Template:Integral/testcases

⧼Purge⧽ Template:Integral/testcases

Test cases for the {{integral}} template.

Gamma function

The code

: {{repeat|2|{{repeat|5|yadda{{nbsp}}}}{{nl}}}}before {{math|{{Gr|Gamma}}(''z'') {{=}} {{int|0|infty|''e''{{^|{{op|-}}{{sp|1}}''t''}}{{sp|2}}''t''{{^|{{sp|2}}''z''{{sp|1}}{{op|-}}1}}{{sp|3}}{{d|''t''}}}}|&}} after{{repeat|2|{{nl}}{{repeat|5|yadda{{nbsp}}}}}}

yields the text style HTML+CSS

yadda yadda yadda yadda yadda 
yadda yadda yadda yadda yadda 
before 

Γ(z) = ∫ ∞ 0  e  −  t  t   z  − 1 d t

after
yadda yadda yadda yadda yadda 
yadda yadda yadda yadda yadda 

The code

: {{repeat|2|{{repeat|5|yadda{{nbsp}}}}{{nl}}}}before {{math|{{Gr|Gamma|tex}}(''z'') {{=}} {{int|0|infty|''e''{{^|{{op|-}}{{sp|1|tex}}''t''|tex}}{{sp|2|tex}}''t''{{^|{{sp|1|tex}}''z''{{op|-}}1|tex}}{{sp|3|tex}}{{d|''t''|tex}}|tex}}|$}} after{{repeat|2|{{nl}}{{repeat|5|yadda{{nbsp}}}}}}

yields the text style LaTeX

yadda yadda yadda yadda yadda 
yadda yadda yadda yadda yadda 
before$\textstyle {\Gamma (z)=\int _{0}^{\infty }\;{e^{{\,}-{\,}{\;\;\!\!\!}t}{\;\!}t^{{\;\;\!\!\!}z{\,}-{\,}1}{\,}d^{}t}}$ after
yadda yadda yadda yadda yadda 
yadda yadda yadda yadda yadda 

The code

{{indent}}{{math|{{Gr|Gamma}}(''z'') {{=}} {{int|0|infty|''e''{{^|{{op|-}}{{sp|1}}''t''}}{{sp|2}}''t''{{^|{{sp|2}}''z''{{sp|1}}{{op|-}}1}}{{sp|3}}{{d|''t''}}|HTM}}|&&}}

yields the display style HTML+CSS

     

Γ(z) = ∫ ∞ 0  e  −  t  t   z  − 1 d t

The code

{{indent}}{{math|{{Gr|Gamma|tex}}(''z'') {{=}} {{int|0|infty|''e''{{^|{{op|-}}{{sp|1|tex}}''t''|tex}}{{sp|2|tex}}''t''{{^|{{sp|1|tex}}''z''{{op|-}}1|tex}}{{sp|3|tex}}{{d|''t''|tex}}|TEX}}|$$}}

yields the display style LaTeX

     ${\begin{array}{l}\displaystyle {\Gamma (z)=\int _{0}^{\infty }\;{e^{-{\;\;\!\!\!}t}{\;\!}t^{{\;\;\!\!\!}z-1}{\,}d^{}t}}\end{array}}$

Maxwell's equations

The code

{{indent}}{{math|
{{begin|align}}
{{oiint|{{d|''V''|part}}||{{vec|E|b}}{{sp|1}}{{op|sdot}}{{sp|1}}{{d|{{vec|S|b}}}}|HTM}} {{&=}} <!--
  -->{{frac|1|''{{Gr|epsilon}}''{{sub|0}}|HTM}} {{iiint|''V''||''{{Gr|rho}}''{{sp|3}}{{d|''V''}}|HTM}} {{\\}}

{{oiint|{{d|''V''|part}}||{{vec|B|b}}{{sp|1}}{{op|sdot}}{{sp|1}}{{d|{{vec|S|b}}}}|HTM}} {{&=}} 0 {{\\}}

{{oint|{{d|''S''|part}}||{{vec|E|b}}{{sp|1}}{{op|sdot}}{{sp|1}}{{d|{{vec|x|b}}}}|HTM}} {{&=}} <!--
  -->{{op|−}}{{iint|''S''||{{partial|{{vec|B|b}}|''t''|HTM}}{{sp|1}}{{op|sdot}}{{sp|1}}{{d|{{vec|S|b}}}}|HTM}} {{\\}}

{{oint|{{sym|partial}}''S''||{{vec|B|b}}{{sp|1}}{{op|sdot}}{{sp|1}}{{d|{{vec|x|b}}}}|HTM}} {{&=}} <!--
  -->{{iint|''S''||{{(|''{{Gr|mu}}''{{sub|0}}{{sp|1}}{{vec|J|b}} + {{frac|1|''c''{{^|2}}|HTM}} <!--
  -->{{partial|{{vec|E|b}}|''t''|HTM}}|)|Big}}{{sp|1}}{{op|sdot}}{{sp|1}}{{d|{{vec|S|b}}}}|HTM}} {{\\}}
{{end|align}}
|tex = 
\begin{align}
\mathit{\oiint}_{\partial V} \ \mathbf{E} \sdot d\mathbf{S} &= <!--
  -->\frac{1}{\epsilon_0} \iiint_V \rho \, dV \\

\mathit{\oiint}_{\partial V} \ \mathbf{B} \sdot d\mathbf{S} &= 0 \\

\oint_{\!\!\!\!\! \partial S} \mathbf{E} \sdot d\mathbf{x} &= <!--
  -->- \iint_S \frac{ \partial \mathbf{B} }{ \partial t } \sdot d\mathbf{S} \\ 

\oint_{\!\!\!\!\! \partial S} \mathbf{B} \sdot d\mathbf{x} &= <!--
  -->\iint_S \left( \mu_0 \, \mathbf{J} + \frac{1}{c^2} <!--
  -->\frac{ \partial \mathbf{E} }{ \partial t } \right) \sdot d\mathbf{S} \\
\end{align}
|&&}}

yields the display style HTML+CSS

     

and with the $$ option, yields the display style LaTeX

     ${\begin{array}{l}\displaystyle {\begin{aligned}{\mathit {\oiint }}_{\partial V}\ \mathbf {E} \cdot d\mathbf {S} &={\frac {1}{\epsilon _{0}}}\iiint _{V}\rho \,dV\\{\mathit {\oiint }}_{\partial V}\ \mathbf {B} \cdot d\mathbf {S} &=0\\\oint _{\!\!\!\!\!\partial S}\mathbf {E} \cdot d\mathbf {x} &=-\iint _{S}{\frac {\partial \mathbf {B} }{\partial t}}\cdot d\mathbf {S} \\\oint _{\!\!\!\!\!\partial S}\mathbf {B} \cdot d\mathbf {x} &=\iint _{S}\left(\mu _{0}\,\mathbf {J} +{\frac {1}{c^{2}}}{\frac {\partial \mathbf {E} }{\partial t}}\right)\cdot d\mathbf {S} \\\end{aligned}}\end{array}}$

The code (now with one equation per {{math}} template call, with manual horizontal alignment of equal signs)

{{indent}}{{math|
{{oiint|{{d|''V''|part}}||{{vec|E|b}}{{sp|1}}{{op|sdot}}{{sp|1}}{{d|{{vec|S|b}}}}|HTM}} {{=}} <!--
  -->{{frac|1|''{{Gr|epsilon}}''{{sub|0}}|HTM}} {{iiint|''V''||''{{Gr|rho}}''{{sp|3}}{{d|''V''}}|HTM}}
|tex = \mathit{\oiint}_{\partial V} \ \mathbf{E} \sdot d\mathbf{S} = <!--
  -->\frac{1}{\epsilon_0} \iiint_V \rho \, dV
|&&}}{{clear}}

{{indent}}{{math|
{{oiint|{{d|''V''|part}}||{{vec|B|b}}{{sp|1}}{{op|sdot}}{{sp|1}}{{d|{{vec|S|b}}}}|HTM}} {{=}} 0
|tex = \mathit{\oiint}_{\partial V} \ \mathbf{B} \sdot d\mathbf{S} = 0 
|&&}}{{clear}}

{{indent}}{{sp|9}}{{sp|4}}{{math|
{{oint|{{d|''S''|part}}||{{vec|E|b}}{{sp|1}}{{op|sdot}}{{sp|1}}{{d|{{vec|x|b}}}}|HTM}} {{=}} <!--
  -->{{op|−}}{{iint|''S''||{{partial|{{vec|B|b}}|''t''|HTM}}{{sp|1}}{{op|sdot}}{{sp|1}}{{d|{{vec|S|b}}}}|HTM}}
|tex = {{sp|6|tex}} \oint_{\!\!\!\!\! \partial S} \mathbf{E} \sdot d\mathbf{x} = <!--
  -->- \iint_S \frac{ \partial \mathbf{B} }{ \partial t } \sdot d\mathbf{S}  
|&&}}{{clear}}

{{indent}}{{sp|9}}{{sp|4}}{{math|
{{oint|{{sym|partial}}''S''||{{vec|B|b}}{{sp|1}}{{op|sdot}}{{sp|1}}{{d|{{vec|x|b}}}}|HTM}} {{=}} <!--
  -->{{iint|''S''||{{(|''{{Gr|mu}}''{{sub|0}}{{sp|1}}{{vec|J|b}} + {{frac|1|''c''{{^|2}}|HTM}} <!--
  -->{{partial|{{vec|E|b}}|''t''|HTM}}|)|Big}}{{sp|1}}{{op|sdot}}{{sp|1}}{{d|{{vec|S|b}}}}|HTM}}
|tex = {{sp|6|tex}} \oint_{\!\!\!\!\! \partial S} \mathbf{B} \sdot d\mathbf{x} = <!--
  -->\iint_S \left( \mu_0 \, \mathbf{J} + \frac{1}{c^2} <!--
  -->\frac{ \partial \mathbf{E} }{ \partial t } \right) \sdot d\mathbf{S} 
|&&}}

yields the display style HTML+CSS

     

∯ ∂ V ∂ V  E  ⋅   d S = 1 ϵ0 ∫ ∫ ∫ V V  ρ d V

     

∯ ∂ V ∂ V  B  ⋅   d S = 0

       

∮ ∂ S ∂ S  E  ⋅   d x = ∫ ∫ S S   ∂B ∂t   ⋅   d S

       

∮ ∂S ∂S  B  ⋅   d x = ∫ ∫ S S   ⎛ ⎝   μ0 J + 1 c 2 ∂E ∂t    ⎞ ⎠   ⋅   d S

and with the $$ option, yields the display style LaTeX

     ${\begin{array}{l}\displaystyle {{\mathit {\oiint }}_{\partial V}\ \mathbf {E} \cdot d\mathbf {S} ={\frac {1}{\epsilon _{0}}}\iiint _{V}\rho \,dV}\end{array}}$

     ${\begin{array}{l}\displaystyle {{\mathit {\oiint }}_{\partial V}\ \mathbf {B} \cdot d\mathbf {S} =0}\end{array}}$

       ${\begin{array}{l}\displaystyle {{\,\,}\oint _{\!\!\!\!\!\partial S}\mathbf {E} \cdot d\mathbf {x} =-\iint _{S}{\frac {\partial \mathbf {B} }{\partial t}}\cdot d\mathbf {S} }\end{array}}$

       ${\begin{array}{l}\displaystyle {{\,\,}\oint _{\!\!\!\!\!\partial S}\mathbf {B} \cdot d\mathbf {x} =\iint _{S}\left(\mu _{0}\,\mathbf {J} +{\frac {1}{c^{2}}}{\frac {\partial \mathbf {E} }{\partial t}}\right)\cdot d\mathbf {S} }\end{array}}$

The code (using the {{integral/nicer}} template) (now with one equation per {{math}} template call, with manual horizontal alignment of equal signs)

{{indent}}{{math|
{{integral/nicer|oiint|{{d|''V''|part}}||{{vec|E|b}}{{sp|1}}{{op|sdot}}{{sp|1}}{{d|{{vec|S|b}}}}|HTM}} {{=}} <!--
  -->{{frac|1|''{{Gr|epsilon}}''{{sub|0}}|HTM}} {{integral/nicer|iiint|''V''||''{{Gr|rho}}''{{sp|3}}{{d|''V''}}|HTM}}
|tex = \mathit{\oiint}_{\partial V} \ \mathbf{E} \sdot d\mathbf{S} = <!--
  -->\frac{1}{\epsilon_0} \iiint_V \rho \, dV
|&&}}{{clear}}

{{indent}}{{math|
{{integral/nicer|oiint|{{d|''V''|part}}||{{vec|B|b}}{{sp|1}}{{op|sdot}}{{sp|1}}{{d|{{vec|S|b}}}}|HTM}} {{=}} 0
|tex = \mathit{\oiint}_{\partial V} \ \mathbf{B} \sdot d\mathbf{S} = 0 
|&&}}{{clear}}

{{indent}}{{sp|9}}{{sp|5}}{{math|
{{integral/nicer|oint|{{d|''S''|part}}||{{vec|E|b}}{{sp|1}}{{op|sdot}}{{sp|1}}{{d|{{vec|x|b}}}}|HTM}} {{=}} <!--
  -->{{op|−}}{{integral/nicer|iint|''S''||{{partial|{{vec|B|b}}|''t''|HTM}}{{sp|1}}{{op|sdot}}{{sp|1}}{{d|{{vec|S|b}}}}|HTM}}
|tex = {{sp|6|tex}} \oint_{\!\!\!\!\! \partial S} \mathbf{E} \sdot d\mathbf{x} = <!--
  -->- \iint_S \frac{ \partial \mathbf{B} }{ \partial t } \sdot d\mathbf{S}  
|&&}}{{clear}}

{{indent}}{{sp|9}}{{sp|5}}{{math|
{{integral/nicer|oint|{{sym|partial}}''S''||{{vec|B|b}}{{sp|1}}{{op|sdot}}{{sp|1}}{{d|{{vec|x|b}}}}|HTM}} {{=}} <!--
  -->{{integral/nicer|iint|''S''||{{(|''{{Gr|mu}}''{{sub|0}}{{sp|1}}{{vec|J|b}} + {{frac|1|''c''{{^|2}}|HTM}} <!--
  -->{{partial|{{vec|E|b}}|''t''|HTM}}|)|Big}}{{sp|1}}{{op|sdot}}{{sp|1}}{{d|{{vec|S|b}}}}|HTM}}
|tex = {{sp|6|tex}} \oint_{\!\!\!\!\! \partial S} \mathbf{B} \sdot d\mathbf{x} = <!--
  -->\iint_S \left( \mu_0 \, \mathbf{J} + \frac{1}{c^2} <!--
  -->\frac{ \partial \mathbf{E} }{ \partial t } \right) \sdot d\mathbf{S} 
|&&}}

yields the display style HTML+CSS ({{integral/nicer|oiint}} needs tweaking: lower limit must be a little higher)

     

∯ ∂ V ∂ V  E  ⋅   d S = 1 ϵ0 ∭ V V  ρ d V

     

∯ ∂ V ∂ V  B  ⋅   d S = 0

        

∮ ∂ S ∂ S  E  ⋅   d x = ∬ S S   ∂B ∂t   ⋅   d S

        

∮ ∂S ∂S  B  ⋅   d x = ∬ S S   ⎛ ⎝   μ0 J + 1 c 2 ∂E ∂t    ⎞ ⎠   ⋅   d S

Quotient of integrals

The code

: {{math|<!-- 
-->{{frac
   | {{int|0|infty|''x''{{^|2{{sp|1}}''n''}}{{sp|1}}''e''{{^|{{op|-}}''a''{{sp|1}}''x''{{^|2}}}}{{sp|3}}{{d|''x''}}}}<!-- 
-->| {{int|0|infty|''x''{{^|2{{sp|1}}(''n''{{sp|1}}{{op|-}}1)}}{{sp|1}}''e''{{^|{{op|-}}''a''{{sp|1}}''x''{{^|2}}}}{{sp|3}}{{d|''x''}}}}<!-- 
-->|HTM}} {{=}} {{frac|2{{sp|1}}''n'' {{op|-}} 1|2{{sp|1}}''a''|HTM}}
|tex = \frac{ \int_{0}^{\infty} x^{2n} e^{-a x^2} dx }{ \int_{0}^{\infty} x^{2(n-1)} e^{-a x^2} dx } = \frac{2n - 1}{2a} 
|&&}}

yields the display style HTML+CSS

∫ ∞ 0  x 2 n e  − a x 2 d x ∫ ∞ 0  x 2 (n  − 1) e  − a x 2 d x = 2 n − 1 2 a

and with the $$ option, yields the display style LaTeX

${\begin{array}{l}\displaystyle {{\frac {\int _{0}^{\infty }x^{2n}e^{-ax^{2}}dx}{\int _{0}^{\infty }x^{2(n-1)}e^{-ax^{2}}dx}}={\frac {2n-1}{2a}}}\end{array}}$

The code

: {{math|<!-- 
-->{{frac
   | {{oiint|{{sym|partial}}{{sp|1}}{{Gr|Omega}}| |{{vec|E|b}}{{op|sdot}}{{d|{{vec|S|b}}}}}}<!-- 
-->| {{iiint|{{Gr|Omega}}| |''{{Gr|rho}}''{{sp|3}}{{d|''V''}}}}<!-- 
-->|HTM}} {{=}} {{frac|1|''{{Gr|epsilon}}''{{sub|0}}|HTM}}
|tex = \frac{ \oiint_{\partial \Omega} {{vec|E|b|tex}} \sdot d{{vec|S|b|tex}} }{ \iiint_{\Omega} \rho \, dV } = \frac{1}{\epsilon_0} 
|&&}}

yields the display style HTML+CSS

∯ ∂ Ω ∂ Ω    E ⋅  d S ∫ ∫ ∫ Ω Ω  ρ d V = 1 ϵ0

and with the $$ option, yields the display style LaTeX

${\begin{array}{l}\displaystyle {{\frac {\oiint _{\partial \Omega }\mathbf {E} \cdot d\mathbf {S} }{\iiint _{\Omega }\rho \,dV}}={\frac {1}{\epsilon _{0}}}}\end{array}}$