Benutzer:Markus Pössel/Sandkiste/Formeln

${\displaystyle {\frac {\Delta P}{P}}={\frac {\Delta N}{N}}={\frac {\sqrt {N}}{N}}={\frac {1}{\sqrt {N}}}.}$

${\displaystyle f={\frac {1}{2\pi }}{\sqrt {\frac {g}{l}}},}$

${\displaystyle T=6{,}17\cdot 10^{-8}\;\left({\frac {M_{\odot }}{M}}\right).}$

${\displaystyle S\;{\stackrel {A}{\to }}\;S'\;{\stackrel {B}{\to }}\;S''}$

${\displaystyle S''\;{\stackrel {B^{-1}}{\to }}\;S'\;{\stackrel {A^{-1}}{\to }}\;S}$

${\displaystyle T={\frac {\hbar c^{3}}{kGM}}\approx 10^{-6}{\frac {M_{\odot }}{M}}}$

${\displaystyle \Delta y'=r\cdot \omega \cdot \Delta t+u'\cdot \omega \cdot (\Delta t)^{2}-\omega \cdot r\cdot \Delta t=u'\cdot \omega \cdot (\Delta t)^{2}.}$

${\displaystyle \cos x\approx 1.}$

${\displaystyle \Delta y'=s_{1}+s_{2}=r\sin(\omega \,\Delta t)+u'\cdot \Delta t\cdot \sin(\omega \,\Delta t)-\omega \cdot r\cdot \Delta t\cdot \cos(\omega \,\Delta t).}$

${\displaystyle \cos \theta ={\frac {u'\cdot \Delta t}{u\cdot \Delta t}}={\frac {u'}{u}}}$

${\displaystyle \sin \theta ={\frac {\omega \cdot r\cdot \Delta t}{u\cdot \Delta t}}={\frac {\omega \cdot r}{u}}.}$

${\displaystyle s_{2}=u\Delta t\cdot \sin(\omega \cdot \Delta t-\theta )=u\Delta t\cdot \sin(\omega \Delta t)\,\cos \theta -u\Delta t\cdot \cos(\omega \Delta t)\,\sin \theta ,}$

${\displaystyle s_{1}=r\cdot \sin \Delta \varphi =r\cdot \sin(\omega \cdot \Delta t).}$

${\displaystyle \tan \theta ={\frac {v}{u'}}={\frac {\omega r}{u'}}.}$

${\displaystyle \Delta \varphi =\omega \;\Delta t}$

${\displaystyle {\sqrt {1+x}}\approx 1+{\frac {x}{2}}}$

${\displaystyle \left({\frac {x}{2}}\right)^{2},}$

${\displaystyle \left(1+{\frac {x}{2}}\right)^{2}=1+x+\left({\frac {x}{2}}\right)^{2}.}$

${\displaystyle ({\sqrt {1+x}})^{2}=1+x}$

${\displaystyle {\begin{array}{lcl}c_{xx}&=&b_{xx}a_{xx}+b_{xy}a_{yx}+b_{xz}a_{zx}+b_{xt}a_{tx}\\[0.5em]c_{xy}&=&b_{xx}a_{xy}+b_{xy}a_{yy}+b_{xz}a_{zy}+b_{xt}a_{ty}\\[0.5em]c_{xz}&=&b_{xx}a_{xz}+b_{xy}a_{yz}+b_{xz}a_{zz}+b_{xt}a_{tz}\\[0.5em]c_{xt}&=&b_{xx}a_{xt}+b_{xy}a_{yt}+b_{xz}a_{zt}+b_{xt}a_{tt}\\[0.5em]c_{yx}&=&b_{yx}a_{xx}+b_{yy}a_{yx}+b_{yz}a_{zx}+b_{yt}a_{tx}\\[0.5em]c_{yy}&=&b_{yx}a_{xy}+b_{yy}a_{yy}+b_{yz}a_{zy}+b_{yt}a_{ty}\\[0.5em]c_{yz}&=&b_{yx}a_{xz}+b_{yy}a_{yz}+b_{yz}a_{zz}+b_{yt}a_{tz}\\[0.5em]c_{yt}&=&b_{yx}a_{xt}+b_{yy}a_{yt}+b_{yz}a_{zt}+b_{yt}a_{tt}\\[0.5em]c_{zx}&=&b_{zx}a_{xx}+b_{zy}a_{yx}+b_{zz}a_{zx}+b_{zt}a_{tx}\\[0.5em]c_{zy}&=&b_{zx}a_{xy}+b_{zy}a_{yy}+b_{zz}a_{zy}+b_{zt}a_{ty}\\[0.5em]c_{zz}&=&b_{zx}a_{xz}+b_{zy}a_{yz}+b_{zz}a_{zz}+b_{zt}a_{tz}\\[0.5em]c_{zt}&=&b_{zx}a_{xt}+b_{zy}a_{yt}+b_{zz}a_{zt}+b_{zt}a_{tt}\\[0.5em]c_{tx}&=&b_{tx}a_{xx}+b_{ty}a_{yx}+b_{tz}a_{zx}+b_{tt}a_{tx}\\[0.5em]c_{ty}&=&b_{tx}a_{xy}+b_{ty}a_{yy}+b_{tz}a_{zy}+b_{tt}a_{ty}\\[0.5em]c_{tz}&=&b_{tx}a_{xz}+b_{ty}a_{yz}+b_{tz}a_{zz}+b_{tt}a_{tz}\\[0.5em]c_{tt}&=&b_{tx}a_{xt}+b_{ty}a_{yt}+b_{tz}a_{zt}+b_{tt}a_{tt}\\[0.5em]c_{xc}&=&b_{xx}a_{xc}+b_{xy}a_{yc}+b_{xz}a_{zc}+b_{xt}a_{tc}+b_{xc}\\[0.5em]c_{yc}&=&b_{yx}a_{xc}+b_{yy}a_{yc}+b_{yz}a_{zc}+b_{yt}a_{tc}+b_{yc}\\[0.5em]c_{zc}&=&b_{zx}a_{xc}+b_{zy}a_{yc}+b_{zz}a_{zc}+b_{zt}a_{tc}+b_{zc}\\[0.5em]c_{tc}&=&b_{tx}a_{xc}+b_{ty}a_{yc}+b_{tz}a_{zc}+b_{tt}a_{tc}+b_{tc}.\end{array}}}$

${\displaystyle {\begin{array}{lclllll}x''&=&c_{xx}\cdot x&+c_{xy}\cdot y&+c_{xz}\cdot z&+c_{xt}\cdot t&+c_{xc}\\[0.3em]y''&=&c_{yx}\cdot x&+c_{yy}\cdot y&+c_{yz}\cdot z&+c_{yt}\cdot t&+c_{yc}\\[0.3em]z''&=&c_{zx}\cdot x&+c_{zy}\cdot y&+c_{zz}\cdot z&+c_{zt}\cdot t&+c_{zc}\\[0.3em]t''&=&c_{tx}\cdot x&+c_{ty}\cdot y&+c_{tz}\cdot z&+c_{tt}\cdot t&+c_{tc},\end{array}}}$

${\displaystyle {\begin{array}{lclllll}x''&=&b_{xx}\cdot x'&+b_{xy}\cdot y'&+b_{xz}\cdot z'&+b_{xt}\cdot t'&+b_{xc}\\[0.3em]y''&=&b_{yx}\cdot x'&+b_{yy}\cdot y'&+b_{yz}\cdot z'&+b_{yt}\cdot t'&+b_{yc}\\[0.3em]z''&=&b_{zx}\cdot x'&+b_{zy}\cdot y'&+b_{zz}\cdot z'&+b_{zt}\cdot t'&+b_{zc}\\[0.3em]t''&=&b_{tx}\cdot x'&+b_{ty}\cdot y'&+b_{tz}\cdot z'&+b_{tt}\cdot t'&+b_{tc},\end{array}}}$

${\displaystyle {\begin{array}{lcr}a_{xx}&=&\cos \varphi \\[0.2em]a_{xy}&=&\sin \varphi \\[0.2em]a_{yx}&=&-\sin \varphi \\[0.2em]a_{yy}&=&\cos \varphi \\[0.2em]a_{tt}&=&1.\end{array}}}$

${\displaystyle a_{cor}=2\,u'\cdot \omega .}$

${\displaystyle T=2\pi {\sqrt {\frac {L}{g}}},}$

${\displaystyle 463\;{\frac {m}{s}},}$

${\displaystyle v=\omega \cdot r.}$

${\displaystyle a_{z}=0{,}034\;{\frac {m}{s^{2}}}=3{,}4\;{\frac {cm}{s^{2}}}.}$

${\displaystyle a_{z}=\omega ^{2}\cdot r}$

${\displaystyle \omega ={\frac {2\pi }{T}}}$

${\displaystyle \omega =7.29211505392569\cdot 10^{-05}{\frac {1}{s}},}$

${\displaystyle b_{K}=\omega ^{2}\cdot r}$

${\displaystyle b_{A}={\frac {F_{A}}{m_{A}}}=F_{B}{\frac {F_{A}}{F_{B}}}{\frac {1}{m_{A}}}=F_{B}{\frac {m_{A}}{m_{B}}}{\frac {1}{m_{A}}}={\frac {F_{B}}{m_{B}}}=b_{B},}$

${\displaystyle F_{A}=m_{A}b_{A}}$

${\displaystyle F_{B}=m_{B}b_{B}.}$

${\displaystyle {\frac {F_{A}}{F_{B}}}={\frac {m_{A}}{m_{B}}}.}$

${\displaystyle {\ddot {\vec {x}}}{\,}'=R\cdot {\ddot {\vec {x}}}+2\,{\vec {\omega }}\times {\dot {\vec {x}}}{\,}'+{\dot {\vec {\omega }}}\times {\vec {x}}{\,}'-{\vec {\omega }}\times ({\vec {\omega }}\times {\vec {x}}{\,}').}$

${\displaystyle {\begin{array}{lcl}{\ddot {x}}'&=&\;{\ddot {x}}\cos \varphi -{\ddot {y}}\sin \varphi -2{\dot {\varphi }}{\dot {y}}'-{\ddot {\varphi }}y'+{\dot {\varphi }}^{2}x'\\[0.5em]{\ddot {y}}'&=&\;{\ddot {y}}\cos \varphi +{\ddot {x}}\sin \varphi +2{\dot {\varphi }}{\dot {x}}'+{\ddot {\varphi }}x'+{\dot {\varphi }}^{2}y'\\[0.5em]{\ddot {z}}'&=&\;{\ddot {z}}.\end{array}}}$

${\displaystyle {\ddot {\vec {x}}}{\,}'=\left({\begin{array}{ccc}{\ddot {x}}'\\{\ddot {y}}'\\{\ddot {z}}'\end{array}}\right)=\left({\begin{array}{ccc}{\ddot {x}}\cos \varphi -{\ddot {y}}\sin \varphi -2{\dot {\varphi }}({\dot {x}}\sin \varphi +{\dot {y}}\cos \varphi )\\-{\ddot {\varphi }}(x\sin \varphi +y\cos \varphi )+{\dot {\varphi }}^{2}(y\sin \varphi -x\cos \varphi )\\[1.5em]{\ddot {y}}\cos \varphi +{\ddot {x}}\sin \varphi -2{\dot {\varphi }}({\dot {y}}\sin \varphi -{\dot {x}}\cos \varphi )\\+{\ddot {\varphi }}(x\cos \varphi -y\sin \varphi )-{\dot {\varphi }}^{2}(x\sin \varphi +y\cos \varphi )\\[1.5em]{\ddot {z}}\end{array}}\right).}$

${\displaystyle {\begin{array}{lcl}{\ddot {x}}'&=&\;{\ddot {x}}\cos \varphi -{\ddot {y}}\sin \varphi -2{\dot {\varphi }}({\dot {x}}\sin \varphi +{\dot {y}}\cos \varphi )\\&&-{\ddot {\varphi }}(x\sin \varphi +y\cos \varphi )+{\dot {\varphi }}^{2}(y\sin \varphi -x\cos \varphi )\\[1.5em]{\ddot {y}}'&=&\;{\ddot {y}}\cos \varphi +{\ddot {x}}\sin \varphi -2{\dot {\varphi }}({\dot {y}}\sin \varphi -{\dot {x}}\cos \varphi )\\&&+{\ddot {\varphi }}(x\cos \varphi -y\sin \varphi )-{\dot {\varphi }}^{2}(x\sin \varphi +y\cos \varphi )\\[1.5em]{\ddot {z}}'&=&\;{\ddot {z}}.\end{array}}}$

${\displaystyle {\dot {\vec {x}}}{\,}'=\left({\begin{array}{ccc}{\dot {x}}'\\{\dot {y}}'\\{\dot {z}}'\end{array}}\right)=\left({\begin{array}{ccc}{\dot {x}}\cos \varphi -{\dot {y}}\sin \varphi -{\dot {\varphi }}\;y'\\{\dot {y}}\cos \varphi +{\dot {x}}\sin \varphi +{\dot {\varphi }}\;x'\\{\dot {z}}\end{array}}\right)=R\cdot {\dot {\vec {x}}}+{\vec {\omega }}\times {\vec {x}}{\,}',}$

${\displaystyle {\vec {\omega }}=\left({\begin{array}{ccc}0\\0\\-{\dot {\varphi }}\end{array}}\right)=\left({\begin{array}{ccc}0\\0\\-\omega \end{array}}\right)}$

${\displaystyle \left({\begin{array}{ccc}x'\\y'\\z'\end{array}}\right)={\vec {x}}{\,}'=R\cdot {\vec {x}}=\left({\begin{array}{ccc}\cos \varphi &-\sin \varphi &0\\\sin \varphi &\cos \varphi &0\\0&0&1\\\end{array}}\right)\cdot \left({\begin{array}{ccc}x\\y\\z\end{array}}\right)=\left({\begin{array}{ccc}x\cos \varphi -y\sin \varphi \\y\cos \varphi +x\sin \varphi \\z\end{array}}\right),}$

${\displaystyle {\vec {x}}(t)=\left({\begin{array}{ccc}x(t)\\y(t)\\z(t)\end{array}}\right)}$

${\displaystyle {\vec {x}}{}'(t')=\left({\begin{array}{ccc}x'(t')\\y'(t')\\z'(t')\end{array}}\right).}$

${\displaystyle {\begin{array}{lcl}{\dot {x}}'&=&{\dot {x}}\cos \varphi -{\dot {y}}\sin \varphi -{\dot {\varphi }}(y\cos \varphi +x\sin \varphi )\\[0.2em]&=&{\dot {x}}\cos \varphi -{\dot {y}}\sin \varphi -{\dot {\varphi }}\;y'\\[0.7em]{\dot {y}}'&=&{\dot {y}}\cos \varphi +{\dot {x}}\sin \varphi +{\dot {\varphi }}(x\cos \varphi -y\sin \varphi )\\[0.2em]&=&{\dot {y}}\cos \varphi +{\dot {x}}\sin \varphi +{\dot {\varphi }}\;x'\\[0.7em]{\dot {z}}'&=&{\dot {z}}.\end{array}}}$

${\displaystyle {\begin{array}{lcl}x'(t')&=&x(t')\cos \varphi (t')-y(t')\sin \varphi (t')\\[0.3em]y'(t')&=&y(t')\cos \varphi (t')+x(t')\sin \varphi (t')\\[0.3em]z'(t')&=&z(t').\end{array}}}$

${\displaystyle {\begin{array}{lcl}x'&=&x\cos \varphi -y\sin \varphi \\[0.3em]y'&=&y\cos \varphi +x\sin \varphi \\[0.3em]z'&=&z\\[0.3em]t'&=&t.\end{array}}}$

${\displaystyle F_{cor}=2\,m\,u'\,\omega .}$

${\displaystyle a_{cor}=2\,u'\,\omega .}$

${\displaystyle \sin \alpha \approx \alpha }$

${\displaystyle a={\dot {\omega }}\cdot r,}$

${\displaystyle F_{cor}=2\,m\,u\,\omega .}$

${\displaystyle a_{cor}=2\,u\,\omega }$

${\displaystyle d=\omega r\cdot \Delta t+u\omega (\Delta t)^{2}.}$

${\displaystyle \sin x\approx x}$

${\displaystyle d=(r+u\cdot \Delta t)\cdot \sin(\omega \Delta t).}$

${\displaystyle {\frac {GM}{r^{2}}}=\omega ^{2}r={\frac {(2\pi )^{2}}{T^{2}}}r\;\;\;\Rightarrow \;\;\;r^{3}={\frac {GM}{(2\pi )^{2}}}\;T^{2}}$

${\displaystyle F_{zent}={\frac {mv^{2}}{r}}=m\omega ^{2}r}$

${\displaystyle b_{zent}={\frac {v^{2}}{r}}=\omega ^{2}r.}$

${\displaystyle b={\frac {v^{2}}{r}}}$

${\displaystyle \Delta s\approx {\frac {1}{2}}{\frac {v^{2}}{r}}(\Delta t)^{2}.}$

${\displaystyle \left(1+{\frac {x}{2}}\right)^{2}=1+x+\left({\frac {x}{2}}\right)^{2}\approx 1+x,}$

${\displaystyle {\sqrt {1+x}}\approx 1+{\frac {x}{2}}.}$

${\displaystyle d={\sqrt {r^{2}+(v\Delta t)^{2}}}=r{\sqrt {1+\left({\frac {v\Delta t}{r}}\right)^{2}}},}$

${\displaystyle \Delta s\equiv d-r=r\left[{\sqrt {1+\left({\frac {v\Delta t}{r}}\right)^{2}}}-1\right]}$

${\displaystyle \omega ={\frac {2\pi }{T}}}$

${\displaystyle F_{x}=-m\cdot a.}$

${\displaystyle -{\frac {1}{2}}a(t')^{2}}$

${\displaystyle {\begin{array}{lcl}x'(t')&=&v_{0}\cdot t'-{\frac {1}{2}}a(t')^{2}\\[0.5em]y'(t')&=&0\\[0.5em]z'(t')&=&0\end{array}}}$

${\displaystyle {\begin{array}{lcl}x(t)&=&v_{0}\cdot t\\[0.5em]y(t)&=&0\\[0.5em]z(t)&=&0\end{array}}}$

${\displaystyle {\begin{array}{lcl}x'&=&x-{\frac {1}{2}}at^{2},\\y'&=&y,\\z'&=&z,\\t'&=&t.\end{array}}}$

${\displaystyle x(t)={\frac {1}{2}}at^{2},}$

${\displaystyle {\begin{array}{lcl}x(t)&=&v_{x}\cdot t+x_{0}\\[0.2em]y(t)&=&v_{y}\cdot t+y_{0}\\[0.2em]z(t)&=&v_{z}\cdot t+z_{0}\end{array}}}$

${\displaystyle s(t)=v\cdot t+const.,}$

${\displaystyle z(t)=v_{z}\cdot t+z_{0},}$

${\displaystyle {\vec {x}}{\,}'\equiv \left({\begin{array}{c}x'\\y'\\z'\\t'\end{array}}\right)=M\cdot {\vec {x}}+{\vec {x}}_{0}\equiv \left({\begin{array}{cccc}a_{xx}&a_{xy}&a_{xz}&a_{xt}\\a_{yx}&a_{yy}&a_{yz}&a_{yt}\\a_{zx}&a_{zy}&a_{zz}&a_{zt}\\a_{tx}&a_{ty}&a_{tz}&a_{tt}\end{array}}\right)\cdot \left({\begin{array}{c}x\\y\\z\\t\end{array}}\right)+\left({\begin{array}{c}a_{xc}\\a_{yc}\\a_{zc}\\a_{tc}\end{array}}\right)}$

${\displaystyle \int \limits _{-\infty }^{\infty }\delta (x)^{2}\;\mathrm {d} x=\int \limits _{-\infty }^{\infty }\delta (x)\cdot \delta (x)\;\mathrm {d} x=\delta (0)=\infty ,}$

${\displaystyle f(0)=\int \limits _{-\infty }^{\infty }f(x)\delta (x)\;\mathrm {d} x.}$

${\displaystyle \partial _{t}h(t,x)=-\lambda (\partial _{x}h(t,x))^{2}+\nu \;\partial _{x}^{2}h(t,x)+{\sqrt {D}}\;\eta (t,x).}$

${\displaystyle r_{p}=a\cdot {\frac {m_{*}}{m_{p}+m_{*}}}}$

${\displaystyle r_{*}=a\cdot {\frac {m_{p}}{m_{p}+m_{*}}}}$

${\displaystyle v_{p}={\frac {2\pi a}{T}}\cdot {\frac {m_{*}}{m_{p}+m_{*}}}}$

${\displaystyle v_{*}={\frac {2\pi a}{T}}\cdot {\frac {m_{p}}{m_{p}+m_{*}}}.}$

${\displaystyle N(t)=N_{0}\cdot \exp(-\lambda \cdot t)}$

${\displaystyle L(t)=L_{0}+\delta L\cdot \sin[\omega (t-t_{0})].}$

${\displaystyle z=h_{0}-{\frac {1}{2}}\,g\,(t-t_{0})^{2}}$

${\displaystyle {\begin{aligned}x'_{0}=&(a_{xx}x_{0}+a_{xy}y_{0}+a_{xz}z_{0}+a_{xc})\\[1em]&-{\frac {(a_{xx}v_{x}+a_{xy}v_{y}+a_{xz}v_{z}+a_{xt})(a_{tx}x_{0}+a_{ty}y_{0}+a_{tz}z_{0}+a_{tc})}{a_{tx}v_{x}+a_{ty}v_{y}+a_{tz}v_{z}+a_{tt}}}.\end{aligned}}}$

${\displaystyle v'_{x}={\frac {a_{xx}v_{x}+a_{xy}v_{y}+a_{xz}v_{z}+a_{xt}}{a_{tx}v_{x}+a_{ty}v_{y}+a_{tz}v_{z}+a_{tt}}}}$

${\displaystyle x'(t')=v'_{x}\cdot t'+x'_{0}}$

${\displaystyle t={\frac {t'-(a_{tc}+a_{tx}x_{0}+a_{ty}y_{0}+a_{tz}z_{0})}{a_{tx}v_{x}+a_{ty}v_{y}+a_{tz}v_{z}+a_{tt}}}}$

${\displaystyle t'=(a_{tx}v_{x}+a_{ty}v_{y}+a_{tz}v_{z}+a_{tt})\cdot t+a_{tc}+a_{tx}x_{0}+a_{ty}y_{0}+a_{tz}z_{0}.}$

${\displaystyle t'=a_{tx}\cdot x+a_{ty}\cdot y+a_{tz}\cdot z+a_{tt}\cdot t+a_{tc},}$

${\displaystyle {\begin{array}{lcl}x(t)&=&d_{x}\cdot t+e_{x}\\[0.3em]y(t)&=&d_{y}\cdot t+e_{y}\\[0.3em]z(t)&=&d_{z}\cdot t+e_{z},\end{array}}}$

${\displaystyle {\begin{array}{lclllll}x'&=&a_{xx}\cdot x&+a_{xy}\cdot y&+a_{xz}\cdot z&+a_{xt}\cdot t&+a_{xc}\\[0.3em]y'&=&a_{yx}\cdot x&+a_{yy}\cdot y&+a_{yz}\cdot z&+a_{yt}\cdot t&+a_{yc}\\[0.3em]z'&=&a_{zx}\cdot x&+a_{zy}\cdot y&+a_{zz}\cdot z&+a_{zt}\cdot t&+a_{zc}\\[0.3em]t'&=&a_{tx}\cdot x&+a_{ty}\cdot y&+a_{tz}\cdot z&+a_{tt}\cdot t&+a_{tc}\end{array}}}$

${\displaystyle {\begin{array}{lcl}x'&=&x\cos(\omega t)+y\sin(\omega t)\\[0.3em]y'&=&y\cos(\omega t)-x\sin(\omega t).\end{array}}}$

${\displaystyle {\begin{array}{lcl}x'&=&x\cos(\varphi (t))+y\sin(\varphi (t))\\[0.3em]y'&=&y\cos(\varphi (t))-x\sin(\varphi (t)).\end{array}}}$

${\displaystyle t'=t}$

${\displaystyle {\begin{array}{lcl}x'&=&x+v_{x}\cdot t\\[0.5em]y'&=&y+v_{y}\cdot t\\[0.5em]z'&=&z+v_{z}\cdot t\end{array}}}$

${\displaystyle {\begin{array}{lcl}x&=&x'-v_{x}\cdot t'\\[0.5em]y&=&y'-v_{y}\cdot t'\\[0.5em]z&=&z'-v_{z}\cdot t'\\[0.5em]t&=&t'.\end{array}}}$

${\displaystyle {\begin{array}{lcl}x'&=&x+v_{x}\cdot t+a_{x}\\[0.5em]y'&=&y+v_{y}\cdot t+a_{y}\\[0.5em]z'&=&z+v_{z}\cdot t+a_{z}.\end{array}}}$

${\displaystyle {\begin{array}{lcl}x'&=&x+a_{x}(t)\\[0.5em]y'&=&y+a_{y}(t)\\[0.5em]z'&=&z+a_{z}(t).\end{array}}}$

${\displaystyle {\begin{array}{lcl}x'&=&x+a_{x}\\[0.5em]y'&=&y+a_{y}\\[0.5em]z'&=&z+a_{z}\end{array}}}$

${\displaystyle t'=t+a(t)}$

${\displaystyle t(E)-t(F)=t'(E)-t'(F)=t(E)-t(F)+a(t(E))-a(t(F))}$

${\displaystyle a(t(E))-a(t(F))=0.}$

${\displaystyle d'={\sqrt {(x'_{E}-x'_{A})^{2}+(y'_{E}-y'_{A})^{2}+(z'_{E}-z'_{A})^{2}}}}$

${\displaystyle d'(t')={\sqrt {[x'_{E}(t')-x'_{A}(t')]^{2}+[y'_{E}(t')-y'_{A}(t')]^{2}+[z'_{E}(t')-z'_{A}(t')]^{2}}}.}$

${\displaystyle {\begin{array}{lcl}x'_{E}(t')&=&x'(d,e,f,t_{E}(t'))\\[0.5em]y'_{E}(t')&=&y'(d,e,f,t_{E}(t'))\\[0.5em]z'_{E}(t')&=&z'(d,e,f,t_{E}(t')).\end{array}}}$

${\displaystyle {\begin{array}{lcl}x'_{A}(t')&=&x'(a,b,c,t_{A}(t'))\\[0.5em]y'_{A}(t')&=&y'(a,b,c,t_{A}(t'))\\[0.5em]z'_{A}(t')&=&z'(a,b,c,t_{A}(t')),\end{array}}}$

${\displaystyle {\begin{array}{lcl}x'&=&x+a\\[0.5em]y'&=&y\\[0.5em]z'&=&z\\[0.5em]t'&=&t.\end{array}}}$

${\displaystyle t_{2}'=t'(a,b,c,t=1\;s)}$

und

${\displaystyle t_{1}'=t'(a,b,c,t=0\;s)}$

${\displaystyle S,\;S',\;S'',\;S''',\;S^{(4)},\;S^{(5)},\ldots }$

${\displaystyle t'=t'(x,y,z,t),}$

${\displaystyle t'=t'(a,b,c,t)}$

${\displaystyle x'=x'(t'),}$

${\displaystyle x'=x'(t).}$

${\displaystyle x'=x'(x,y,z,t)}$

${\displaystyle t'=t'(x(t),y(t),z(t),t),}$

${\displaystyle t'=t'(x,y,z,t)}$

${\displaystyle {\begin{array}{lcl}x'&=&x'(x,y,z,t)\\[0.5em]y'&=&y'(x,y,z,t)\\[0.5em]z'&=&z'(x,y,z,t)\\[0.5em]t'&=&t'(x,y,z,t).\end{array}}}$

${\displaystyle {\begin{array}{lcl}x&=&x(x',y',z',t')\\[0.5em]y&=&y(x',y',z',t')\\[0.5em]z&=&z(x',y',z',t')\\[0.5em]t&=&t(x',y',z',t').\end{array}}}$

${\displaystyle t'=t'(x,y,z,t)}$

${\displaystyle F=k\cdot \Delta l.}$

${\displaystyle F_{G}=G{\frac {m_{1}m_{2}}{r^{2}}}=m_{2}b_{2}\;\;\;\;\Rightarrow \;\;\;\;b_{2}={\frac {G\,m_{1}}{r^{2}}},}$

${\displaystyle F_{G}=G{\frac {m_{1}m_{2}}{r^{2}}}.}$

${\displaystyle {\frac {{\mathrm {d} }T}{\mathrm {d} t}}=m(v_{x}\cdot b_{x}+v_{y}\cdot b_{y}+v_{z}\cdot b_{z}).}$

${\displaystyle T={\frac {1}{2}}mv^{2}={\frac {1}{2}}m\left(v_{x}^{2}+v_{y}^{2}+v_{z}^{2}\right).}$

${\displaystyle {\frac {{\mathrm {d} }U}{\mathrm {d} t}}=-m(v_{x}\cdot b_{x}+v_{y}\cdot b_{y}+v_{z}\cdot b_{z}).}$

${\displaystyle {\frac {{\mathrm {d} }U}{\mathrm {d} t}}=-(v_{x}\cdot F_{x}+v_{y}\cdot F_{y}+v_{z}\cdot F_{z}).}$

${\displaystyle {\frac {F_{x}}{F}}=-{\frac {x}{r}}.}$

${\displaystyle {\frac {|F_{x}|}{F}}={\frac {|x|}{r}},}$

${\displaystyle {\frac {\mathrm {d} U}{\mathrm {d} t}}={\frac {GmM}{r^{3}}}(v_{x}\cdot x+v_{y}\cdot y+v_{z}\cdot z).}$

${\displaystyle {\frac {1}{r(t_{0}+\Delta t)}}-{\frac {1}{r(t_{0})}}.}$

${\displaystyle x(t_{0}+\Delta t)=[v_{x}(t_{0})+R_{x}(t_{0},\Delta t)]\cdot \Delta t+x(t_{0}),}$

${\displaystyle {\frac {r(t_{0})^{2}-r(t_{0}+\Delta t)^{2}}{r(t_{0})+r(t_{0}+\Delta t)}},}$

${\displaystyle 1={\frac {r(t_{0})+r(t_{0}+\Delta t)}{r(t_{0})+r(t_{0}+\Delta t)}}}$

${\displaystyle r(t_{0})-r(t_{0}+\Delta t)}$

${\displaystyle r={\sqrt {x^{2}+y^{2}+z^{2}}},}$

${\displaystyle {\frac {1}{r(t_{0}+\Delta t)}}-{\frac {1}{r(t_{0})}}={\frac {r(t_{0})-r(t_{0}+\Delta t)}{r(t_{0})r(t_{0}+\Delta t)}}.}$

${\displaystyle {\frac {\mathrm {d} (1/r)}{\mathrm {d} t}}=\lim _{\Delta t\to 0}{\frac {[1/r(t_{0}+\Delta t)]-[1/r(t_{0})]}{\Delta t}}\;\;?}$

${\displaystyle {\frac {\mathrm {d} U}{\mathrm {d} t}}=\lim _{\Delta t\to 0}{\frac {U(t_{0}+\Delta t)-U(t_{0})}{\Delta t}}.}$

${\displaystyle {\frac {\mathrm {d} U}{\mathrm {d} t}}=-m\,v_{x}\,b_{x}.}$

${\displaystyle F=-kx,}$

${\displaystyle {\frac {\mathrm {d} U}{\mathrm {d} t}}=m\,k\,x\,v_{x}.}$

${\displaystyle {\begin{array}{lcl}U(t_{0}+\Delta t)&=&{\frac {1}{2}}\,m\,k\,x(t_{0}+\Delta t)^{2}\\[1.0em]&=&{\frac {1}{2}}\,m\,k{\big [}x(t_{0})^{2}+v_{x}(t_{0})^{2}\Delta t^{2}+R(t_{0},\Delta t)^{2}\Delta t^{2}\\[0.75em]&&\;\;\;\;\;\;\;\;\;\;\;+2x(t_{0})v_{x}(t_{0})\Delta t+2x(t_{0})R(t_{0},\Delta t)\Delta t\\[0.75em]&&\;\;\;\;\;\;\;\;\;\;\;+2v_{x}(t_{0})R(t_{0},\Delta t)\Delta t^{2}{\big ]}.\end{array}}}$

${\displaystyle x(t_{0}+\Delta t)=x(t_{0})+v_{x}(t_{0})\Delta t+R(t_{0},\Delta t)\Delta t.}$

${\displaystyle R(t_{0},\Delta t)={\frac {x(t_{0}+\Delta t)-x(t_{0})}{\Delta t}}-v_{x}(t_{0}).}$

${\displaystyle U={\frac {1}{2}}\,m\,k\,x^{2}}$

${\displaystyle m\,v_{x}\,b_{x},}$

${\displaystyle T={\frac {1}{2}}\,m\,(v_{x})^{2}.}$

${\displaystyle U={\frac {1}{2}}\,m\,k\,(\Delta l)^{2}}$

${\displaystyle t_{f}={\sqrt {\frac {3\pi }{40\,G\,\rho }}}\left({\frac {x_{0}}{r}}\right)^{3/2}.}$

${\displaystyle m=\rho \cdot {\frac {4}{3}}\cdot \pi r^{3}.}$

${\displaystyle t_{f}={\frac {\pi }{\sqrt {10Gm}}}\;(x_{0})^{3/2}.}$

${\displaystyle t_{f}^{2}={\frac {\pi ^{2}}{8G(M+m)}}x_{0}^{3}.}$

${\displaystyle T^{2}={\frac {4\pi ^{2}}{G(M+m)}}a^{3}.}$

${\displaystyle {\frac {m_{1}}{m_{2}}}={\frac {b_{2}}{b_{1}}}.}$

${\displaystyle U=-{\frac {G\cdot m\cdot M}{r}}}$

${\displaystyle F_{G}={\frac {G\cdot m_{1}\cdot m_{2}}{r^{2}}},}$

${\displaystyle G=6{,}674\,\cdot 10^{-11}\mathrm {\frac {m^{3}}{kg\cdot s^{2}}} .}$

${\displaystyle v_{a}={\sqrt {2gh_{0}}}.}$

${\displaystyle T+U={\frac {1}{2}}mv_{a}^{2}.}$

${\displaystyle T+U=mgh_{0}.}$

${\displaystyle T+U={\frac {1}{2}}mv^{2}+mgh}$

${\displaystyle a\cdot \sin(\omega \cdot t+b)}$

${\displaystyle S_{E}={\frac {s}{\sqrt {n}}}}$

${\displaystyle r={\sqrt {(x_{2}-x_{1})^{2}+(y_{2}-y_{1})^{2}}}}$

${\displaystyle z'=z.}$

${\displaystyle z'=z-a_{3}}$

${\displaystyle {\begin{array}{lcl}x'&=&x\cos \varphi +y\sin \varphi \\[0.3em]y'&=&y\cos \varphi -x\sin \varphi \end{array}}}$

${\displaystyle {\begin{array}{lcl}x&=&x'\cos \varphi -y'\sin \varphi \\[0.3em]y&=&y'\cos \varphi +x'\sin \varphi .\end{array}}}$

${\displaystyle {\begin{array}{lcl}y'&=&(y-x/\cot \varphi )\cos \varphi =y\cos \varphi -x\sin \varphi \\[0.3em]x'&=&x/\cos \varphi +y'/\cot \varphi =(x[1-\sin ^{2}\varphi ])/\cos \varphi +y\sin \varphi \\[0.3em]&=&x\cos \varphi +y\sin \varphi ,\end{array}}}$

${\displaystyle {\begin{array}{lcl}x&=&g_{1}\cos \varphi =h_{1}\cot \varphi \\[0.3em]y'&=&h_{2}\cos \varphi =g_{2}\cot \varphi \\[0.3em]x'&=&g_{1}+g_{2}\\[0.3em]y&=&h_{1}+h_{2}.\end{array}}}$

${\displaystyle {\begin{array}{lcl}x'&=&x-a_{1}\\[0.3em]y'&=&y-a_{2},\end{array}}}$

${\displaystyle {\begin{array}{lcl}x&=&x'+a_{1}\\[0.3em]y&=&y'+a_{2}.\end{array}}}$

${\displaystyle F_{x}(2\to 1)}$

${\displaystyle t''\;=\;t+c_{0},}$

${\displaystyle t''\;=\;t'+b_{0},}$

${\displaystyle t'\;=\;t+a_{0}}$

${\displaystyle t\;=\;t'-a_{0}.}$

${\displaystyle t\;=\;t'-a_{0}=(t+a_{0})-a_{0}=t.}$

${\displaystyle {\begin{array}{lcl}x'&=&x'(x,y,z,t)\\[0.5em]y'&=&y'(x,y,z,t)\\[0.5em]z'&=&z'(x,y,z,t)\\[0.5em]t'&=&t'(x,y,z,t).\end{array}}}$

${\displaystyle {\begin{array}{lcl}x&=&x(x',y',z',t')\\[0.5em]y&=&y(x',y',z',t')\\[0.5em]z&=&z(x',y',z',t')\\[0.5em]t&=&t(x',y',z',t').\end{array}}}$

${\displaystyle {\begin{array}{lcl}x(E)&=&e'(x'(E),y'(E),z'(E),t'(E))\\[0.5em]y(E)&=&f'(x'(E),y'(E),z'(E),t'(E))\\[0.5em]z(E)&=&g'(x'(E),y'(E),z'(E),t'(E))\\[0.5em]t(E)&=&h'(x'(E),y'(E),z'(E),t'(E)).\\[0.5em]\end{array}}}$

${\displaystyle {\begin{array}{lcl}x'(E)&=&e(x(E),y(E),z(E),t(E))\\[0.5em]y'(E)&=&f(x(E),y(E),z(E),t(E))\\[0.5em]z'(E)&=&g(x(E),y(E),z(E),t(E))\\[0.5em]t'(E)&=&h(x(E),y(E),z(E),t(E))\\[0.5em]\end{array}}}$

${\displaystyle {\begin{array}{lcl}{\frac {[v_{y}(t_{0}+\Delta t)]^{2}-[v_{y}(t_{0})]^{2}}{\Delta t}}&=&\Delta t\cdot b_{y}(t_{0})^{2}+\Delta t\cdot [R(t_{0},\Delta t)]^{2}\\[0.5em]&&+2\Delta tb_{y}(t_{0})R(t_{0},\Delta t)+2\,b_{y}(t_{0})v_{y}(t_{0})\\[0.5em]&&+2\,R(t_{0},\Delta t)v_{y}(t_{0}).\end{array}}}$

${\displaystyle v_{y}(t_{0}+\Delta t)=\Delta t\cdot b_{y}(t_{0})+\Delta t\cdot R(t_{0},\Delta t)+v_{y}(t_{0}).}$

${\displaystyle R(t_{0},\Delta t)={\frac {v_{y}(t_{0}+\Delta t)-v_{y}(t_{0})}{\Delta t}}-b_{y}(t_{0})}$

${\displaystyle b_{y}(t_{0})=\lim _{\Delta t\to 0}{\frac {v_{y}(t_{0}+\Delta t)-v_{y}(t_{0})}{\Delta t}}.}$

${\displaystyle {\frac {\mathrm {d} (T+U)}{\mathrm {d} t}}=-mv_{y}g+mgv_{y}=0.}$

${\displaystyle {\frac {\mathrm {d} T}{\mathrm {d} t}}=mv_{y}{\frac {\mathrm {d} v_{y}}{\mathrm {d} t}}}$

${\displaystyle {\frac {\mathrm {d} T}{\mathrm {d} t}}=mv_{y}b_{y}=-mv_{y}g.}$

${\displaystyle F_{y}=-mg}$

${\displaystyle F_{y}=m\cdot b_{y}}$

${\displaystyle b_{x}=-g.}$

${\displaystyle T+U={\frac {1}{2}}mv_{y}^{2}+mgy}$

${\displaystyle F_{x}=m\cdot b_{x}}$

${\displaystyle {\frac {m_{1}}{m_{2}}}=\left({\frac {T_{1}}{T_{2}}}\right)^{2}.}$

${\displaystyle {\frac {m_{1}}{m_{2}}}={\frac {a_{2}}{a_{1}}}.}$

${\displaystyle p_{1x}+p_{2x}=p'_{1x}+p'_{2x},}$

${\displaystyle m_{1}v_{1x}+m_{2}v_{2x}=m_{1}v'_{1x}+m_{2}v'_{2x},}$

${\displaystyle {\frac {m_{1}}{m_{2}}}={\frac {(v'_{2x}-v_{2x})}{(v_{1x}-v'_{1x})}}.}$

${\displaystyle F_{xij}=-F_{xji},}$

${\displaystyle {\dot {P}}_{x}={\dot {p}}_{1x}+\cdots +{\dot {p}}_{nx}=F_{x1}+\cdots +F_{xn}.}$

${\displaystyle {\dot {P}}_{x}=0,}$

${\displaystyle F_{x12}=-F_{x21}.}$

${\displaystyle F_{x13}=-F_{x31}}$

${\displaystyle F_{x23}=-F_{x32}.}$

${\displaystyle {\begin{array}{lcl}{\dot {P}}_{x}&=&{\dot {p}}_{1x}+{\dot {p}}_{2x}+{\dot {p}}_{3x}=F_{x1}+F_{x2}+F_{x3}\\[0.5em]&=&F_{x12}+F_{x13}+F_{x21}+F_{x23}+F_{x31}+F_{x32}.\end{array}}}$

${\displaystyle F_{x1}=F_{x12}+F_{x13},}$

${\displaystyle F_{x2}=F_{x21}+F_{x23}}$

${\displaystyle F_{x3}=F_{x31}+F_{x32}.}$

${\displaystyle {\dot {P}}_{x}={\dot {p}}_{1x}+{\dot {p}}_{2x}+{\dot {p}}_{3x}.}$

${\displaystyle P_{x}=p_{1x}+p_{2x}+p_{3x},}$

${\displaystyle F_{xi}=F_{xi1}+F_{xi2}+\cdots +F_{xi(i-1)}+F_{xi(i+1)}+\cdots +F_{xin}.}$

${\displaystyle {\dot {P}}_{x}={\dot {p}}_{1x}+{\dot {p}}_{2x}+\cdots +{\dot {p}}_{nx}.}$

${\displaystyle P_{x}=p_{1x}+p_{2x}+\cdots +p_{nx},}$

${\displaystyle {\dot {x}}(t)={\frac {\mathrm {d} x(t)}{\mathrm {d} t}}}$

${\displaystyle \left.{\frac {\mathrm {d} y}{\mathrm {d} x}}\right|_{x=x_{0}}=\left[\lim _{\Delta x\to 0}{\frac {\Delta y}{\Delta x}}\right]_{x=x_{0}}=\lim _{\Delta x\to 0}{\frac {f(x_{0}+\Delta x)-f(x_{0})}{\Delta x}}.}$

${\displaystyle {\frac {f(x_{0}+\Delta x)-f(x_{0})}{x_{0}+\Delta x-x_{0}}}={\frac {f(x_{0}+\Delta x)-f(x_{0})}{\Delta x}}.}$

${\displaystyle a={\frac {y_{2}-y_{1}}{x_{2}-x_{1}}}}$

${\displaystyle b={\frac {y_{1}x_{2}-y_{2}x_{1}}{x_{2}-x_{1}}}}$

${\displaystyle y_{1}=ax_{1}+b}$

${\displaystyle y_{2}=ax_{2}+b.}$

${\displaystyle {\begin{array}{lcl}y_{1}=ax_{1}+b\\y_{2}=ax_{2}+b.\end{array}}}$

${\displaystyle v_{x}={\frac {\Delta x}{\Delta t}}}$

${\displaystyle \Delta x=v_{x}\cdot (t_{2}-t_{1})=v_{x}\cdot \Delta t}$

${\displaystyle {\vec {L}}={\vec {r}}\times {\vec {p}}.}$

${\displaystyle {\begin{array}{lcl}L_{x}&=&y\cdot p_{z}-z\cdot p_{y}\\L_{y}&=&z\cdot p_{x}-x\cdot p_{z}\\L_{z}&=&x\cdot p_{y}-y\cdot p_{x}\end{array}}}$

${\displaystyle U={\frac {1}{2}}\,m\cdot k\cdot (\Delta l)^{2},}$

${\displaystyle U=m\cdot g\cdot y,}$

${\displaystyle m\cdot v_{y0}\cdot g\cdot \Delta t}$

${\displaystyle U=m\cdot g\cdot (y_{y0}+v_{y0}\cdot \Delta t)}$

${\displaystyle U=m\cdot g\cdot y_{y0}}$

${\displaystyle U=m\cdot g\cdot y}$

${\displaystyle -m\cdot v_{y0}\cdot g\cdot \Delta t.}$

${\displaystyle T={\frac {1}{2}}m\cdot v^{2}}$

${\displaystyle T={\frac {1}{2}}m\cdot v_{y0}^{2}}$

${\displaystyle T={\frac {1}{2}}m\cdot \left(v_{y0}-g\cdot \Delta t\right)^{2}={\frac {1}{2}}m\cdot \left(v_{y0}-2v_{y0}\cdot g\cdot \Delta t\right)+O(\Delta t^{2})}$

Ab hier in Reihenfolge:

${\displaystyle {\begin{array}{l}x(t)=v_{x}\cdot t+x_{0}\\[0.2em]y(t)=v_{y}\cdot t+y_{0}\\[0.2em]z(t)=v_{z}\cdot t+z_{0}\end{array}}}$

${\displaystyle v={\sqrt {v_{x}^{2}+v_{y}^{2}+v_{z}^{2}}}}$

${\displaystyle d={\sqrt {[x(t_{2})-x(t_{1})]^{2}+[y(t_{2})-y(t_{1})]^{2}+[z(t_{2})-z(t_{1})]^{2}}}}$

${\displaystyle d={\sqrt {v_{x}^{2}+v_{y}^{2}+v_{z}^{2}}}\cdot (t_{2}-t_{1})=v\cdot (t_{2}-t_{1}).}$

${\displaystyle v_{x}={\frac {\mathrm {d} x(t)}{\mathrm {d} t}}}$

${\displaystyle x(t)={\frac {1}{2}}\cdot b_{x}\cdot t^{2}+v_{0x}\cdot t+x_{0}.}$

${\displaystyle x(t)=\left[{\frac {1}{2}}\cdot b_{x}\cdot t+v_{0x}\right]\cdot t+x_{0}.}$

${\displaystyle v_{x}(t)={\frac {\mathrm {d} x(t)}{\mathrm {d} t}}=b_{x}\cdot t+v_{0x}}$

${\displaystyle {\begin{array}{rl}x(t+\Delta t)=&{\frac {1}{2}}b_{x}\cdot (t+\Delta t)^{2}+v_{0x}\cdot (t+\Delta t)+x_{0}\\[0.5em]=&{\frac {1}{2}}b_{x}\cdot t^{2}+b_{x}\cdot t\cdot \Delta t+{\frac {1}{2}}b_{x}\cdot (\Delta t)^{2}\\[0.25em]&+v_{0x}\cdot t+v_{0x}\Delta t+x_{0}.\end{array}}}$

${\displaystyle {\begin{array}{rl}x(t+\Delta t)-x(t)=&b_{x}\cdot t\cdot \Delta t+{\frac {1}{2}}b_{x}\cdot (\Delta t)^{2}+v_{0x}\Delta t.\end{array}}}$

${\displaystyle {\frac {x(t+\Delta t)-x(t)}{\Delta t}}=b_{x}\cdot t+{\frac {1}{2}}b_{x}\cdot \Delta t+v_{0x}.}$

${\displaystyle v_{x}(t)=b_{x}\cdot t+v_{0x}.}$

${\displaystyle v_{x}(t)={\frac {\mathrm {d} x(t)}{\mathrm {d} t}},\;\;\;b_{x}(t)={\frac {\mathrm {d} v_{x}(t)}{\mathrm {d} t}}}$

${\displaystyle y={\frac {v_{y}}{v_{x}}}\cdot (x-x_{0})+y_{0}}$

${\displaystyle F_{x}=m\cdot b_{x},}$

${\displaystyle {\begin{array}{l}F(2\to 1)_{x}=-F(1\to 2)_{x}\\[0.5em]F(2\to 1)_{y}=-F(1\to 2)_{y}\\[0.5em]F(2\to 1)_{z}=-F(1\to 2)_{z}\end{array}}}$

${\displaystyle p_{x}=m\cdot v_{x},}$

${\displaystyle {\dot {p}}_{x},}$

${\displaystyle {\dot {p}}_{x}=m\cdot b_{x}}$

${\displaystyle F_{x}={\dot {p}}_{x}.}$

${\displaystyle {\dot {p}}_{1x}+{\dot {p}}_{2x}+{\dot {p}}_{3x}+\ldots }$

${\displaystyle F=k\cdot \Delta l}$

${\displaystyle F=-{\frac {G\cdot m_{1}\cdot m_{2}}{r^{2}}}}$

${\displaystyle M=m_{1}+m_{2}+\ldots +m_{n}}$

${\displaystyle {\overline {x}}={\frac {1}{M}}(m_{1}\cdot x_{1}+m_{2}\cdot x_{2}+\ldots +m_{n}\cdot x_{n})}$

${\displaystyle {\overline {v}}_{x}={\frac {1}{M}}(p_{1x}+p_{2x}\ldots +p_{nx}),}$

${\displaystyle {\overline {P}}_{x}=M\cdot {\overline {v}}_{x},}$

${\displaystyle T={\frac {1}{2}}m\cdot v^{2},}$

${\displaystyle {\begin{array}{rcl}{\frac {1}{(6\,370\,000)^{2}}}&=&2,46446\cdot 10^{-14}\\[0.4em]{\frac {1}{(6\,370\,000+1)^{2}}}&=&2,46446\cdot 10^{-14}\\[0.4em]{\frac {1}{(6\,370\,000+10)^{2}}}&=&2,46445\cdot 10^{-14}\\[0.4em]{\frac {1}{(6\,370\,000+100)^{2}}}&=&2,46438\cdot 10^{-14}\\[0.4em]{\frac {1}{(6\,370\,000+1000)^{2}}}&=&2,46368\cdot 10^{-14}\\[0.4em]{\frac {1}{(6\,370\,000+10000)^{2}}}&=&2,45674\cdot 10^{-14}\end{array}}}$

${\displaystyle g={\frac {G\cdot m_{E}}{r^{2}}},}$