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3. Calculus

3.1 Integrals

3.1.1 Arithmetic rules

The primitive function

$F(x)$ of

$f(x)$ obeys the rule

$F'(x)=f(x)$ . With

$F(x)$ the primitive of

$f(x)$ holds for the definite integral

$\int\limits_a^bf(x)dx=F(b)-F(a)$ If

$u=f(x)$ holds:

$\int\limits_a^bg(f(x))df(x)=\int\limits_{f(a)}^{f(b)}g(u)du$ Partial integration: with

$F$ and

$G$ the primitives of

$f$ and

$g$ holds:

$\int f(x)\cdot g(x)dx=f(x)G(x)-\int G(x)\frac{df(x)}{dx}dx$ A derivative can be brought under the intergral sign (see section 1.8.3 for the required conditions):

$\frac{d}{dy}\left[\int\limits_{x=g(y)}^{x=h(y)}f(x,y)dx\right]= \int\limits_{x=g(y)}^{x=h(y)}\frac{\partial f(x,y)}{\partial y}dx-f(g(y),y)\frac{dg(y)}{dy}+f(h(y),y)\frac{dh(y)}{dy}$

3.1.2 Arc lengts, surfaces and volumes

The arc length

$\ell$ of a curve

$y(x)$ is given by:

$\ell=\int\sqrt{1+\left(\frac{dy(x)}{dx}\right)^2}dx$ The arc length

$\ell$ of a parameter curve

$F(\vec{x}(t))$ is:

$\ell=\int Fds=\int F(\vec{x}(t))|\dot{\vec{x}}(t)|dt$ with

$\vec{t}=\frac{d\vec{x}}{ds}=\frac{\dot{\vec{x}}(t)}{|\dot{\vec{x}}(t)|}~~~,~~|\vec{t}~|=1$

$\int(\vec{v},\vec{t})ds=\int(\vec{v},\dot{\vec{t}}(t))dt=\int(v_1dx+v_2dy+v_3dz)$ The surface

$A$ of a solid of revolution is:

$A=2\pi\int y\sqrt{1+\left(\frac{dy(x)}{dx}\right)^2}dx$ The volume

$V$ of a solid of revolution is:

$V=\pi\int f^2(x)dx$

3.1.3 Separation of quotients

Every rational function

$P(x)/Q(x)$ where

$P$ and

$Q$ are polynomials can be written as a linear combination of functions of the type

$(x-a)^k$ with

$k\in\mathbb{Z}$ , and of functions of the type

$\frac{px+q}{((x-a)^2+b^2)^n}$ with

$b>0$ and

$n\in I\hspace{-1mm}N$ . So:

$\frac{p(x)}{(x-a)^n}=\sum_{k=1}^n\frac{A_k}{(x-a)^k}~~,~~~ \frac{p(x)}{((x-b)^2+c^2)^n}=\sum_{k=1}^n \frac{A_kx+B}{((x-b)^2+c^2)^k}$ Recurrent relation: for

$n\neq0$ holds:

$\int\frac{dx}{(x^2+1)^{n+1}}=\frac{1}{2n}\frac{x}{(x^2+1)^n}+\frac{2n-1}{2n}\int\frac{dx}{(x^2+1)^n}$

3.1.4 Special functions

3.1.4.1 Elliptic functions

Elliptic functions can be written as a power series as follows:

$\sqrt{1-k^2\sin^2(x)}=1-\sum_{n=1}^\infty\frac{(2n-1)!!}{(2n)!!(2n-1)}k^{2n}\sin^{2n}(x)$

$\frac{1}{\sqrt{1-k^2\sin^2(x)}}=1+\sum_{n=1}^\infty\frac{(2n-1)!!}{(2n)!!}k^{2n}\sin^{2n}(x)$ with

$n!!=n(n-2)!!$ .

3.1.4.2 The Gamma function

The gamma function

$\Gamma(y)$ is defined by:

$\Gamma(y)=\int\limits_0^\infty{\rm e}^{-x}x^{y-1}dx$ One can derive that

$\Gamma(y+1)=y\Gamma(y)=y!$ . This is a way to define faculties for non-integers. Further one can derive that

$\Gamma(n+\frac{1}{2})=\frac{\sqrt{\pi}}{2^n}(2n-1)!!~~\mbox{and}~~ \Gamma^{(n)}(y)=\int\limits_0^\infty{\rm e}^{-x}x^{y-1}\ln^n(x)dx$

3.1.4.3 The Beta function

The betafunction

$\beta(p,q)$ is defined by:

$\beta(p,q)=\int\limits_0^1x^{p-1}(1-x)^{q-1}dx$ with

$p$ and

$q$

$>0$ . The beta and gamma functions are related by the following equation:

$\beta(p,q)=\frac{\Gamma(p)\Gamma(q)}{\Gamma(p+q)}$

3.1.4.4 The Delta function

The delta function

$\delta(x)$ is an infinitely thin peak function with surface 1. It can be defined by:

$\delta(x)=\lim_{\varepsilon\rightarrow0}P(\varepsilon,x)~~\mbox{with}~~ P(\varepsilon,x)=\left\{ \begin{array}{l} 0~~~\mbox{for}~|x|>\varepsilon\\ \displaystyle\frac{1}{2\varepsilon}~~~\mbox{when}~|x|<\varepsilon \end{array}\right.$ Some properties are:

$\int\limits_{-\infty}^\infty\delta(x)dx=1~~,~~~ \int\limits_{-\infty}^\infty F(x)\delta(x)dx=F(0)$

3.1.5 Goniometric integrals

When solving goniometric integrals it can be useful to change variables. The following holds if one defines

$\tan(\frac{1}{2}x):=t$ :

$dx=\frac{2dt}{1+t^2}~,~~\cos(x)=\frac{1-t^2}{1+t^2}~,~~\sin(x)=\frac{2t}{1+t^2}$ Each integral of the type

$\int R(x,\sqrt{ax^2+bx+c})dx$ can be converted into one of the types that were treated in section 3.1.3. After this conversion one can substitute in the integrals of the type:

$\begin{eqnarray*} \int R(x,\sqrt{x^2+1})dx&~:~~&x=\tan(\varphi) ~,dx=\frac{d\varphi}{\cos(\varphi)} ~~\mbox{or}~~\sqrt{x^2+1}=t+x\\ \int R(x,\sqrt{1-x^2})dx&~:~~&x=\sin(\varphi) ~,dx=\cos(\varphi)d\varphi ~~\mbox{or}~~\sqrt{1-x^2}=1-tx\\ \int R(x,\sqrt{x^2-1})dx&~:~~&x=\frac{1}{\cos(\varphi)}~,dx=\frac{\sin(\varphi)}{\cos^2(\varphi)}d\varphi~~\mbox{or}~~\sqrt{x^2-1}=x-t \end{eqnarray*}$ These definite integrals are easily solved:

$\int\limits_0^{\pi/2}\cos^n(x)\sin^m(x)dx=\frac{(n-1)!!(m-1)!!}{(m+n)!!}\cdot \left\{\begin{array}{l} \pi/2\;\;\mbox{when $m$ and $n$ are both even}\\ 1\;\;\;\;\mbox{in all other cases} \end{array}\right.$ Some important integrals are:

$\int\limits_0^\infty\frac{xdx}{{\rm e}^{ax}+1}=\frac{\pi^2}{12a^2}~~,~~ \int\limits_{-\infty}^\infty\frac{x^2dx}{({\rm e}^x+1)^2}=\frac{\pi^2}{3}~~,~~ \int\limits_0^\infty\frac{x^3dx}{{\rm e}^x-1}=\frac{\pi^4}{15}$

3.2 Functions with more variables

3.2.1 Derivatives

The partial derivative with respect to

$x$ of a function

$f(x,y)$ is defined by:

$\left(\frac{\partial f}{\partial x}\right)_{x_0}=\lim_{h\rightarrow0}\frac{f(x_0+h,y_0)-f(x_0,y_0)}{h}$ The directional derivative in the direction of

$\alpha$ is defined by:

$\frac{\partial f}{\partial\alpha}=\lim_{r\downarrow0}\frac{f(x_0+r\cos(\alpha),y_0+r\sin(\alpha))-f(x_0,y_0)}{r}= (\vec{\nabla}f,(\sin\alpha,\cos\alpha))=\frac{\nabla f\cdot\vec{v}}{|\vec{v}|}$ When one changes to coordinates

$f(x(u,v),y(u,v))$ holds:

$\frac{\partial f}{\partial u}=\frac{\partial f}{\partial x}\frac{\partial x}{\partial u}+\frac{\partial f}{\partial y}\frac{\partial y}{\partial u}$ If

$x(t)$ and

$y(t)$ depend only on one parameter

$t$ holds:

$\frac{\partial f}{\partial t}=\frac{\partial f}{\partial x}\frac{dx}{dt}+\frac{\partial f}{\partial y}\frac{dy}{dt}$ The total differential

$df$ of a function of 3 variables is given by:

$df=\frac{\partial f}{\partial x}dx+\frac{\partial f}{\partial y}dy+\frac{\partial f}{\partial z}dz$ So

$\frac{df}{dx}=\frac{\partial f}{\partial x}+\frac{\partial f}{\partial y}\frac{dy}{dx}+\frac{\partial f}{\partial z}\frac{dz}{dx}$ The tangent in point

$\vec{x}_0$ at the surface

$f(x,y)=0$ is given by the equation

$f_x(\vec{x}_0)(x-x_0)+f_y(\vec{x}_0)(y-y_0)=0$ .

The tangent plane in $\vec{x}_0$ is given by: $f_x(\vec{x}_0)(x-x_0)+f_y(\vec{x}_0)(y-y_0)=z-f(\vec{x}_0)$ .

3.2.2 Taylor series

A function of two variables can be expanded as follows in a Taylor series:

$f(x_0+h,y_0+k)=\sum\limits_{p=0}^n \frac{1}{p!} \left(h\frac{\partial^p}{\partial x^p}+k\frac{\partial^p}{\partial y^p}\right)f(x_0,y_0)+R(n)$ with

$R(n)$ the residual error and

$\left(h\frac{\partial^p}{\partial x^p}+k\frac{\partial^p}{\partial y^p}\right)f(a,b)=\sum\limits_{m=0}^p{p\choose m} h^mk^{p-m}\frac{\partial^pf(a,b)}{\partial x^m\partial y^{p-m}}$

3.2.3 Extrema

When

$f$ is continuous on a compact boundary

$V$ there exists a global maximum and a global minumum for

$f$ on this boundary. A boundary is called compact if it is limited and closed.

Possible extrema of $f(x,y)$ on a boundary $V\in I\hspace{-1mm}R^2$ are:

Points on $V$ where $f(x,y)$ is not differentiable,
Points where $\vec{\nabla}f=\vec{0}$ ,
If the boundary $V$ is given by $\varphi(x,y)=0$ , than all points where $\vec{\nabla}f(x,y)+\lambda\vec{\nabla}\varphi(x,y)=0$ are possible for extrema. This is the multiplicator method of Lagrange, $\lambda$ is called a multiplicator.

The same as in $I\hspace{-1mm}R^2$ holds in $I\hspace{-1mm}R^3$ when the area to be searched is constrained by a compact $V$ , and $V$ is defined by $\varphi_1(x,y,z)=0$ and $\varphi_2(x,y,z)=0$ for extrema of $f(x,y,z)$ for points (1) and (2). Point (3) is rewritten as follows: possible extrema are points where $\vec{\nabla}f(x,y,z)+\lambda_1\vec{\nabla}\varphi_1(x,y,z)+\lambda_2\vec{\nabla}\varphi_2(x,y,z)=0$ .

3.2.4 The $\nabla$ -operator

In cartesian coordinates

$(x,y,z)$ holds:

$\begin{eqnarray*} \vec{\nabla} &=&\frac{\partial}{\partial x}\vec{e}_{x}+\frac{\partial}{\partial y}\vec{e}_{y}+\frac{\partial}{\partial z}\vec{e}_{z}\\ {\rm grad}f &=&\frac{\partial f}{\partial x}\vec{e}_{x}+\frac{\partial f}{\partial y}\vec{e}_{y}+\frac{\partial f}{\partial z}\vec{e}_{z}\\ {\rm div}~\vec{a}&=&\frac{\partial a_x}{\partial x}+\frac{\partial a_y}{\partial y}+\frac{\partial a_z}{\partial z}\\ {\rm curl}~\vec{a}&=&\left(\frac{\partial a_z}{\partial y}-\frac{\partial a_y}{\partial z}\right)\vec{e}_{x}+ \left(\frac{\partial a_x}{\partial z}-\frac{\partial a_z}{\partial x}\right)\vec{e}_{y}+ \left(\frac{\partial a_y}{\partial x}-\frac{\partial a_x}{\partial y}\right)\vec{e}_{z}\\ \nabla^2 f &=&\frac{\partial^2f}{\partial x^2}+\frac{\partial^2f}{\partial y^2}+\frac{\partial^2f}{\partial z^2} \end{eqnarray*}$

In cylindrical coordinates $(r,\varphi,z)$ holds: $\begin{eqnarray*} \vec{\nabla} &=&\frac{\partial}{\partial r}\vec{e}_{r}+\frac{1}{r}\frac{\partial}{\partial\varphi}\vec{e}_{\varphi}+\frac{\partial}{\partial z}\vec{e}_{z}\\ {\rm grad}f &=&\frac{\partial f}{\partial r}\vec{e}_{r}+\frac{1}{r}\frac{\partial f}{\partial\varphi}\vec{e}_{\varphi}+\frac{\partial f}{\partial z}\vec{e}_{z}\\ {\rm div}~\vec{a}&=&\frac{\partial a_r}{\partial r}+\frac{a_r}{r}+\frac{1}{r}\frac{\partial a_\varphi}{\partial\varphi}+\frac{\partial a_z}{\partial z}\\ {\rm curl}~\vec{a}&=&\left(\frac{1}{r}\frac{\partial a_z}{\partial\varphi}-\frac{\partial a_\varphi}{\partial z}\right)\vec{e}_{r}+ \left(\frac{\partial a_r}{\partial z}-\frac{\partial a_z}{\partial r}\right)\vec{e}_{\varphi}+ \left(\frac{\partial a_\varphi}{\partial r}+\frac{a_\varphi}{r}-\frac{1}{r}\frac{\partial a_r}{\partial\varphi}\right)\vec{e}_{z}\\ \nabla^2 f &=&\frac{\partial^2f}{\partial r^2}+\frac{1}{r}\frac{\partial f}{\partial r}+\frac{1}{r^2}\frac{\partial^2f}{\partial\varphi^2}+\frac{\partial^2f}{\partial z^2} \end{eqnarray*}$

In spherical coordinates $(r,\theta,\varphi)$ holds: $\begin{eqnarray*} \vec{\nabla} &=&\frac{}\partial{\partial r}\vec{e}_{r}+\frac{1}{r}\frac{\partial}{\partial\theta}\vec{e}_{\theta}+\frac{1}{r\sin\theta}\frac{\partial}{\partial\varphi}\vec{e}_{\varphi}\\ {\rm grad}f &=&\frac{\partial f}{\partial r}\vec{e}_{r}+\frac{1}{r}\frac{\partial f}{\partial\theta}\vec{e}_{\theta}+\frac{1}{r\sin\theta}\frac{\partial f}{\partial\varphi}\vec{e}_{\varphi}\\ {\rm div}~\vec{a}&=&\frac{\partial a_r}{\partial r}+\frac{2a_r}{r}+\frac{1}{r}\frac{\partial a_\theta}{\partial\theta}+\frac{a_\theta}{r\tan\theta}+\frac{1}{r\sin\theta}\frac{\partial a_\varphi}{\partial\varphi}\\ {\rm curl}~\vec{a}&=&\left(\frac{1}{r}\frac{\partial a_\varphi}{\partial\theta}+\frac{a_\theta}{r\tan\theta}-\frac{1}{r\sin\theta}\frac{\partial a_\theta}{\partial\varphi}\right)\vec{e}_{r}+ \left(\frac{1}{r\sin\theta}\frac{\partial a_r}{\partial\varphi}-\frac{\partial a_\varphi}{\partial r}-\frac{a_\varphi}{r}\right)\vec{e}_{\theta}+ \left(\frac{\partial a_\theta}{\partial r}+\frac{a_\theta}{r}-\frac{1}{r}\frac{\partial a_r}{\partial\theta}\right)\vec{e}_{\varphi}\\ \nabla^2 f &=&\frac{\partial^2f}{\partial r^2}+\frac{2}{r}\frac{\partial f}{\partial r}+\frac{1}{r^2}\frac{\partial^2f}{\partial\theta^2}+\frac{1}{r^2\tan\theta}\frac{\partial f}{\partial\theta}+\frac{1}{r^2\sin^2\theta}\frac{\partial^2f}{\partial\varphi^2} \end{eqnarray*}$

General orthonormal curvilinear coordinates $(u,v,w)$ can be derived from cartesian coordinates by the transformation $\vec{x}=\vec{x}(u,v,w)$ . The unit vectors are given by: $\vec{e}_{u}=\frac{1}{h_1}\frac{\partial\vec{x}}{\partial u}~,~~ \vec{e}_{v}=\frac{1}{h_2}\frac{\partial\vec{x}}{\partial v}~,~~ \vec{e}_{w}=\frac{1}{h_3}\frac{\partial\vec{x}}{\partial w}$ where the terms $h_i$ give normalization to length 1. The differential operators are than given by: $\begin{eqnarray*} {\rm grad}f &=&\frac{1}{h_1}\frac{\partial f}{\partial u}\vec{e}_{u}+\frac{1}{h_2}\frac{\partial f}{\partial v}\vec{e}_{v}+\frac{1}{h_3}\frac{\partial f}{\partial w}\vec{e}_{w}\\ {\rm div}~\vec{a}&=&\frac{1}{h_1h_2h_3}\left(\frac{\partial}{\partial u}(h_2h_3a_u)+\frac{\partial}{\partial v}(h_3h_1a_v)+\frac{\partial}{\partial w}(h_1h_2a_w)\right)\\ {\rm curl}~\vec{a}&=&\frac{1}{h_2h_3}\left(\frac{\partial(h_3a_w)}{\partial v}-\frac{\partial(h_2a_v)}{\partial w}\right)\vec{e}_{u}+ \frac{1}{h_3h_1}\left(\frac{\partial(h_1a_u)}{\partial w}-\frac{\partial(h_3a_w)}{\partial u}\right)\vec{e}_{v}+ \frac{1}{h_1h_2}\left(\frac{\partial(h_2a_v)}{\partial u}-\frac{\partial(h_1a_u)}{\partial v}\right)\vec{e}_{w}\\ \nabla^2 f &=&\frac{1}{h_1h_2h_3}\left[\frac{\partial}{\partial u}\left(\frac{h_2h_3}{h_1}\frac{\partial f}{\partial u}\right)+ \frac{\partial}{\partial v}\left(\frac{h_3h_1}{h_2}\frac{\partial f}{\partial v}\right)+ \frac{\partial}{\partial w}\left(\frac{h_1h_2}{h_3}\frac{\partial f}{\partial w}\right)\right] \end{eqnarray*}$

Some properties of the $\nabla$ -operator are: $\begin{array}{l@{~~~~~}l@{~~~~~}l} {\rm div}(\phi\vec{v})=\phi{\rm div}\vec{v}+{\rm grad}\phi\cdot\vec{v}& {\rm curl}(\phi\vec{v})=\phi{\rm curl}\vec{v}+({\rm grad}\phi)\times\vec{v}&{\rm curl~grad}\phi=\vec{0}\\ {\rm div}(\vec{u}\times\vec{v})=\vec{v}\cdot({\rm curl}\vec{u})-\vec{u}\cdot({\rm curl}\vec{v})& {\rm curl~curl}\vec{v}={\rm grad~div}\vec{v}-\nabla^2\vec{v}&{\rm div~curl}\vec{v}=0\\ {\rm div~grad}\phi=\nabla^2\phi&\nabla^2\vec{v}\equiv(\nabla^2v_1,\nabla^2v_2,\nabla^2v_3) \end{array}$ Here, $\vec{v}$ is an arbitrary vectorfield and $\phi$ an arbitrary scalar field.

3.2.5 Integral theorems

Some important integral theorems are:

Gauss: $\displaystyle\int\hspace{-2ex}\int\hspace{-2.8ex}\bigcirc (\vec{v}\cdot\vec{n})d^2A=\int\hspace{-1.5ex}\int\hspace{-1.5ex}\int({\rm div}\vec{v})d^3V$
Stokes for a scalar field: $\displaystyle\oint(\phi\cdot\vec{e}_t)ds=\int\hspace{-1.5ex}\int(\vec{n}\times{\rm grad}\phi)d^2A$
Stokes for a vector field: $\displaystyle\oint(\vec{v}\cdot\vec{e}_t)ds=\iint({\rm curl}\vec{v}\cdot\vec{n})d^2A$
this gives: $\displaystyle\int\hspace{-2ex}\int\hspace{-2.8ex}\bigcirc({\rm curl}\vec{v}\cdot\vec{n})d^2A=0$
Ostrogradsky: $\displaystyle\int\hspace{-2ex}\int\hspace{-2.8ex}\bigcirc(\vec{n}\times\vec{v})d^2A=\iiint({\rm curl}\vec{v})d^3A$
$\displaystyle\int\hspace{-2ex}\int\hspace{-2.8ex}\bigcirc(\phi\vec{n})d^2A=\int\hspace{-1.5ex}\int\hspace{-1.5ex}\int({\rm grad}\phi)d^3V$

Here the orientable surface

$\int\hspace{-1mm}\int d^2A$ is bounded by the Jordan curve

$s(t)$ .

3.2.6 Multiple integrals

Let

$A$ be a closed curve given by

$f(x,y)=0$ , than the surface

$A$ inside the curve in

$I\hspace{-1mm}R^2$ is given by

$A=\int\hspace{-1.5ex}\int d^2A=\int\hspace{-1.5ex}\int dxdy$ Let the surface

$A$ be defined by the function

$z=f(x,y)$ . The volume

$V$ bounded by

$A$ and the

$xy$ plane is than given by:

$V=\int\hspace{-1.5ex}\int f(x,y)dxdy$ The volume inside a closed surface defined by

$z=f(x,y)$ is given by:

$V=\int\hspace{-1.5ex}\int\hspace{-1.5ex}\int d^3V=\iint f(x,y)dxdy=\int\hspace{-1.5ex}\int\hspace{-1.5ex}\int dxdydz$

3.2.7 Coordinate transformations

The expressions

$d^2A$ and

$d^3V$ transform as follows when one changes coordinates to

$\vec{u}=(u,v,w)$ through the transformation

$x(u,v,w)$ :

$V=\int\hspace{-1.5ex}\int\hspace{-1.5ex}\int f(x,y,z)dxdydz=\int\hspace{-1.5ex}\int\hspace{-1.5ex}\int f(\vec{x}(\vec{u}))\left|\frac{\partial\vec{x}}{\partial\vec{u}}\right|dudvdw$ In

$I\hspace{-1mm}R^2$ holds:

$\frac{\partial\vec{x}}{\partial\vec{u}}=\left|\begin{array}{cc}x_u&x_v\\ y_u&y_v\end{array}\right|$ Let the surface

$A$ be defined by

$z=F(x,y)=X(u,v)$ . Than the volume bounded by the

$xy$ plane and

$F$ is given by:

$\int\hspace{-1.5ex}\int\limits_Sf(\vec{x})d^2A=\int\hspace{-1.5ex}\int\limits_Gf(\vec{x}(\vec{u})) \left|\frac{\partial X}{\partial u}\times\frac{\partial X}{\partial v}\right|dudv= \int\hspace{-1.5ex}\int\limits_Gf(x,y,F(x,y))\sqrt{1+\partial_xF^2+\partial_yF^2}dxdy$

3.3 Orthogonality of functions

The inner product of two functions

$f(x)$ and

$g(x)$ on the interval

$[a,b]$ is given by:

$(f,g)=\int\limits_a^bf(x)g(x)dx$ or, when using a weight function

$p(x)$ , by:

$(f,g)=\int\limits_a^bp(x)f(x)g(x)dx$ The norm

$\|f\|$ follows from:

$\|f\|^2=(f,f)$ . A set functions

$f_i$ is orthonormal if

$(f_i,f_j)=\delta_{ij}$ .

Each function $f(x)$ can be written as a sum of orthogonal functions: $f(x)=\sum_{i=0}^\infty c_ig_i(x)$ and $\sum c_i^2\leq\|f\|^2$ . Let the set $g_i$ be orthogonal, than it follows: $c_i=\frac{(f,g_i)}{(g_i,g_i)}$

3.4 Fourier series

Each function can be written as a sum of independent base functions. When one chooses the orthogonal basis

$(\cos(nx),\sin(nx))$ we have a Fourier series.

A periodical function $f(x)$ with period $2L$ can be written as: $f(x)=a_0+\sum_{n=1}^\infty\left[a_n\cos\left(\frac{n\pi x}{L}\right)+b_n\sin\left(\frac{n\pi x}{L}\right)\right]$ Due to the orthogonality follows for the coefficients: $a_0=\frac{1}{2L}\int\limits_{-L}^Lf(t)dt~~,~~ a_n=\frac{1}{L}\int\limits_{-L}^Lf(t)\cos\left(\frac{n\pi t}{L}\right)dt~~,~~ b_n=\frac{1}{L}\int\limits_{-L}^Lf(t)\sin\left(\frac{n\pi t}{L}\right)dt$ A Fourier series can also be written as a sum of complex exponents: $f(x)=\sum_{n=-\infty}^\infty c_n{\rm e}^{inx}$ with $c_n=\frac{1}{2\pi}\int\limits_{-\pi}^\pi f(x){\rm e}^{-inx}dx$ The Fourier transform of a function $f(x)$ gives the transformed function $\hat{f}(\omega)$ : $\hat{f}(\omega)=\frac{1}{\sqrt{2\pi}}\int\limits_{-\infty}^\infty f(x){\rm e}^{-i\omega x}dx$ The inverse transformation is given by: $\frac{1}{2}\left[f(x^+)+f(x^-)\right]=\frac{1}{\sqrt{2\pi}}\int\limits_{-\infty}^\infty\hat{f}(\omega){\rm e}^{i\omega x}d\omega$ where $f(x^+)$ and $f(x^-)$ are defined by the lower - and upper limit: $f(a^-)=\lim_{x\uparrow a}f(x)~~,~~f(a^+)=\lim_{x\downarrow a}f(x)$ For continuous functions is $\frac{1}{2}\left[f(x^+)+f(x^-)\right]=f(x)$ .

Gauss:	$\displaystyle\int\hspace{-2ex}\int\hspace{-2.8ex}\bigcirc (\vec{v}\cdot\vec{n})d^2A=\int\hspace{-1.5ex}\int\hspace{-1.5ex}\int({\rm div}\vec{v})d^3V$
Stokes for a scalar field:	$\displaystyle\oint(\phi\cdot\vec{e}_t)ds=\int\hspace{-1.5ex}\int(\vec{n}\times{\rm grad}\phi)d^2A$
Stokes for a vector field:	$\displaystyle\oint(\vec{v}\cdot\vec{e}_t)ds=\iint({\rm curl}\vec{v}\cdot\vec{n})d^2A$
this gives:	$\displaystyle\int\hspace{-2ex}\int\hspace{-2.8ex}\bigcirc({\rm curl}\vec{v}\cdot\vec{n})d^2A=0$
Ostrogradsky:	$\displaystyle\int\hspace{-2ex}\int\hspace{-2.8ex}\bigcirc(\vec{n}\times\vec{v})d^2A=\iiint({\rm curl}\vec{v})d^3A$
	$\displaystyle\int\hspace{-2ex}\int\hspace{-2.8ex}\bigcirc(\phi\vec{n})d^2A=\int\hspace{-1.5ex}\int\hspace{-1.5ex}\int({\rm grad}\phi)d^3V$