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<big><big>Please do not use this template.  Instead go to [[Physics equations/Equations]] or a subpage and transclude from there. </big></big>


==Sample Superstructure==
===Sample Substructure===
====SampleName====
<!--Copy and paste before modifying.-->
<onlyinclude>{{#ifeq:{{{transcludesection|SampleName}}}|SampleName|
* Foo
<section end=SampleName/>
}}</onlyinclude>
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<!--Rename in 5 places-->
=====SampleSubname=====
used for alternative versions of the same equation
===Geometry and algebra===
====CircleSphere====
<onlyinclude>{{#ifeq:{{{transcludesection|CircleSphere}}}|CircleSphere|
*<math>C_\odot = \,2\pi r</math> is the circumference of '''[[w:circle|circle]]'''; &nbsp;<math>A_\odot = \,\pi r^2</math> is its area.
*<math>A_\bigcirc = 4\pi r^2</math> is the surface area of a '''[[w:sphere|sphere]]'''; &nbsp; <math>V_\bigcirc = \frac{4}{3}\pi r^3</math> is its volume.
<section end=CircleSphere/>
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====SimpleDefinitionThetaRadians====
<onlyinclude>{{#ifeq:{{{transcludesection|SimpleDefinitionThetaRadians}}}|SimpleDefinitionThetaRadians|
[[File:Theta in radians minimalistic.svg|right|140px]]
*<math>\theta = \frac s r</math> defines angle (in radians), where ''s'' is arclength and ''r'' is radius.
<section end=SimpleDefinitionThetaRadians/>
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====TrigWithoutVectors====
<onlyinclude>{{#ifeq:{{{transcludesection|TrigWithoutVectors}}}|TrigWithoutVectors|
[[File:TrigonometryTriangle.svg|280px|right]]
*<math>\sin A=\frac{\textrm{opposite}}{\textrm{hypotenuse}}=\frac{a}{\,c\,}\,.</math> (where A is the angle shown)
*<math>\cos A=\frac{\textrm{adjacent}}{\textrm{hypotenuse}}=\frac{b}{\,c\,}\,.</math> 
*<math>\tan A=\frac{\textrm{opposite}}{\textrm{adjacent}}=\frac{a}{\,b\,}=\frac{\sin A}{\cos A}\,.</math>
<section end=TrigWithoutVectors/>
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=====InverseTrigFunctions=====
<onlyinclude>{{#ifeq:{{{transcludesection|InverseTrigFunctions}}}|InverseTrigFunctions|
 *<math>\sin\left(\sin^{-1}(z)\right)=z</math> and <math>sin^{-1}\left(\sin\theta\right)=\theta</math> defines the '''[[w:Inverse trigonometric functions|arcsine]]''' function as the inverse of the sine.  Similarly, <math>\tan^{-1}</math> is called the ''arctangent'', or the ''inverse tangent'',  and <math>\cos^{-1}</math> is called ''arccosine'', or the ''inverse cosine'' and so forth.  In general, <math>f(f^{-1}(y))=y</math> and <math>f^{-1}((f(x))=x</math> for any function and its inverse.  Complexities occur whenever the inverse is not a true function; for example, since <math>\tan(\theta)=\tan(\theta+\pi)</math>, the inverse is multi-valued: <math>\tan^{-1}(\tan \theta) = \theta\; or\; \theta+\pi\,.</math>
<section end=InverseTrigFunctions/>
}}</onlyinclude>
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===First year calculus===
=====CalculusBasic=====
<onlyinclude>{{#ifeq:{{{transcludesection|CalculusBasic}}}|CalculusBasic|
* <math>\Delta f = f\left(x+\Delta x\right)-f(x)</math>, &nbsp; &nbsp; and the derivative is  <math>\frac{\Delta f}{\Delta x}\rightarrow \frac{df}{dx}</math> in the limit that <math>\Delta x \rightarrow 0</math>
* <math>\frac{d}{dx}f\left(g(x)\right)=\frac{df}{dg}\frac{dg}{dx}</math> is the chain rule.
* <math>\frac{d}{dx}Ax^p=(p-1)Ax^{p-1}</math>, &nbsp; &nbsp; &nbsp;<math>\frac{d}{dx}\ln x = \frac 1 x</math>,&nbsp;  &nbsp; &nbsp;<math>\frac{d}{dx}e^x = e^x</math>,&nbsp;  &nbsp; &nbsp;<math>\frac{d}{dx}\sin x = \cos x</math>,&nbsp;  &nbsp; &nbsp;<math>\frac{d}{dx}\cos x = -\sin x</math>
*<math>\int\frac {df}{dx}dx=f(x)+c</math> expresses the fundamental theorem of calculus.
<section end=CalculusBasic/>
}}</onlyinclude>
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====RiemannSum====
<onlyinclude>{{#ifeq:{{{transcludesection|RiemannSum}}}|RiemannSum|
[[File:Riemann sum convergence.png|right|240px]]
* <math>\int_a^bf(x)\mathrm{d}x\approx\sum f(x_j)\Delta x_j</math> is the [[w:Riemann sum|Riemann sum]] representation of the integral of f(x) from x=a to x=b.  It is the area under the curve, with contributions from f(x)<0 being negative (if a>b).  The sum equals the integral in the limit that the widths of all the intervals vanish (Δx<sub>j</sub>→0).
<gallery>
Image:Riemann sum (leftbox).gif|Left sum
Image:Riemann sum (rightbox).gif|Right sum
Image:Riemann sum (middlebox).gif|Middle sum
<!--Image:Riemann sum (y=x^2).gif|With <math>y=x^2</math>-->
</gallery>
<section end=RiemannSum/>
}}</onlyinclude>
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====RiemannSumShort====
<onlyinclude>{{#ifeq:{{{transcludesection|RiemannSumShort}}}|RiemannSumShort|
* <math>\int_a^bf(x)\mathrm{d}x\approx\sum f(x_j)\Delta x_j</math> is the Riemann sum representation of the integral of f(x) from x=a to x=b. 
<section end=RiemannSumShort/>
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====IntegrateFundamentalTheorem====
<onlyinclude>{{#ifeq:{{{transcludesection|IntegrateFundamentalTheorem}}}|IntegrateFundamentalTheorem|
*The fundamental theorem of calculus allows us to construct integrals from known derivatives:
* <math>\int_a^bAx^n\mathrm{d}x=\left\{\frac{A}{n+1}b^{n+1}-\frac{A}{n+1}a^{n+1}\right\}</math>
* <math>\int_a^bAx^{-1}\mathrm{d}x=A\ln b-A\ln a = A\ln\frac b a</math>
<section end=IntegrateFundamentalTheorem/>
}}</onlyinclude>
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===Vector algebra===

====VectorComponents====
<onlyinclude>{{#ifeq:{{{transcludesection|VectorComponents}}}|VectorComponents|
[[Image:Polar to cartesian.svg|right|180px]]
* <math>x = r \cos \theta \quad</math> and <math>y = r \sin \theta \quad</math> are the x and y components of a displacement from the origin to some point.  The inverse transformations are:

:*<math>r = \sqrt{x^2 + y^2} \quad</math>, and
 
:*<math>\theta = \arctan{(y/x)} \quad</math> , which is multi-valued and therefore not a true function. 
 
* <math>A_x=A\cos\theta</math> &nbsp; and &nbsp; <math>A_y=A\sin\theta</math> &nbsp; are called the x and y '''[[w:Vector_component#Decomposition|components]]''' of vector A, respectively.  

* <math>|\vec A|=\sqrt{A_x^2 + A_y^2}=A \quad</math> is called the ''magnitude'',''norm'' (or sometimes '''[[w:Absolute_value#Vector_spaces|absolute value]]''') of vector A.  

*Omission of the arrow indicates that the quantity is the scalar magnitude of that vector.  Another notation that distinguishes between a '''[[w:vector|vector]]''' and  a '''[[w:Scalar (physics)|scalar]]''' is '''''boldface''''' font: '''''A''''' is a vector and ''A''=&nbsp;<math>\,|</math>&nbsp;'''''A'''''&nbsp;<math>\,|</math> is the scalar magnitude.
<section end=VectorComponents/>
}}</onlyinclude>
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====VectorAddition====
<onlyinclude>{{#ifeq:{{{transcludesection|VectorAddition}}}|VectorAddition|
[[File:Vectors A+B=C.png|right|150px]]
* <math>\vec A +\vec B =\vec C</math> and <math>\vec B = \vec C -\vec A</math> have geometric interpretation as vector addition and subtraction as shown in the figure.  Vector addition and subtraction can also be defined through the components.  For example, the following two statements are equivalent:
*<math>\vec A +\vec B =\vec C \Leftrightarrow</math> <math>A_x + B_x =C_x</math> AND <math>A_y + B_y =C_y</math>
<section end=VectorAddition/>
}}</onlyinclude>
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====CrossProductVisual====
<onlyinclude>{{#ifeq:{{{transcludesection|CrossProductVisual}}}|CrossProductVisual|
[[Image:Right hand rule cross product.svg|right|thumb|180px|[[w:right-hand rule|right-hand rule]]]]
*<math>\vec A \times\vec B = \vec C</math> is the '''[[w:cross product|cross product]]''' of <math>\vec A</math> and <math>\vec B</math>.  The cross product, <math>\vec C</math> is directed perpendicular to <math>\vec A</math> and <math>\vec B</math> by the right hand rule.  
*<math>|\vec A \times\vec B| = C = |AB\sin\theta|</math> wehre <math>\theta</math> is the angle between vectors <math>\vec A</math> and <math>\vec B</math>.  
*<math>|\vec A \times\vec B| = C</math> is also the magnitude of the of the parallelogram defined by the vectors <math>\vec A</math> and <math>\vec B</math>.  
*<math>\vec A\times\vec B=0</math> if <math>\vec A</math> and <math>\vec B</math> are either parallel or [[w:Antiparallel_(mathematics)#Antiparallel_vectors|antiparallel]].
*The unit vectors obey <math>\hat x\times\hat y = \hat z</math>, <math>\hat y\times\hat z = \hat x</math>, and <math>\hat z\times\hat x = \hat y</math>.
<section end=CrossProductVisual/>
}}</onlyinclude>
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====DotProduct====
<onlyinclude>{{#ifeq:{{{transcludesection|DotProduct}}}|DotProduct|
* <math>\vec{A} \cdot \vec{B} = AB\cos\theta = A_xB_x + A_yB_y + A_zB_z\,</math> is the '''[[dot product]]''' between two vectors separated in angle by &theta;.
<section end=DotProduct/>
}}</onlyinclude>
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====CrossProductComponents====
<onlyinclude>{{#ifeq:{{{transcludesection|CrossProductComponents}}}|CrossProductComponents|
<math>\begin{align}
\vec A\times\vec B  \Leftrightarrow & C_x & = & \ A_yB_z-A_xB_y\ \text{and}\\
                                     & C_y & = & \ A_zB_x-A_xB_z\ \text{and}\\
                                    & C_z & = & \ A_xB_y-A_yB_x\ .
\end{align}</math>
<section end=CrossProductComponents/>
}}</onlyinclude>
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====UnitVectors====
<onlyinclude>{{#ifeq:{{{transcludesection|UnitVectors}}}|UnitVectors|
*A '''[[w:unit vector|unit vector]]''' is any vector with unit magnitude equal to one.  For any nonzero vector, <math>\hat V =\vec V/V</math> is a unit vector. An important set of unit vectors is the [[w:orthonormality|orthonormal]] basis associated with Cartesian coordinates:
*<math>\mathbf\hat i\cdot \mathbf\hat  i = \mathbf\hat j\cdot \mathbf\hat j =\mathbf\hat k\cdot \mathbf\hat k =1</math>
*<math>\mathbf\hat j\cdot  \mathbf\hat k = \mathbf\hat k\cdot \mathbf\hat i =\mathbf\hat j\cdot\mathbf\hat k =0</math>
* The basis vectors <math>(\mathbf\hat i, \mathbf\hat j, \mathbf\hat k)</math> are also written as <math>(\hat x, \hat y, \hat z)</math>, so that any vector may be written <math>\vec A = A_x\hat x +A_y\hat y +A_z\hat z</math>.  Even more elegance is achieved by labeling the directions with integers: <math>\vec A = A_1\hat{e_1} +A_2\hat{e_2}+A_3\hat{e_3}</math> <math>=\Sigma A_j\hat{e_j}\,.</math>
<section end=UnitVectors/>
}}</onlyinclude>
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==Removed templates==
=====PathIntegralOpenClosedSurface=====
<onlyinclude>{{#ifeq:{{{transcludesection|PathIntegralOpenClosedSurface}}}|PathIntegralOpenClosedSurface|
THIS TEMPLATE HAS BEEN REMOVED
<section end=PathIntegralOpenClosedSurface/>
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[[Category:Template:Physeq]] [[Category:Physics equations/Equations/Transcluded]]