User:-jkb-/Helpers 5

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Mathematische Probleme orig PDF http://www.mathematik.uni-bielefeld.de/~kersten/hilbert/rede.pdf
Tex de.wiki http://de.wikipedia.org/wiki/Hilfe:TeX

space: \quad \qquad

kleine Formeln im Text

α, β   ƒ(x)   e^{i π z}  a^β   ζ(s)  φ(x)

α, β ƒ(x) e^{i π z} a^β ζ(s) φ(x)

 $a β$    2^√2    $e π = i - 2 i$     $ζ(s)$

$a β$ 2^√2 $e π = i - 2 i$ $ζ(s)$

Kapitel 5

$x'_i = f\!\,_i(x_1,...,x_n;a_1,...,a_r) \qquad \qquad (i = 1,...,n)$

$x'_i = f\!\,_i(x_1,...,x_n;a_1,...,a_r)$

$x''_i = f\!\,_i(x'_1,...,x'_n;b_1,...,b_r)$

$x''_i = f\!\,_i(f\!\,_1(x,a),...f\!\,_n(x,a);b_1,...,b_r) = f\!\,_i(x_1,...,x_n;c_1,...,c_r)$

$f\!\,(x_1 + y_1,...,x_n + y_n) \geq f\!\,(x_1,...,x_n) + f\!\,(y_1,...,y_n)$

$\varphi(x+\alpha) - \varphi(x) = f\!\,(x)$

$\varphi(x+\beta) - \varphi(x) = 0$

Kapitel 8

$a β$

Kap. 9 ++

$\frac{\alpha}{\Im} = + 1$

$f^7 + x f^3 y f^2 + z f + 1 = 0 \quad$

$X_1 = f\!\,_1(x_1,...,x_n)$

$X_2 = f\!\,_2(x_1,...,x_n)$

$(S)\qquad ............$

$X_m = f\!\,_m(x_1,...,x_n)$

gegebenen Functionen $f'$ ₁,..., $f'$ _m als ganze

$\frac{G(X1,X2,...,Xm)}{p^h}$

Kap 15+

$\frac{dy}{dx} = \frac{Y}{X}$

$X \bigg(y \frac{dz}{dt} - z \frac{dy}{dt} \bigg) + Y \bigg(z \frac{dx}{dt} - x \frac{dz}{dt} \bigg) + Z \bigg(x \frac{dy}{dt} - y \frac{dx}{dt} \bigg) = 0$

$x \frac{\delta \zeta (s.x)}{\delta x} = \zeta (s - 1.x)$

$\frac{\delta d^2 f\!\,}{\delta x^2} + \frac{\delta^2 f\!\,}{\delta y^2} = 0$

$\frac{\delta d^2 f\!\,}{\delta x^2} + \frac{\delta^2 f\!\,}{\delta y^2} = e^f\!\,$

p 29:

$\iint F (p,q,z;x,y) dx dy = \rm{Minimum,} \qquad\qquad \bigg[ p = \frac{\delta z}{\delta x}, q = \frac{\delta z}{\delta y} \bigg]$

$\frac{\delta d^2 F}{\delta p^2} \quad \frac{\delta d^2 F}{\delta q^2} - \bigg( \frac{\delta d^2 F}{\delta p \delta q} \bigg)^2 > 0$

++

$\zeta (s, x) = x + \frac{x^2}{2^s} + \frac{x^3}{3^s} + \frac{x^4}{4^s} + ...$

Kap 21++

.

$J = \int_a^b F (y_x,y;x)dx, \qquad \bigg [ y_x = \frac{dy}{dx} \bigg ]$

$\delta J = 0 \quad$

$(\!\,1)$

$\frac{{dF_y}_x}{dx} - F_y = 0,\qquad\bigg[{F_y}_x = \frac{\delta F}{\delta y_x}\bigg]$

$J^* = \int_a^b \{F + (y_x - p)F_p \}dx, \quad \bigg [ F = F (p,y;x), F_p = \frac{\delta F(p,y;x)}{\delta p} \bigg ]$

$J^* = \int_a \{Ay_x - B\}dx, \quad$

$\delta J^* = 0 \quad$

$\frac{\delta A}{\delta x} + \frac{\delta B}{\delta y} = 0 \quad$

$(\!\,1^*)$

$\frac{\delta F_p}{\delta x} + \frac{\delta (pF_p - F)}{\delta y} = 0$ ‎

$\delta J^* \quad$	$= \int_a^b \{ F_y \delta y + F_p \delta p + (\delta y_x - \delta p)F_p + (y_x - p)\delta F_p \} dx$
	$= \int_a^b \{ F_y \delta y + \delta y_x F_p + (y_x - p)\delta F_p \} dx$
	$= \delta J + \int_a^b (y_x - p)\delta F_p dx$