }}^{\mathrm{\infty }}(E_1)\stackrel{D_1}{⟶}\dots \stackrel{D_{m-1}}{⟶}{\mathrm{\Gamma }}^{\mathrm{\infty }}(E_m)⟶0$
3. $\bar{\partial }f:=\sum _{j=1}^n\frac{\partial }{\partial {\bar{z}}_j}f\mathrm{d}{\bar{z}}_j$
4. $\underset{i\in I}{\bigvee }{\mathcal{𝒜}}_i=\sigma \left(\bigcup _{i\in I}{\mathcal{𝒜}}_i\right)$
5. ${\bigcap }_{n\in \mathbb{\mathbb{N}}}{A}_n=({\bigcup }_{n\in \mathbb{\mathbb{N}}}{A}_n^{\mathsf{c}}{)}^{\mathsf{c}}$
6. $\{\begin{array}{cc}\frac{1}{2\pi }\ln \left|x\right|,& n=2,\\ -\frac{1}{(n-2){\omega }_n}\frac{1}{|x{|}^{n-2}},& n>2\\ \end{array}$

Buchstaben

MathML

1. $\mathbb {N} ,\mathbb {Z} ,\mathbb {Q} ,\mathbb {R} ,\mathbb {R} ,\mathbb {C} ,\mathbb {C} ,\mathbb {H} ,\mathbb {a}$
2. ${\mathcal {ABCDEFGHIJKLM}}{\mathcal {N}}$

Native

1. $\mathbb{\mathbb{N}},\mathbb{\mathbb{Z}},\mathbb{\mathbb{Q}},\mathbb{ℝ},\mathbb{ℝ},\mathbb{ℂ},\mathbb{ℂ},\mathbb{ℍ},\mathbb{𝕒}$
2. $\mathcal{𝒜}\mathcal{ℬ}\mathcal{𝒞}\mathcal{𝒟}\mathcal{ℰ}\mathcal{ℱ}\mathcal{𝒢}\mathcal{\mathscr{H}}\mathcal{ℐ}\mathcal{𝒥}\mathcal{𝒦}\mathcal{ℒ}\mathcal{ℳ}\mathcal{𝒩}$

Matrizen

MathMl

1. $A={\begin{pmatrix}3&2&1\\1&0&2\end{pmatrix}}\in \mathbb {R} ^{2\times 3}$

Native

1. $A=\left(\begin{array}{ccc}3& 2& 1\\ 1& 0& 2\end{array}\right)\in {\mathbb{ℝ}}^{2\times 3}$

Fließtext (Satz von Atkinson)

MathMl

Nach dem Satz von Atkinson ist ein Operator $A\colon X\to Y$ genau dann ein Fredholm-Operator, wenn es Operatoren $B_{1},B_{2}$ und kompakte Operatoren $K_{1},K_{2}$ gibt, so dass $AB_{1}=I_{Y}-K_{1}$ und $B_{2}A=I_{X}-K_{2}$ gilt, das heißt wenn $A$ modulo kompakter Operatoren invertierbar ist. Insbesondere ist ein beschränkter Operator $A\colon X\to X$ genau dann ein Fredholm-Operator, wenn seine Klasse $[A]_{{\mathcal {C}}(X)}$ in der Calkin-Algebra ${\mathcal {B}}(X)/{\mathcal {C}}(X)$ invertierbar ist.

Native

Nach dem Satz von Atkinson ist ein Operator $A:X\to Y$ genau dann ein Fredholm-Operator, wenn es Operatoren $B_1,B_2$ und kompakte Operatoren $K_1,K_2$ gibt, so dass $A{B}_1=I_Y-K_1$ und $B_{2}A=I_X-K_2$ gilt, das heißt wenn $A$ modulo kompakter Operatoren invertierbar ist. Insbesondere ist ein beschränkter Operator $A:X\to X$ genau dann ein Fredholm-Operator, wenn seine Klasse $[A{]}_{\mathcal{𝒞}(X)}$ in der Calkin-Algebra $\mathcal{ℬ}(X)/\mathcal{𝒞}(X)$ invertierbar ist.

Algin

MathMl

1. ${\begin{aligned}&\left({\frac {\partial ^{2}}{\partial r^{2}}}+{\frac {2}{r}}{\frac {\partial }{\partial r}}+{\frac {1}{r^{2}}}{\frac {\partial ^{2}}{\partial \theta ^{2}}}+{\frac {1}{r^{2}\tan \theta }}{\frac {\partial }{\partial \theta }}+{\frac {1}{r^{2}\sin ^{2}\theta }}{\frac {\partial ^{2}}{\partial \varphi ^{2}}}\right)\cdot (v_{r}{\hat {e}}_{r}+v_{\theta }{\hat {e}}_{\theta }+v_{\varphi }{\hat {e}}_{\varphi })\\=&{\frac {\partial ^{2}}{\partial r^{2}}}(v_{r}{\hat {e}}_{r}+v_{\theta }{\hat {e}}_{\theta }+v_{\varphi }{\hat {e}}_{\varphi })+{\frac {2}{r}}{\frac {\partial }{\partial r}}(v_{r}{\hat {e}}_{r}+v_{\theta }{\hat {e}}_{\theta }+v_{\varphi }{\hat {e}}_{\varphi })+{\frac {1}{r^{2}}}{\frac {\partial ^{2}}{\partial \theta ^{2}}}(v_{r}{\hat {e}}_{r}+v_{\theta }{\hat {e}}_{\theta }+v_{\varphi }{\hat {e}}_{\varphi })\\&+{\frac {1}{r^{2}\tan \theta }}{\frac {\partial }{\partial \theta }}(v_{r}{\hat {e}}_{r}+v_{\theta }{\hat {e}}_{\theta }+v_{\varphi }{\hat {e}}_{\varphi })+{\frac {1}{r^{2}\sin ^{2}\theta }}{\frac {\partial ^{2}}{\partial \varphi ^{2}}}(v_{r}{\hat {e}}_{r}+v_{\theta }{\hat {e}}_{\theta }+v_{\varphi }{\hat {e}}_{\varphi })\\=&v_{r,rr}{\hat {e}}_{r}+v_{\theta ,rr}{\hat {e}}_{\theta }+v_{\varphi ,rr}{\hat {e}}_{\varphi }+{\frac {2}{r}}v_{r,r}{\hat {e}}_{r}+{\frac {2}{r}}v_{\theta ,r}{\hat {e}}_{\theta }+{\frac {2}{r}}v_{\varphi ,r}{\hat {e}}_{\varphi }\\&+{\frac {1}{r^{2}}}{\frac {\partial }{\partial \theta }}(v_{r,\theta }{\hat {e}}_{r}+v_{r}{\hat {e}}_{\theta }+v_{\theta ,\theta }{\hat {e}}_{\theta }-v_{\theta }{\hat {e}}_{r}+v_{\varphi ,\theta }{\hat {e}}_{\varphi })\\&+{\frac {1}{r^{2}\tan \theta }}(v_{r,\theta }{\hat {e}}_{r}+v_{r}{\hat {e}}_{\theta }+v_{\theta ,\theta }{\hat {e}}_{\theta }-v_{\theta }{\hat {e}}_{r}+v_{\varphi ,\theta }{\hat {e}}_{\varphi })\\&+{\frac {1}{r^{2}\sin ^{2}\theta }}{\frac {\partial }{\partial \varphi }}(v_{r,\varphi }{\hat {e}}_{r}+\sin \theta v_{r}{\hat {e}}_{\varphi }+v_{\theta ,\varphi }{\hat {e}}_{\theta }+\cos \theta v_{\theta }{\hat {e}}_{\varphi }+v_{\varphi ,\varphi }{\hat {e}}_{\varphi }-\sin \theta v_{\varphi }{\hat {e}}_{r}-\cos \theta v_{\varphi }{\hat {e}}_{\theta })\\=&v_{r,rr}{\hat {e}}_{r}+v_{\theta ,rr}{\hat {e}}_{\theta }+v_{\varphi ,rr}{\hat {e}}_{\varphi }+{\frac {2}{r}}v_{r,r}{\hat {e}}_{r}+{\frac {2}{r}}v_{\theta ,r}{\hat {e}}_{\theta }+{\frac {2}{r}}v_{\varphi ,r}{\hat {e}}_{\varphi }\\&+{\frac {1}{r^{2}}}(v_{r,\theta \theta }{\hat {e}}_{r}+v_{r,\theta }{\hat {e}}_{\theta }+v_{r,\theta }{\hat {e}}_{\theta }-v_{r}{\hat {e}}_{r}+v_{\theta ,\theta \theta }{\hat {e}}_{\theta }-v_{\theta ,\theta }{\hat {e}}_{r}-v_{\theta ,\theta }{\hat {e}}_{r}-v_{\theta }{\hat {e}}_{\theta }+v_{\varphi ,\theta \theta }{\hat {e}}_{\varphi })\\&+{\frac {1}{r^{2}\tan \theta }}(v_{r,\theta }{\hat {e}}_{r}+v_{r}{\hat {e}}_{\theta }+v_{\theta ,\theta }{\hat {e}}_{\theta }-v_{\theta }{\hat {e}}_{r}+v_{\varphi ,\theta }{\hat {e}}_{\varphi })\\&+{\frac {1}{r^{2}\sin ^{2}\theta }}(v_{r,\varphi \varphi }{\hat {e}}_{r}+\sin \theta v_{r,\varphi }{\hat {e}}_{\varphi }+\sin \theta v_{r,\varphi }{\hat {e}}_{\varphi }-\sin ^{2}\theta v_{r}{\hat {e}}_{r}-\sin \theta \cos \theta v_{r}{\hat {e}}_{\theta }\\&+v_{\theta ,\varphi \varphi }{\hat {e}}_{\theta }+\cos \theta v_{\theta ,\varphi }{\hat {e}}_{\varphi }+\cos \theta v_{\theta ,\varphi }{\hat {e}}_{\varphi }-\sin \theta \cos \theta v_{\theta }{\hat {e}}_{r}-\cos ^{2}\theta v_{\theta }{\hat {e}}_{\theta }\\&+v_{\varphi ,\varphi \varphi }{\hat {e}}_{\varphi }-\sin \theta v_{\varphi ,\varphi }{\hat {e}}_{r}-\cos \theta v_{\varphi ,\varphi }{\hat {e}}_{\theta }-\sin \theta v_{\varphi ,\varphi }{\hat {e}}_{r}-\sin ^{2}\theta v_{\varphi }{\hat {e}}_{\varphi }\\&-\cos \theta v_{\varphi ,\varphi }{\hat {e}}_{\theta }-\cos ^{2}\theta v_{\varphi }{\hat {e}}_{\varphi })\\=&{\Bigl (}v_{r,rr}+{\frac {2}{r}}v_{r,r}+{\frac {1}{r^{2}}}v_{r,\theta \theta }+{\frac {1}{r^{2}\tan \theta }}v_{r,\theta }+{\frac {1}{r^{2}\sin ^{2}\theta }}v_{r,\varphi \varphi }\\&\qquad -{\frac {1}{r^{2}}}v_{r}-{\frac {1}{r^{2}}}v_{\theta ,\theta }-{\frac {1}{r^{2}}}v_{\theta ,\theta }-{\frac {1}{r^{2}\tan \theta }}v_{\theta }-{\frac {1}{r^{2}}}v_{r}-{\frac {\cos \theta }{r^{2}\sin \theta }}v_{\theta }-{\frac {1}{r^{2}\sin \theta }}v_{\varphi ,\varphi }-{\frac {1}{r^{2}\sin \theta }}v_{\varphi ,\varphi }{\Bigr )}{\hat {e}}_{r}\\&+{\Bigl (}v_{\theta ,rr}+{\frac {2}{r}}v_{\theta ,r}+{\frac {1}{r^{2}}}v_{\theta ,\theta \theta }+{\frac {1}{r^{2}\tan \theta }}v_{\theta ,\theta }+{\frac {1}{r^{2}\sin ^{2}\theta }}v_{\theta ,\varphi \varphi }\\&\qquad +{\frac {2}{r^{2}}}v_{r,\theta }-{\frac {1}{r^{2}}}v_{\theta }+{\frac {1}{r^{2}\tan \theta }}v_{r}-{\frac {\cos \theta }{r^{2}\sin \theta }}v_{r}-{\frac {\cos ^{2}\theta }{r^{2}\sin ^{2}\theta }}v_{\theta }-{\frac {2\cos \theta }{r^{2}\sin ^{2}\theta }}v_{\varphi ,\varphi }{\Bigr )}{\hat {e}}_{\theta }\\&+{\Bigl (}v_{\varphi ,rr}+{\frac {2}{r}}v_{\varphi ,r}+{\frac {1}{r^{2}}}v_{\varphi ,\theta \theta }+{\frac {1}{r^{2}\tan \theta }}v_{\varphi ,\theta }+{\frac {1}{r^{2}\sin ^{2}\theta }}v_{\varphi ,\varphi \varphi }\\&\qquad +{\frac {2}{r^{2}\sin \theta }}v_{r,\varphi }+{\frac {2\cos \theta }{r^{2}\sin ^{2}\theta }}v_{\theta ,\varphi }-{\frac {\sin ^{2}\theta +\cos ^{2}\theta }{r^{2}\sin ^{2}\theta }}v_{\varphi }{\Bigr )}{\hat {e}}_{\varphi }\\=&\left(\Delta v_{r}-{\frac {2}{r^{2}}}v_{r}-{\frac {2}{r^{2}}}v_{\theta ,\theta }-{\frac {2}{r^{2}\tan \theta }}v_{\theta }-{\frac {2}{r^{2}\sin \theta }}v_{\varphi ,\varphi }\right){\hat {e}}_{r}\\&+\left(\Delta v_{\theta }+{\frac {2}{r^{2}}}v_{r,\theta }-{\frac {1}{r^{2}\sin ^{2}\theta }}v_{\theta }-{\frac {2\cos \theta }{r^{2}\sin ^{2}\theta }}v_{\varphi ,\varphi }\right){\hat {e}}_{\theta }\\&+\left(\Delta v_{\varphi }+{\frac {2\cos \theta }{r^{2}\sin ^{2}\theta }}v_{\theta ,\varphi }-{\frac {1}{r^{2}\sin ^{2}\theta }}v_{\varphi }+{\frac {2}{r^{2}\sin \theta }}v_{r,\varphi }\right){\hat {e}}_{\varphi }\end{aligned}}$

Native

1. $\begin{array}{ll}& (\frac{{\partial }^2}{\partial r^2}+\frac{2}{r}\frac{\partial }{\partial r}+\frac{1}{r^2}\frac{{\partial }^2}{\partial {\theta }^2}+\frac{1}{r^2\tan \theta }\frac{\partial }{\partial \theta }+\frac{1}{r^2{\sin }^2\theta }\frac{{\partial }^2}{\partial {\phi }^2})\cdot (v_r{\hat{e}}_r+v_{\theta }{\hat{e}}_{\theta }+v_{\phi }{\hat{e}}_{\phi })\\ =& \frac{{\partial }^2}{\partial r^2}(v_r{\hat{e}}_r+v_{\theta }{\hat{e}}_{\theta }+v_{\phi }{\hat{e}}_{\phi })+\frac{2}{r}\frac{\partial }{\partial r}(v_r{\hat{e}}_r+v_{\theta }{\hat{e}}_{\theta }+v_{\phi }{\hat{e}}_{\phi })+\frac{1}{r^2}\frac{{\partial }^2}{\partial {\theta }^2}(v_r{\hat{e}}_r+v_{\theta }{\hat{e}}_{\theta }+v_{\phi }{\hat{e}}_{\phi })\\ & +\frac{1}{r^2\tan \theta }\frac{\partial }{\partial \theta }(v_r{\hat{e}}_r+v_{\theta }{\hat{e}}_{\theta }+v_{\phi }{\hat{e}}_{\phi })+\frac{1}{r^2{\sin }^2\theta }\frac{{\partial }^2}{\partial {\phi }^2}(v_r{\hat{e}}_r+v_{\theta }{\hat{e}}_{\theta }+v_{\phi }{\hat{e}}_{\phi })\\ =& v_{r,rr}{\hat{e}}_r+v_{\theta ,rr}{\hat{e}}_{\theta }+v_{\phi ,rr}{\hat{e}}_{\phi }+\frac{2}{r}{v}_{r,r}{\hat{e}}_r+\frac{2}{r}{v}_{\theta ,r}{\hat{e}}_{\theta }+\frac{2}{r}{v}_{\phi ,r}{\hat{e}}_{\phi }\\ & +\frac{1}{r^2}\frac{\partial }{\partial \theta }(v_{r,\theta }{\hat{e}}_r+v_r{\hat{e}}_{\theta }+v_{\theta ,\theta }{\hat{e}}_{\theta }-v_{\theta }{\hat{e}}_r+v_{\phi ,\theta }{\hat{e}}_{\phi })\\ & +\frac{1}{r^2\tan \theta }(v_{r,\theta }{\hat{e}}_r+v_r{\hat{e}}_{\theta }+v_{\theta ,\theta }{\hat{e}}_{\theta }-v_{\theta }{\hat{e}}_r+v_{\phi ,\theta }{\hat{e}}_{\phi })\\ & +\frac{1}{r^2{\sin }^2\theta }\frac{\partial }{\partial \phi }(v_{r,\phi }{\hat{e}}_r+\sin \theta v_r{\hat{e}}_{\phi }+v_{\theta ,\phi }{\hat{e}}_{\theta }+\cos \theta v_{\theta }{\hat{e}}_{\phi }+v_{\phi ,\phi }{\hat{e}}_{\phi }-\sin \theta v_{\phi }{\hat{e}}_r-\cos \theta v_{\phi }{\hat{e}}_{\theta })\\ =& v_{r,rr}{\hat{e}}_r+v_{\theta ,rr}{\hat{e}}_{\theta }+v_{\phi ,rr}{\hat{e}}_{\phi }+\frac{2}{r}{v}_{r,r}{\hat{e}}_r+\frac{2}{r}{v}_{\theta ,r}{\hat{e}}_{\theta }+\frac{2}{r}{v}_{\phi ,r}{\hat{e}}_{\phi }\\ & +\frac{1}{r^2}(v_{r,\theta \theta }{\hat{e}}_r+v_{r,\theta }{\hat{e}}_{\theta }+v_{r,\theta }{\hat{e}}_{\theta }-v_r{\hat{e}}_r+v_{\theta ,\theta \theta }{\hat{e}}_{\theta }-v_{\theta ,\theta }{\hat{e}}_r-v_{\theta ,\theta }{\hat{e}}_r-v_{\theta }{\hat{e}}_{\theta }+v_{\phi ,\theta \theta }{\hat{e}}_{\phi })\\ & +\frac{1}{r^2\tan \theta }(v_{r,\theta }{\hat{e}}_r+v_r{\hat{e}}_{\theta }+v_{\theta ,\theta }{\hat{e}}_{\theta }-v_{\theta }{\hat{e}}_r+v_{\phi ,\theta }{\hat{e}}_{\phi })\\ & +\frac{1}{r^2{\sin }^2\theta }(v_{r,\phi \phi }{\hat{e}}_r+\sin \theta v_{r,\phi }{\hat{e}}_{\phi }+\sin \theta v_{r,\phi }{\hat{e}}_{\phi }-{\sin }^{}\theta v_r{\hat{e}}_r-\sin \theta \cos \theta v_r{\hat{e}}_{\theta }\\ & +v_{\theta ,\phi \phi }{\hat{e}}_{\theta }+\cos \theta v_{\theta ,\phi }{\hat{e}}_{\phi }+\cos \theta v_{\theta ,\phi }{\hat{e}}_{\phi }-\sin \theta \cos \theta v_{\theta }{\hat{e}}_r-{\cos }^{}\theta v_{\theta }{\hat{e}}_{\theta }\\ & +v_{\phi ,\phi \phi }{\hat{e}}_{\phi }-\sin \theta v_{\phi ,\phi }{\hat{e}}_r-\cos \theta v_{\phi ,\phi }{\hat{e}}_{\theta }-\sin \theta v_{\phi ,\phi }{\hat{e}}_r-{\sin }^{}\theta v_{\phi }{\hat{e}}_{\phi }\\ & -\cos \theta v_{\phi ,\phi }{\hat{e}}_{\theta }-{\cos }^{}\theta v_{\phi }{\hat{e}}_{\phi })\\ =& (v_{r,rr}+\frac{2}{r}{v}_{r,r}+\frac{1}{r^2}{v}_{r,\theta \theta }+\frac{1}{r^2\tan \theta }{v}_{r,\theta }+\frac{1}{r^2{\sin }^2\theta }{v}_{r,\phi \phi }\\ & -\frac{1}{r^2}{v}_r-\frac{1}{r^2}{v}_{\theta ,\theta }-\frac{1}{r^2}{v}_{\theta ,\theta }-\frac{1}{r^2\tan \theta }{v}_{\theta }-\frac{1}{r^2}{v}_r-\frac{\cos \theta }{r^2\sin \theta }{v}_{\theta }-\frac{1}{r^2\sin \theta }{v}_{\phi ,\phi }-\frac{1}{r^2\sin \theta }{v}_{\phi ,\phi }){\hat{e}}_r\\ & +(v_{\theta ,rr}+\frac{2}{r}{v}_{\theta ,r}+\frac{1}{r^2}{v}_{\theta ,\theta \theta }+\frac{1}{r^2\tan \theta }{v}_{\theta ,\theta }+\frac{1}{r^2{\sin }^2\theta }{v}_{\theta ,\phi \phi }\\ & +\frac{2}{r^2}{v}_{r,\theta }-\frac{1}{r^2}{v}_{\theta }+\frac{1}{r^2\tan \theta }{v}_r-\frac{\cos \theta }{r^2\sin \theta }{v}_r-\frac{{\cos }^2\theta }{r^2{\sin }^2\theta }{v}_{\theta }-\frac{2\cos \theta }{r^2{\sin }^2\theta }{v}_{\phi ,\phi }){\hat{e}}_{\theta }\\ & +(v_{\phi ,rr}+\frac{2}{r}{v}_{\phi ,r}+\frac{1}{r^2}{v}_{\phi ,\theta \theta }+\frac{1}{r^2\tan \theta }{v}_{\phi ,\theta }+\frac{1}{r^2{\sin }^2\theta }{v}_{\phi ,\phi \phi }\\ & +\frac{2}{r^2\sin \theta }{v}_{r,\phi }+\frac{2\cos \theta }{r^2{\sin }^2\theta }{v}_{\theta ,\phi }-\frac{{\sin }^2\theta +{\cos }^2\theta }{r^2{\sin }^2\theta }{v}_{\phi }){\hat{e}}_{\phi }\\ =& (\mathrm{\Delta }{v}_r-\frac{2}{r^2}{v}_r-\frac{2}{r^2}{v}_{\theta ,\theta }-\frac{2}{r^2\tan \theta }{v}_{\theta }-\frac{2}{r^2\sin \theta }{v}_{\phi ,\phi }){\hat{e}}_r\\ & +(\mathrm{\Delta }{v}_{\theta }+\frac{2}{r^2}{v}_{r,\theta }-\frac{1}{r^2{\sin }^2\theta }{v}_{\theta }-\frac{2\cos \theta }{r^2{\sin }^2\theta }{v}_{\phi ,\phi }){\hat{e}}_{\theta }\\ & +(\mathrm{\Delta }{v}_{\phi }+\frac{2\cos \theta }{r^2{\sin }^2\theta }{v}_{\theta ,\phi }-\frac{1}{r^2{\sin }^2\theta }{v}_{\phi }+\frac{2}{r^2\sin \theta }{v}_{r,\phi }){\hat{e}}_{\phi }\end{array}$

Widetilde & Overline

MathMl

1. ${\widetilde {\operatorname {div} }}T=\operatorname {\widetilde {div}} T=\operatorname {div} (T^{\top })$
2. ${\frac {1}{7}}=0,{\overline {142857}}$

Native

1. $\widetilde{\mathrm{d}\mathrm{i}\mathrm{v}}T=\widetilde{\mathrm{d}\mathrm{i}\mathrm{v}}T=\mathrm{d}\mathrm{i}\mathrm{v}(T^{\mathrm{\top }})$ phab:T352609
2. $\frac{1}{7}=0,\overline{142857}$ phab:T352698

Potenzen & Indizes

MathMl

1. $\sigma _{a}^{2}$
2. $z=\operatorname {erf} ^{-1}(p)$

Native

1. ${\sigma }_a^2$
2. $z={\mathrm{e}\mathrm{r}\mathrm{f}}^{-1}(p)$