siehe Hilfe:TeX
![{\displaystyle Saturn\quad \alpha \ {\text{Saturn}}\qquad \mathbf {Saturn} \ {\mathtt {Saturn}}\ {\boldsymbol {Saturn}}\ {\mathfrak {Saturn}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/1108a4ec28eb1b340560f1855694968b030af586)
![{\displaystyle \zeta =2\,\cdot \,\arcsin \left(\,{\frac {1}{2}}\cdot {\sqrt {{\Big (}\sin(\phi _{A})-\sin(\phi _{B}){\Big )}^{2}+{\Big (}\cos(\phi _{A})\cdot \cos(\lambda _{A})+\cos(\phi _{B})\cdot \cos(\lambda _{B}){\Big )}^{2}+{\Big (}\cos(\phi _{A})\cdot \sin(\lambda _{A})-\cos(\phi _{B})\cdot \sin(\lambda _{B}){\Big )}^{2}}}\ \right)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/d2fd25ff6cc26b1428cfaf7c1c6c931661784a42)
![{\displaystyle \zeta =2\,\cdot \,\arcsin \left(\,{\frac {1}{2}}\cdot {\sqrt {{\Big (}\sin(\phi _{A})-\sin(\phi _{B}){\Big )}^{2}+{\Big (}\cos(\phi _{A})\cdot \cos(\lambda _{A})+\cos(\phi _{B})\cdot \cos(\lambda _{B}){\Big )}^{2}+{\Big (}\cos(\phi _{A})\cdot \sin(\lambda _{A})-\cos(\phi _{B})\cdot \sin(\lambda _{B}){\Big )}^{2}}}\ \right)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/d2fd25ff6cc26b1428cfaf7c1c6c931661784a42)
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Exponentiation is a mathematical operation, written as
, scriptstyle:
, displaystyle:
, textstyle:
, involving two numbers, the base
and the exponent
. When
is a positive integer, exponentiation corresponds to repeated multiplication of the base: that is,
is the product of multiplying
bases:
, scriptstyle:
, displaystyle:
, textstyle:
![{\displaystyle \displaystyle b^{n}=\underbrace {b\times \cdots \times b} _{n}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/36d60b0305fa496630e54be177b5e9062987fb2e)
![{\displaystyle {\frac {2+{\frac {1/2+{\frac {5}{3}}}{5^{3}}}}{\sqrt {\frac {2}{3}}}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/082ac5e4d0088f69db5ea3bc13cbf96b39a7bba3)
The exponent is usually shown as a superscript to the right of the base. In that case,
is called “b raised to the n-th power”, b raised to the power of n, or the n-th power of b.
When
is a positive integer and
is not zero,
is naturally defined as
or
, or
preserving the property
. With exponent
,
is equal to
, and is the reciprocal of
.
![{\displaystyle {\begin{alignedat}{2}{\text{es sei}}\\y&=\log _{b}x&\qquad &{\big |}\ {\text{auf beiden Seiten}}\ b^{(\ )}\ {\text{und mit}}\ b^{\log _{b}x}=x\\b^{y}&=x&&{\big |}\ {\text{auf beiden Seiten}}\log _{a}{(\ )}\\\log _{a}b^{y}&=\log _{a}x&&{\big |}\ {\text{mit}}\log b^{y}=y\cdot \log a\\y\log _{a}b&=\log _{a}x&&{\big |}\ {\text{auf beiden Seiten}}\div \log _{a}b\\{\text{dann ist}}\\y=\log _{b}x&={\frac {\log _{a}x}{\log _{a}b}}\end{alignedat}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/21c50c92015ec18596f5d59756341f63d29b195f)
- Find the equation of the line that is tangent to the following curve at
:
![{\displaystyle y=x^{3}-12x^{2}-42.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/6d958b0b744df43fdb503c170738f45486bdbcac)
- Begin by dividing the polynomial by
:
![{\displaystyle {\begin{array}{rclcl}x^{3}-12x^{2}{\color {Red}\,+\,21x}-42&\div &x^{2}-2x+1&=&x\\x^{3}-12x^{2}+21x-42\\{\underline {x^{3}-{\color {Yellow}0}2x^{2}+{\color {Red}1}x\,{\color {Red}-\,42}}}\\-10x^{2}-{\color {White}01}x-42\\{\underline {-10x^{2}+20x-10}}\\-21x-32\end{array}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/ee91dafdb8b371fd107e2b259fcfca9ca48f3a97)
Drehung um die
-Achse:
![{\displaystyle R_{x}(\alpha )={\begin{pmatrix}1&0&0\\0&\cos \alpha &-\sin \alpha \\0&\sin \alpha &\cos \alpha \end{pmatrix}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/70c39b1c4aba6804b353d7e214c945707941698b)
Drehung um die
- und die
-Achse:
![{\displaystyle R_{y}(\alpha )={\begin{pmatrix}\cos \alpha &0&\sin \alpha \\0&1&0\\-\sin \alpha &0&\cos \alpha \end{pmatrix}};\quad R_{z}(\alpha )={\begin{pmatrix}\cos \alpha &-\sin \alpha &0\\\sin \alpha &\cos \alpha &0\\0&0&1\end{pmatrix}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/40f1503d1651f93ed66a79a235d505c555b5aa60)
![{\displaystyle {\begin{aligned}&a^{(i)}=[{}_{i}^{k}A]\cdot a^{(k)}\\&a^{(k)}=[{}_{i}^{k}A]^{-1}\cdot a^{(i)}\end{aligned}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/895cc1b5f1580929f5833db0c49306ee8288e6b9)
Inverse Drehmatrizen:
![{\displaystyle (A\cdot B\cdot C)^{-1}=C^{-1}\cdot B^{-1}\cdot A^{-1}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/82b8297bd684ef89f66c03e61b647b6000cbd2d0)
Kombinierte, hintereinander ausgeführte (innere) Drehungen um die Achsen
,
und
:
![{\displaystyle {\begin{pmatrix}r_{00}&r_{01}&r_{02}\\r_{10}&r_{11}&r_{12}\\r_{20}&r_{21}&r_{22}\end{pmatrix}}={\begin{pmatrix}\cos \theta _{y}\cdot \cos \theta _{z}&-\cos \theta _{y}\cdot \sin \theta _{z}&\sin \theta _{y}\\\sin \theta _{x}\cdot \sin \theta _{y}\cdot \cos \theta _{z}+\cos \theta _{x}\cdot \sin \theta _{z}&-\sin \theta _{x}\cdot \sin \theta _{y}\cdot \sin \theta _{z}+\cos \theta _{x}\cdot \cos \theta _{z}&-\sin \theta _{x}\cdot \cos \theta _{y}\\-\cos \theta _{x}\cdot \sin \theta _{y}\cdot \cos \theta _{z}+\sin \theta _{x}\cdot \sin \theta _{z}&\cos \theta _{x}\cdot \sin \theta _{y}\cdot \sin \theta _{z}+\sin \theta _{x}\cdot \cos \theta _{z}&\cos \theta _{x}\cdot \cos \theta _{y}\end{pmatrix}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/3c6df7ca0b7e1da245acf106e8e3053bd0cb66e5)
![{\displaystyle r_{02}=\sin \theta _{y}\quad \Longrightarrow \quad \theta _{y}={\text{asin}}(r_{02})}](https://wikimedia.org/api/rest_v1/media/math/render/svg/33c55d996b9f0a4f13580df9b0a34a16c71af09f)
![{\displaystyle \Longrightarrow \quad \sin \theta _{x}={\frac {-r_{12}}{\cos \theta _{y}}}\quad {\text{und}}\quad \cos \theta _{x}={\frac {r_{22}}{\cos \theta _{y}}};\quad \sin \theta _{z}={\frac {-r_{01}}{\cos \theta _{y}}}\quad {\text{und}}\quad \cos \theta _{z}={\frac {r_{00}}{\cos \theta _{y}}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/c0d48806a56f358bba7932ff324735d064d74409)
![{\displaystyle \Longrightarrow \quad \theta _{x}={\text{atan2}}(-r_{12},r_{22});\quad \theta _{z}={\text{atan2}}(-r_{01},r_{00})\quad {\text{mit}}\quad \theta ={\text{atan2}}(\sin \theta ,\cos \theta )}](https://wikimedia.org/api/rest_v1/media/math/render/svg/7edbae1986c83546b2a1718a211dd79f51ff468a)
![{\displaystyle {\begin{alignedat}{3}&r_{10}=r_{21}&&=\sin \theta _{x}\cdot \cos \theta _{z}+\cos \theta _{x}\cdot \sin \theta _{z}\quad &&\Longrightarrow \quad r_{10}=r_{21}=\sin(\theta _{x}+\theta _{z})\\&r_{11}=-r_{20}&&=\cos \theta _{x}\cdot \cos \theta _{z}-\sin \theta _{x}\cdot \sin \theta _{z}\quad &&\Longrightarrow \quad r_{11}=-r_{20}=\cos(\theta _{x}+\theta _{z})\end{alignedat}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/067ec35bf65f255c63190ed61ce0c5c66647c05a)
![{\displaystyle \Longrightarrow \quad \theta _{x}+\theta _{z}={\text{atan2}}(r_{10},r_{11})={\text{atan2}}(r_{21},-r_{20})\quad {\text{mit}}\quad \theta ={\text{atan2}}(\sin \theta ,\cos \theta )\quad {\text{Lösung mit einem Freiheitsgrad,}}\ \theta _{x}\ {\text{oder}}\ \theta _{z}\ {\text{ist frei wählbar}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/1ffa3d6d1099467b04a1ec970b7be71b58c7fdae)
![{\displaystyle {\begin{alignedat}{5}-r_{10}&=r_{21}&&=\sin \theta _{x}\cdot \cos \theta _{z}-\cos \theta _{x}\cdot \sin \theta _{z}\quad &&\Longrightarrow \quad -r_{10}&&=r_{21}&&=\sin(\theta _{x}-\theta _{z})\\r_{11}&=r_{20}&&=\cos \theta _{x}\cdot \cos \theta _{z}+\sin \theta _{x}\cdot \sin \theta _{z}\quad &&\Longrightarrow \quad r_{11}&&=r_{20}&&=\cos(\theta _{x}-\theta _{z})\end{alignedat}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/b3f9c738d015b6ac4c8838a4122dfbce8cad741b)
![{\displaystyle \Longrightarrow \quad \theta _{x}-\theta _{z}={\text{atan2}}(-r_{10},r_{11})={\text{atan2}}(r_{21},r_{20})\quad {\text{mit}}\quad \theta ={\text{atan2}}(\sin \theta ,\cos \theta )\quad {\text{Lösung mit einem Freiheitsgrad,}}\ \theta _{x}\ {\text{oder}}\ \theta _{z}\ {\text{ist frei wählbar}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/550ec343188ebbd0446ebe17be59af0f35785c28)
Kombinierte, hintereinander ausgeführte (innere) Drehungen um die Achsen
,
und
:
![{\displaystyle {\begin{pmatrix}r_{00}&r_{01}&r_{02}\\r_{10}&r_{11}&r_{12}\\r_{20}&r_{21}&r_{22}\end{pmatrix}}={\begin{pmatrix}\cos \theta _{y}\cdot \cos \theta _{z}&\sin \theta _{x}\cdot \sin \theta _{y}\cdot \cos \theta _{z}-\cos \theta _{x}\cdot \sin \theta _{z}&\cos \theta _{x}\cdot \sin \theta _{y}\cdot \cos \theta _{z}+\sin \theta _{x}\cdot \sin \theta _{z}\\\cos \theta _{y}\cdot \sin \theta _{z}&\sin \theta _{x}\cdot \sin \theta _{y}\cdot \sin \theta _{z}+\cos \theta _{x}\cdot \cos \theta _{z}&\cos \theta _{x}\cdot \sin \theta _{y}\cdot \sin \theta _{z}-\sin \theta _{x}\cdot \cos \theta _{z}\\-\sin \theta _{y}&\sin \theta _{x}\cdot \cos \theta _{y}&\cos \theta _{x}\cdot \cos \theta _{y}\end{pmatrix}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/38fbdbb92f6b17332891dfc1fcc20a1cc1ef456b)
![{\displaystyle -r_{20}=\sin \theta _{y}\quad \Longrightarrow \quad \theta _{y}={\text{asin}}(-r_{20})}](https://wikimedia.org/api/rest_v1/media/math/render/svg/552e953627bcb85021c2e6184ad5a409b6e30fb7)
![{\displaystyle \Longrightarrow \quad \sin \theta _{x}={\frac {r_{21}}{\cos \theta _{y}}}\quad {\text{und}}\quad \cos \theta _{x}={\frac {r_{22}}{\cos \theta _{y}}};\quad \sin \theta _{z}={\frac {r_{10}}{\cos \theta _{y}}}\quad {\text{und}}\quad \cos \theta _{z}={\frac {r_{00}}{\cos \theta _{y}}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/65ae27aa5bedeb22a046c56241fd3ef096a3e603)
![{\displaystyle \Longrightarrow \quad \theta _{x}={\text{atan2}}(r_{21},r_{22});\quad \theta _{z}={\text{atan2}}(r_{10},r_{00})\quad {\text{mit}}\quad \theta ={\text{atan2}}(\sin \theta ,\cos \theta )}](https://wikimedia.org/api/rest_v1/media/math/render/svg/b7730fc4ba1c17cb5c4894a10f34a31f1b0c4f25)
![{\displaystyle {\begin{alignedat}{3}&r_{01}=-r_{12}&&=\sin \theta _{x}\cdot \cos \theta _{z}-\cos \theta _{x}\cdot \sin \theta _{z}\quad &&\Longrightarrow \quad r_{01}=-r_{12}=\sin(\theta _{x}-\theta _{z})\\&r_{11}=r_{02}&&=\cos \theta _{x}\cdot \cos \theta _{z}+\sin \theta _{x}\cdot \sin \theta _{z}\quad &&\Longrightarrow \quad r_{11}=r_{02}=\cos(\theta _{x}-\theta _{z})\end{alignedat}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/e28c05161d056fae47203e7abbb8fab5975d3133)
![{\displaystyle \Longrightarrow \quad \theta _{x}-\theta _{z}={\text{atan2}}(r_{01},r_{11})={\text{atan2}}(-r_{12},r_{02})\quad {\text{mit}}\quad \theta ={\text{atan2}}(\sin \theta ,\cos \theta )\quad {\text{Lösung mit einem Freiheitsgrad,}}\ \theta _{x}\ {\text{oder}}\ \theta _{z}\ {\text{ist frei wählbar}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/96ec70aa88e396bb946f59e715df245802b7b663)
![{\displaystyle {\begin{alignedat}{5}-r_{01}&=-r_{12}&&=\sin \theta _{x}\cdot \cos \theta _{z}+\cos \theta _{x}\cdot \sin \theta _{z}\quad &&\Longrightarrow \quad -r_{01}&&=-r_{12}&&=\sin(\theta _{x}+\theta _{z})\\r_{11}&=-r_{02}&&=\cos \theta _{x}\cdot \cos \theta _{z}-\sin \theta _{x}\cdot \sin \theta _{z}\quad &&\Longrightarrow \quad r_{11}&&=-r_{02}&&=\cos(\theta _{x}+\theta _{z})\end{alignedat}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/e50ad332464350de1d699d1201909af777169ef4)
![{\displaystyle \Longrightarrow \quad \theta _{x}+\theta _{z}={\text{atan2}}(-r_{01},r_{11})={\text{atan2}}(-r_{12},-r_{02})\quad {\text{mit}}\quad \theta ={\text{atan2}}(\sin \theta ,\cos \theta )\quad {\text{Lösung mit einem Freiheitsgrad,}}\ \theta _{x}\ {\text{oder}}\ \theta _{z}\ {\text{ist frei wählbar}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/51a478621e52bb723d5c7d44ea793a88430dddd8)
Kombinierte, hintereinander ausgeführte (innere) Drehungen um die Achsen
,
und
:
![{\displaystyle {\begin{pmatrix}r_{00}&r_{01}&r_{02}\\r_{10}&r_{11}&r_{12}\\r_{20}&r_{21}&r_{22}\end{pmatrix}}={\begin{pmatrix}\cos \theta _{z_{1}}\cdot \cos \theta _{y}\cdot \cos \theta _{z_{2}}-\sin \theta _{z_{1}}\cdot \sin \theta _{z_{2}}&-\cos \theta _{z_{1}}\cdot \cos \theta _{y}\cdot \sin \theta _{z_{2}}-\sin \theta _{z_{1}}\cdot \cos \theta _{z_{2}}&\cos \theta _{z_{1}}\cdot \sin \theta _{y}\\\sin \theta _{z_{1}}\cdot \cos \theta _{y}\cdot \cos \theta _{z_{2}}+\cos \theta _{z_{1}}\cdot \sin \theta _{z_{2}}&-\sin \theta _{z_{1}}\cdot \cos \theta _{y}\cdot \sin \theta _{z_{2}}+\cos \theta _{z_{1}}\cdot \cos \theta _{z_{2}}&\sin \theta _{z_{1}}\cdot \sin \theta _{y}\\-\sin \theta _{y}\cdot \cos \theta _{z_{2}}&\sin \theta _{y}\cdot \sin \theta _{z_{2}}&\cos \theta _{y}\end{pmatrix}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/0806403b03c762ca39ae50e2f135133cd4ce7b2c)
![{\displaystyle r_{22}=\cos \theta _{y}\quad \Longrightarrow \quad \theta _{y}={\text{acos}}(r_{22})}](https://wikimedia.org/api/rest_v1/media/math/render/svg/ce74877f6935d7eb0b0cd2cd822bbf33c49f4cb5)
![{\displaystyle \Longrightarrow \quad \sin \theta _{z_{1}}={\frac {r_{12}}{\sin \theta _{y}}}\quad {\text{und}}\quad \cos \theta _{z_{1}}={\frac {r_{02}}{\sin \theta _{y}}};\quad \sin \theta _{z_{2}}={\frac {r_{21}}{\sin \theta _{y}}}\quad {\text{und}}\quad \cos \theta _{z_{2}}={\frac {-r_{20}}{\sin \theta _{y}}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/444d12a46ee13f9077fb27f33f43fde61dae686d)
![{\displaystyle \Longrightarrow \quad \theta _{z_{1}}={\text{atan2}}(r_{12},r_{02});\quad \theta _{z_{2}}={\text{atan2}}(r_{21},-r_{20})\quad {\text{mit}}\quad \theta ={\text{atan2}}(\sin \theta ,\cos \theta )}](https://wikimedia.org/api/rest_v1/media/math/render/svg/95a110e11612355d3657ddcb3a43494de93d27e5)
![{\displaystyle {\begin{alignedat}{8}-r_{01}&=r_{10}&&=\sin \theta _{z_{1}}\cdot \cos \theta _{z_{2}}+\cos \theta _{z_{1}}\cdot \sin \theta _{z_{2}}\quad &&\Longrightarrow \quad -r_{01}&&=r_{10}&&=\sin(\theta _{z_{1}}+\theta _{z_{2}})\\r_{00}&=r_{11}&&=\cos \theta _{z_{1}}\cdot \cos \theta _{z_{2}}-\sin \theta _{z_{1}}\cdot \sin \theta _{z_{2}}\quad &&\Longrightarrow \quad r_{00}&&=r_{11}&&=\cos(\theta _{z_{1}}+\theta _{z_{2}})\end{alignedat}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/a70c33b806b65f5509b67246f078d32742e7a5a1)
![{\displaystyle \Longrightarrow \quad \theta _{z_{1}}+\theta _{z_{2}}={\text{atan2}}(-r_{01},r_{00})={\text{atan2}}(r_{10},r_{11})\quad {\text{mit}}\quad \theta ={\text{atan2}}(\sin \theta ,\cos \theta )\quad {\text{Lösung mit einem Freiheitsgrad,}}\ \theta _{x}\ {\text{oder}}\ \theta _{z}\ {\text{ist frei wählbar}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/6371e967c842b0dba8a4a9f719dbf6793b7bc455)
![{\displaystyle {\begin{alignedat}{8}-r_{01}&=-r_{10}&&=\sin \theta _{z_{1}}\cdot \cos \theta _{z_{2}}-\cos \theta _{z_{1}}\cdot \sin \theta _{z_{2}}\quad &&\Longrightarrow \quad -r_{01}&&=-r_{10}&&=\sin(\theta _{z_{1}}-\theta _{z_{2}})\\-r_{00}&=r_{11}&&=\cos \theta _{z_{1}}\cdot \cos \theta _{z_{2}}+\sin \theta _{z_{1}}\cdot \sin \theta _{z_{2}}\quad &&\Longrightarrow \quad r_{00}&&=r_{11}&&=\cos(\theta _{z_{1}}-\theta _{z_{2}})\end{alignedat}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/c112147d3281d0bbf4e7abf7688a341defa1f1ea)
![{\displaystyle \Longrightarrow \quad \theta _{z_{1}}-\theta _{z_{2}}={\text{atan2}}(-r_{01},r_{00})={\text{atan2}}(-r_{10},r_{11})\quad {\text{mit}}\quad \theta ={\text{atan2}}(\sin \theta ,\cos \theta )\quad {\text{Lösung mit einem Freiheitsgrad,}}\ \theta _{x}\ {\text{oder}}\ \theta _{z}\ {\text{ist frei wählbar}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/b62dd1f73da9dc0b54bc448c3b78081b4db2bce0)