# Template:PhyseqM

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

## Sample Superstructure

### Sample Substructure

#### SampleName

• Foo

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##### SampleSubname

used for alternative versions of the same equation

### Geometry and algebra

#### CircleSphere

• ${\displaystyle C_{\odot }=\,2\pi r}$ is the circumference of circle;  ${\displaystyle A_{\odot }=\,\pi r^{2}}$ is its area.
• ${\displaystyle A_{\bigcirc }=4\pi r^{2}}$ is the surface area of a sphere;   ${\displaystyle V_{\bigcirc }={\frac {4}{3}}\pi r^{3}}$ is its volume.

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• ${\displaystyle \theta ={\frac {s}{r}}}$ defines angle (in radians), where s is arclength and r is radius.

#### TrigWithoutVectors

• ${\displaystyle \sin A={\frac {\textrm {opposite}}{\textrm {hypotenuse}}}={\frac {a}{\,c\,}}\,.}$ (where A is the angle shown)
• ${\displaystyle \cos A={\frac {\textrm {adjacent}}{\textrm {hypotenuse}}}={\frac {b}{\,c\,}}\,.}$
• ${\displaystyle \tan A={\frac {\textrm {opposite}}{\textrm {adjacent}}}={\frac {a}{\,b\,}}={\frac {\sin A}{\cos A}}\,.}$

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##### InverseTrigFunctions
• ${\displaystyle \sin \left(\sin ^{-1}(z)\right)=z}$ and ${\displaystyle sin^{-1}\left(\sin \theta \right)=\theta }$ defines the arcsine function as the inverse of the sine. Similarly, ${\displaystyle \tan ^{-1}}$ is called the arctangent, or the inverse tangent, and ${\displaystyle \cos ^{-1}}$ is called arccosine, or the inverse cosine and so forth. In general, ${\displaystyle f(f^{-1}(y))=y}$ and ${\displaystyle f^{-1}((f(x))=x}$ for any function and its inverse. Complexities occur whenever the inverse is not a true function; for example, since ${\displaystyle \tan(\theta )=\tan(\theta +\pi )}$, the inverse is multi-valued: ${\displaystyle \tan ^{-1}(\tan \theta )=\theta \;or\;\theta +\pi \,.}$

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### First year calculus

##### CalculusBasic
• ${\displaystyle \Delta f=f\left(x+\Delta x\right)-f(x)}$,     and the derivative is ${\displaystyle {\frac {\Delta f}{\Delta x}}\rightarrow {\frac {df}{dx}}}$ in the limit that ${\displaystyle \Delta x\rightarrow 0}$
• ${\displaystyle {\frac {d}{dx}}f\left(g(x)\right)={\frac {df}{dg}}{\frac {dg}{dx}}}$ is the chain rule.
• ${\displaystyle {\frac {d}{dx}}Ax^{p}=(p-1)Ax^{p-1}}$,      ${\displaystyle {\frac {d}{dx}}\ln x={\frac {1}{x}}}$,     ${\displaystyle {\frac {d}{dx}}e^{x}=e^{x}}$,     ${\displaystyle {\frac {d}{dx}}\sin x=\cos x}$,     ${\displaystyle {\frac {d}{dx}}\cos x=-\sin x}$
• ${\displaystyle \int {\frac {df}{dx}}dx=f(x)+c}$ expresses the fundamental theorem of calculus.

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#### RiemannSum

• ${\displaystyle \int _{a}^{b}f(x)\mathrm {d} x\approx \sum f(x_{j})\Delta x_{j}}$ is the 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 (Δxj→0).

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#### RiemannSumShort

• ${\displaystyle \int _{a}^{b}f(x)\mathrm {d} x\approx \sum f(x_{j})\Delta x_{j}}$ is the Riemann sum representation of the integral of f(x) from x=a to x=b.

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#### IntegrateFundamentalTheorem

• The fundamental theorem of calculus allows us to construct integrals from known derivatives:
• ${\displaystyle \int _{a}^{b}Ax^{n}\mathrm {d} x=\left\{{\frac {A}{n+1}}b^{n+1}-{\frac {A}{n+1}}a^{n+1}\right\}}$
• ${\displaystyle \int _{a}^{b}Ax^{-1}\mathrm {d} x=A\ln b-A\ln a=A\ln {\frac {b}{a}}}$

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### Vector algebra

#### VectorComponents

• ${\displaystyle x=r\cos \theta \quad }$ and ${\displaystyle y=r\sin \theta \quad }$ are the x and y components of a displacement from the origin to some point. The inverse transformations are:
• ${\displaystyle r={\sqrt {x^{2}+y^{2}}}\quad }$, and
• ${\displaystyle \theta =\arctan {(y/x)}\quad }$ , which is multi-valued and therefore not a true function.
• ${\displaystyle A_{x}=A\cos \theta }$   and   ${\displaystyle A_{y}=A\sin \theta }$   are called the x and y components of vector A, respectively.
• ${\displaystyle |{\vec {A}}|={\sqrt {A_{x}^{2}+A_{y}^{2}}}=A\quad }$ is called the magnitude,norm (or sometimes 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 vector and a scalar is boldface font: A is a vector and A${\displaystyle \,|}$ A ${\displaystyle \,|}$ is the scalar magnitude.

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• ${\displaystyle {\vec {A}}+{\vec {B}}={\vec {C}}}$ and ${\displaystyle {\vec {B}}={\vec {C}}-{\vec {A}}}$ 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:
• ${\displaystyle {\vec {A}}+{\vec {B}}={\vec {C}}\Leftrightarrow }$ ${\displaystyle A_{x}+B_{x}=C_{x}}$ AND ${\displaystyle A_{y}+B_{y}=C_{y}}$

#### CrossProductVisual

• ${\displaystyle {\vec {A}}\times {\vec {B}}={\vec {C}}}$ is the cross product of ${\displaystyle {\vec {A}}}$ and ${\displaystyle {\vec {B}}}$. The cross product, ${\displaystyle {\vec {C}}}$ is directed perpendicular to ${\displaystyle {\vec {A}}}$ and ${\displaystyle {\vec {B}}}$ by the right hand rule.
• ${\displaystyle |{\vec {A}}\times {\vec {B}}|=C=|AB\sin \theta |}$ wehre ${\displaystyle \theta }$ is the angle between vectors ${\displaystyle {\vec {A}}}$ and ${\displaystyle {\vec {B}}}$.
• ${\displaystyle |{\vec {A}}\times {\vec {B}}|=C}$ is also the magnitude of the of the parallelogram defined by the vectors ${\displaystyle {\vec {A}}}$ and ${\displaystyle {\vec {B}}}$.
• ${\displaystyle {\vec {A}}\times {\vec {B}}=0}$ if ${\displaystyle {\vec {A}}}$ and ${\displaystyle {\vec {B}}}$ are either parallel or antiparallel.
• The unit vectors obey ${\displaystyle {\hat {x}}\times {\hat {y}}={\hat {z}}}$, ${\displaystyle {\hat {y}}\times {\hat {z}}={\hat {x}}}$, and ${\displaystyle {\hat {z}}\times {\hat {x}}={\hat {y}}}$.

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#### DotProduct

• ${\displaystyle {\vec {A}}\cdot {\vec {B}}=AB\cos \theta =A_{x}B_{x}+A_{y}B_{y}+A_{z}B_{z}\,}$ is the dot product between two vectors separated in angle by θ.

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#### CrossProductComponents

{\displaystyle {\begin{aligned}{\vec {A}}\times {\vec {B}}\Leftrightarrow &C_{x}&=&\ A_{y}B_{z}-A_{x}B_{y}\ {\text{and}}\\&C_{y}&=&\ A_{z}B_{x}-A_{x}B_{z}\ {\text{and}}\\&C_{z}&=&\ A_{x}B_{y}-A_{y}B_{x}\ .\end{aligned}}}

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#### UnitVectors

• A unit vector is any vector with unit magnitude equal to one. For any nonzero vector, ${\displaystyle {\hat {V}}={\vec {V}}/V}$ is a unit vector. An important set of unit vectors is the orthonormal basis associated with Cartesian coordinates:
• ${\displaystyle \mathbf {\hat {i}} \cdot \mathbf {\hat {i}} =\mathbf {\hat {j}} \cdot \mathbf {\hat {j}} =\mathbf {\hat {k}} \cdot \mathbf {\hat {k}} =1}$
• ${\displaystyle \mathbf {\hat {j}} \cdot \mathbf {\hat {k}} =\mathbf {\hat {k}} \cdot \mathbf {\hat {i}} =\mathbf {\hat {j}} \cdot \mathbf {\hat {k}} =0}$
• The basis vectors ${\displaystyle (\mathbf {\hat {i}} ,\mathbf {\hat {j}} ,\mathbf {\hat {k}} )}$ are also written as ${\displaystyle ({\hat {x}},{\hat {y}},{\hat {z}})}$, so that any vector may be written ${\displaystyle {\vec {A}}=A_{x}{\hat {x}}+A_{y}{\hat {y}}+A_{z}{\hat {z}}}$. Even more elegance is achieved by labeling the directions with integers: ${\displaystyle {\vec {A}}=A_{1}{\hat {e_{1}}}+A_{2}{\hat {e_{2}}}+A_{3}{\hat {e_{3}}}}$ ${\displaystyle =\Sigma A_{j}{\hat {e_{j}}}\,.}$

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## Removed templates

##### PathIntegralOpenClosedSurface

THIS TEMPLATE HAS BEEN REMOVED

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