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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
b∫af(x)dx=F(b)−F(a)
If u=f(x) holds:
b∫ag(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)∂ydx−f(g(y),y)dg(y)dy+f(h(y),y)dh(y)dy
The arc length ℓ of a curve y(x) is given by:
ℓ=∫√1+(dy(x)dx)2dx
The arc length ℓ of a parameter curve F(→x(t)) is:
ℓ=∫Fds=∫F(→x(t))|˙→x(t)|dt
with
→t=d→xds=˙→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=2π∫y√1+(dy(x)dx)2dx
The volume V of a solid of revolution is:
V=π∫f2(x)dx
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∈Z, and of functions of the type
px+q((x−a)2+b2)n
with b>0 and n∈IN. So:
p(x)(x−a)n=n∑k=1Ak(x−a)k , p(x)((x−b)2+c2)n=n∑k=1Akx+B((x−b)2+c2)k
Recurrent relation: for n≠0 holds:
∫dx(x2+1)n+1=12nx(x2+1)n+2n−12n∫dx(x2+1)n
Elliptic functions can be written as a power series as follows:
√1−k2sin2(x)=1−∞∑n=1(2n−1)!!(2n)!!(2n−1)k2nsin2n(x)
1√1−k2sin2(x)=1+∞∑n=1(2n−1)!!(2n)!!k2nsin2n(x)
with n!!=n(n−2)!!.
The gamma function Γ(y) is defined by:
Γ(y)=∞∫0e−xxy−1dx
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(2n−1)!! and Γ(n)(y)=∞∫0e−xxy−1lnn(x)dx
The betafunction β(p,q) is defined by:
β(p,q)=1∫0xp−1(1−x)q−1dx
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)
The delta function δ(x) is an infinitely thin peak function with
surface 1. It can be defined by:
δ(x)=limε→0P(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)
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)=1−t21+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+x∫R(x,√1−x2)dx : x=sin(φ) ,dx=cos(φ)dφ or √1−x2=1−tx∫R(x,√x2−1)dx : x=1cos(φ) ,dx=sin(φ)cos2(φ)dφ or √x2−1=x−t
These definite integrals are easily solved:
π/2∫0cosn(x)sinm(x)dx=(n−1)!!(m−1)!!(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 , ∞∫0x3dxex−1=π415
The partial derivative with respect to x of a function f(x,y) is defined by:
(∂f∂x)x0=limh→0f(x0+h,y0)−f(x0,y0)h
The directional derivative in the direction of α is defined by:
∂f∂α=limr↓0f(x0+rcos(α),y0+rsin(α))−f(x0,y0)r=(→∇f,(sinα,cosα))=∇f⋅→v|→v|
When one changes to coordinates f(x(u,v),y(u,v)) holds:
∂f∂u=∂f∂x∂x∂u+∂f∂y∂y∂u
If x(t) and y(t) depend only on one parameter t holds:
∂f∂t=∂f∂xdxdt+∂f∂ydydt
The total differential df of a function of 3 variables is given by:
df=∂f∂xdx+∂f∂ydy+∂f∂zdz
So
dfdx=∂f∂x+∂f∂ydydx+∂f∂zdzdx
The tangent in point →x0 at the surface f(x,y)=0 is given by
the equation fx(→x0)(x−x0)+fy(→x0)(y−y0)=0.
The tangent plane in →x0 is given by:
fx(→x0)(x−x0)+fy(→x0)(y−y0)=z−f(→x0).
A function of two variables can be expanded as follows in a Taylor series:
f(x0+h,y0+k)=n∑p=01p!(h∂p∂xp+k∂p∂yp)f(x0,y0)+R(n)
with R(n) the residual error and
(h∂p∂xp+k∂p∂yp)f(a,b)=p∑m=0(pm)hmkp−m∂pf(a,b)∂xm∂yp−m
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∈IR2 are:
- Points on V where f(x,y) is not differentiable,
- Points where →∇f=→0,
- If the boundary V is given by φ(x,y)=0, than all points where
→∇f(x,y)+λ→∇φ(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)+λ1→∇φ1(x,y,z)+λ2→∇φ2(x,y,z)=0.
In cartesian coordinates (x,y,z) holds:
→∇=∂∂x→ex+∂∂y→ey+∂∂z→ezgradf=∂f∂x→ex+∂f∂y→ey+∂f∂z→ezdiv →a=∂ax∂x+∂ay∂y+∂az∂zcurl →a=(∂az∂y−∂ay∂z)→ex+(∂ax∂z−∂az∂x)→ey+(∂ay∂x−∂ax∂y)→ez∇2f=∂2f∂x2+∂2f∂y2+∂2f∂z2
In cylindrical coordinates (r,φ,z) holds:
→∇=∂∂r→er+1r∂∂φ→eφ+∂∂z→ezgradf=∂f∂r→er+1r∂f∂φ→eφ+∂f∂z→ezdiv →a=∂ar∂r+arr+1r∂aφ∂φ+∂az∂zcurl →a=(1r∂az∂φ−∂aφ∂z)→er+(∂ar∂z−∂az∂r)→eφ+(∂aφ∂r+aφr−1r∂ar∂φ)→ez∇2f=∂2f∂r2+1r∂f∂r+1r2∂2f∂φ2+∂2f∂z2
In spherical coordinates (r,θ,φ) holds:
→∇=∂∂r→er+1r∂∂θ→eθ+1rsinθ∂∂φ→eφgradf=∂f∂r→er+1r∂f∂θ→eθ+1rsinθ∂f∂φ→eφdiv →a=∂ar∂r+2arr+1r∂aθ∂θ+aθrtanθ+1rsinθ∂aφ∂φcurl →a=(1r∂aφ∂θ+aθrtanθ−1rsinθ∂aθ∂φ)→er+(1rsinθ∂ar∂φ−∂aφ∂r−aφr)→eθ+(∂aθ∂r+aθr−1r∂ar∂θ)→eφ∇2f=∂2f∂r2+2r∂f∂r+1r2∂2f∂θ2+1r2tanθ∂f∂θ+1r2sin2θ∂2f∂φ2
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=1h1∂→x∂u , →ev=1h2∂→x∂v , →ew=1h3∂→x∂w
where the terms hi give normalization to length 1. The differential operators are than given by:
gradf=1h1∂f∂u→eu+1h2∂f∂v→ev+1h3∂f∂w→ewdiv →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)→ew∇2f=1h1h2h3[∂∂u(h2h3h1∂f∂u)+∂∂v(h3h1h2∂f∂v)+∂∂w(h1h2h3∂f∂w)]
Some properties of the ∇-operator are:
div(ϕ→v)=ϕdiv→v+gradϕ⋅→vcurl(ϕ→v)=ϕcurl→v+(gradϕ)×→vcurl gradϕ=→0div(→u×→v)=→v⋅(curl→u)−→u⋅(curl→v)curl curl→v=grad div→v−∇2→vdiv curl→v=0div gradϕ=∇2ϕ∇2→v≡(∇2v1,∇2v2,∇2v3)
Here, →v is an arbitrary vectorfield and ϕ an arbitrary scalar field.
Some important integral theorems are:
Gauss: | ∫∫◯(→v⋅→n)d2A=∫∫∫(div→v)d3V |
Stokes for a scalar field: | ∮(ϕ⋅→et)ds=∫∫(→n×gradϕ)d2A |
Stokes for a vector field: | ∮(→v⋅→et)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).
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
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
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)}
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).