Loading [MathJax]/jax/element/mml/optable/MathOperators.js

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 baf(x)dx=F(b)F(a) If u=f(x) holds: bag(f(x))df(x)=f(b)f(a)g(u)du Partial integration: with F and G the primitives of f and g holds: f(x)g(x)dx=f(x)G(x)G(x)df(x)dxdx A derivative can be brought under the intergral sign (see section 1.8.3 for the required conditions): ddy[x=h(y)x=g(y)f(x,y)dx]=x=h(y)x=g(y)f(x,y)ydxf(g(y),y)dg(y)dy+f(h(y),y)dh(y)dy

3.1.2 Arc lengts, surfaces and volumes

The arc length of a curve y(x) is given by: =1+(dy(x)dx)2dx The arc length of a parameter curve F(x(t)) is: =Fds=F(x(t))|˙x(t)|dt with t=dxds=˙x(t)|˙x(t)|   ,  |t |=1 (v,t)ds=(v,˙t(t))dt=(v1dx+v2dy+v3dz) The surface A of a solid of revolution is: A=2πy1+(dy(x)dx)2dx The volume V of a solid of revolution is: V=πf2(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 (xa)k with kZ, and of functions of the type px+q((xa)2+b2)n with b>0 and nIN. So: p(x)(xa)n=nk=1Ak(xa)k  ,   p(x)((xb)2+c2)n=nk=1Akx+B((xb)2+c2)k Recurrent relation: for n0 holds: dx(x2+1)n+1=12nx(x2+1)n+2n12ndx(x2+1)n

3.1.4 Special functions

3.1.4.1 Elliptic functions

Elliptic functions can be written as a power series as follows: 1k2sin2(x)=1n=1(2n1)!!(2n)!!(2n1)k2nsin2n(x) 11k2sin2(x)=1+n=1(2n1)!!(2n)!!k2nsin2n(x) with n!!=n(n2)!!.

3.1.4.2 The Gamma function

The gamma function Γ(y) is defined by: Γ(y)=0exxy1dx One can derive that Γ(y+1)=yΓ(y)=y!. This is a way to define faculties for non-integers. Further one can derive that Γ(n+12)=π2n(2n1)!!  and  Γ(n)(y)=0exxy1lnn(x)dx

3.1.4.3 The Beta function

The betafunction β(p,q) is defined by: β(p,q)=10xp1(1x)q1dx with p and q >0. The beta and gamma functions are related by the following equation: β(p,q)=Γ(p)Γ(q)Γ(p+q)

3.1.4.4 The Delta function

The delta function δ(x) is an infinitely thin peak function with surface 1. It can be defined by: δ(x)=limε0P(ε,x)  with  P(ε,x)={0   for |x|>ε12ε   when |x|<ε Some properties are: δ(x)dx=1  ,   F(x)δ(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(12x):=t: dx=2dt1+t2 ,  cos(x)=1t21+t2 ,  sin(x)=2t1+t2 Each integral of the type R(x,ax2+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: R(x,x2+1)dx :  x=tan(φ) ,dx=dφcos(φ)  or  x2+1=t+xR(x,1x2)dx :  x=sin(φ) ,dx=cos(φ)dφ  or  1x2=1txR(x,x21)dx :  x=1cos(φ) ,dx=sin(φ)cos2(φ)dφ  or  x21=xt These definite integrals are easily solved: π/20cosn(x)sinm(x)dx=(n1)!!(m1)!!(m+n)!!{π/2when m and n are both even1in all other cases Some important integrals are: 0xdxeax+1=π212a2  ,  x2dx(ex+1)2=π23  ,  0x3dxex1=π415

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: (fx)x0=limh0f(x0+h,y0)f(x0,y0)h The directional derivative in the direction of α is defined by: fα=limr0f(x0+rcos(α),y0+rsin(α))f(x0,y0)r=(f,(sinα,cosα))=fv|v| When one changes to coordinates f(x(u,v),y(u,v)) holds: fu=fxxu+fyyu If x(t) and y(t) depend only on one parameter t holds: ft=fxdxdt+fydydt The total differential df of a function of 3 variables is given by: df=fxdx+fydy+fzdz So dfdx=fx+fydydx+fzdzdx The tangent in point x0 at the surface f(x,y)=0 is given by the equation fx(x0)(xx0)+fy(x0)(yy0)=0.

The tangent plane in x0 is given by: fx(x0)(xx0)+fy(x0)(yy0)=zf(x0).

3.2.2 Taylor series

A function of two variables can be expanded as follows in a Taylor series: f(x0+h,y0+k)=np=01p!(hpxp+kpyp)f(x0,y0)+R(n) with R(n) the residual error and (hpxp+kpyp)f(a,b)=pm=0(pm)hmkpmpf(a,b)xmypm

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 VIR2 are:

  1. Points on V where f(x,y) is not differentiable,
  2. Points where f=0,
  3. If the boundary V is given by φ(x,y)=0, than all points where f(x,y)+λφ(x,y)=0 are possible for extrema. This is the multiplicator method of Lagrange, λ is called a multiplicator.

The same as in IR2 holds in IR3 when the area to be searched is constrained by a compact V, and V is defined by φ1(x,y,z)=0 and φ2(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 f(x,y,z)+λ1φ1(x,y,z)+λ2φ2(x,y,z)=0.

3.2.4 The -operator

In cartesian coordinates (x,y,z) holds: =xex+yey+zezgradf=fxex+fyey+fzezdiv a=axx+ayy+azzcurl a=(azyayz)ex+(axzazx)ey+(ayxaxy)ez2f=2fx2+2fy2+2fz2

In cylindrical coordinates (r,φ,z) holds: =rer+1rφeφ+zezgradf=frer+1rfφeφ+fzezdiv a=arr+arr+1raφφ+azzcurl a=(1razφaφz)er+(arzazr)eφ+(aφr+aφr1rarφ)ez2f=2fr2+1rfr+1r22fφ2+2fz2

In spherical coordinates (r,θ,φ) holds: =rer+1rθeθ+1rsinθφeφgradf=frer+1rfθeθ+1rsinθfφeφdiv a=arr+2arr+1raθθ+aθrtanθ+1rsinθaφφcurl a=(1raφθ+aθrtanθ1rsinθaθφ)er+(1rsinθarφaφraφr)eθ+(aθr+aθr1rarθ)eφ2f=2fr2+2rfr+1r22fθ2+1r2tanθfθ+1r2sin2θ2fφ2

General orthonormal curvilinear coordinates (u,v,w) can be derived from cartesian coordinates by the transformation x=x(u,v,w). The unit vectors are given by: eu=1h1xu ,  ev=1h2xv ,  ew=1h3xw where the terms hi give normalization to length 1. The differential operators are than given by: gradf=1h1fueu+1h2fvev+1h3fwewdiv a=1h1h2h3(u(h2h3au)+v(h3h1av)+w(h1h2aw))curl a=1h2h3((h3aw)v(h2av)w)eu+1h3h1((h1au)w(h3aw)u)ev+1h1h2((h2av)u(h1au)v)ew2f=1h1h2h3[u(h2h3h1fu)+v(h3h1h2fv)+w(h1h2h3fw)]

Some properties of the -operator are: div(ϕv)=ϕdivv+gradϕvcurl(ϕv)=ϕcurlv+(gradϕ)×vcurl gradϕ=0div(u×v)=v(curlu)u(curlv)curl curlv=grad divv2vdiv curlv=0div gradϕ=2ϕ2v(2v1,2v2,2v3) Here, v is an arbitrary vectorfield and ϕ an arbitrary scalar field.

3.2.5 Integral theorems

Some important integral theorems are:

Gauss: (vn)d2A=(divv)d3V
Stokes for a scalar field: (ϕet)ds=(n×gradϕ)d2A
Stokes for a vector field: (vet)ds=
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).