MathML Example document

produced by "iMathGeo" & "Softwares" generator under ©

Sunday on June 12, 2011 at 09:01AM

$typographical news$

$α, β, χ, δ, η, γ, κ, ι, ω, σ, θ, τ, ξ, π, ψ, ρ, μ, ν, ϖ, λ, Φ, φ, ε, ο, υ, ϑ, ζ$

$Α, Β, Χ, Δ, Η, Γ, Κ, Ι, Ω, Σ, Θ, Τ, Ξ, π, Ψ, Ρ, Μ, Ν, Λ, Φ, Ε, Ο, Υ, Ζ$

$C \in (AB]$

$C \in [AB)$

$i \in ⟦ 1; 2 ⟧$

$A \notin [AB)$

$A ⊄ B$

$A \supset B$

$A \supseteq B$

$A ⊅ B$

$A ⊉ B$

$A ∋ B$

$A ∌ B$

$(AB) ∦ (CD)$

$(AB) ⊥ (CD)$

$a ≯ b$

$a ≮ b$

$a ≱ b$

$a ≰ b$

$ℤ / 2 ℤ^{*}$

$ℤ / 2 ℤ [X]$

$ℤ / 2 ℤ^{*} [X]$

$(f * g) (x) = \int_{− \infty}^{+ \infty} f(x - t) . g(t) . d t$

$(f * g) (x) = ∑_{m = − \infty}^{+ \infty} f(n - m) . g(m) = ∑_{m = − \infty}^{+ \infty} f(m) . g(n - m)$

$(f * (g + h)) (x) \begin{matrix} def \\ = \end{matrix} \int_{− \infty}^{+ \infty} f(x - t) . (g(t) + h(t)) . d t$

$‖ \vec{AB} ‖ + ‖ \vec{BC} ‖ \leq ‖ \vec{AC} ‖$

$Π(x), Π_{x}$

$e(x), e_{x}$

$\cos^{2} x + \sin^{2} x = 1$

$\log^{2} x$

$x \in ℜ and y \in ℑ$

$\partial, ℏ, ƛ$

${a \land b} \Rightarrow C$

${a and b} \Rightarrow C$

${a \lor b} \Rightarrow C$

${a or b} \Rightarrow C$

${a xor b} \Rightarrow C$

$\Leftrightarrow x + 2 = 0 \Leftrightarrow x = − 2$

$(A \cup B) \cap C$

$(d') ∥ (d) and (d'') ∥ (d) \Rightarrow (d'') ∥ (d)$

$(d') ⊥ (d)$

$\exists (x; y) \in ℕ \times ℕ$

$\exists! x | toto$

$\forall ε > 0; \exists η > 0 tel que si ∣ x - x0 ∣ < η \Rightarrow ∣ f(x) - f(x0) ∣ < ε$

$P (ω \in Ω | x(ω) \leq \frac{3}{2}) = \int_{− \infty}^{\frac{3}{2}} e^{− t^{2}} d t$

$A \subset B \cup C \cap D$

$A \subseteq B$

$P(A \cup B) = P(A) + P(B) - P(A \cap B)$

$x \equiv 1 \mod 2$

$f \sim g + h$

$f + f_{x} + f(x)$

$f + {f_{x}}^{2} + f(x)$

$\int_{ℝ}^{} f d x$

$\int_{x \in ℍ \cup ℂ}^{} ψ(x; t) d x$

$\int_{ℍ \cup ℂ}^{} ψ(x; t) d x$

$\frac{1}{\sqrt{2 π}} \int_{x \in [1; 2] \cap [− \frac{1}{2}; + \infty [}^{} ψ(x; t) \sqrt[3]{2} d x$

$\frac{1}{\sqrt{2 π}} \int_{}^{x \in [1; 2]} ψ(x; t) \sqrt[3]{2} d t$

$∑_{n \in ℕ \cap [1; 2]} \frac{1}{n^{2}}$

$∑_{n \in ℕ} \frac{1}{n^{2}}$

$∏_{n \in ℕ} \frac{1}{n^{2}}$

$lim_{x \to + \infty} \frac{sin(x)}{x}$

$lim_{x \to + \infty} x$

$lim_{\begin{matrix} x \to 0 \\ x < 0 \end{matrix}} \frac{sin(x)}{x}$

$lim_{x \to 0^{+}} \frac{sin(x)}{x}$

$(\begin{matrix} 1 & \dots & \dots & \dots & N \\ ⋮ & ⋱ & ⋮ \\ ⋮ & ⋱ & ⋮ \\ ⋮ & ⋱ & ⋮ \\ N & \dots & \dots & \dots & N \end{matrix})$

$(\begin{matrix} a_{11} & \dots & \dots & \dots & a_{n1} \\ ⋮ & ⋱ & ⋮ \\ ⋮ & ⋱ & ⋮ \\ ⋮ & ⋱ & ⋮ \\ a_{1 n} & \dots & \dots & \dots & a_{nn} \end{matrix})$

${2^{d} log(2)} = {∑_{n = 1}^{d} \frac{2^{d - n}}{n}} + ∑_{n = d + 1}^{+ \infty} \frac{2^{d - n}}{n}$

$∑_{k = 0}^{+ \infty} {(− 1)}^{k} u_{k} = ∑_{k = 0}^{m - 1} {(− 1)}^{k} u_{k} + ∑_{k = 0}^{\infty} {(− 1)}^{k} Δ^{k} u_{n}$

$Δ^{1} u_{n} = u_{n + 1} - u_{n}$

$Δ^{2} u_{n} = Δ^{1} u_{n + 1} - Δ^{1} u_{n} = u_{n + 2} - 2 u_{n + 1} + u_{n}$

$Δ^{3} u_{n} = Δ^{2} u_{n + 2} - Δ^{2} u_{n + 1} = u_{n + 3} - 3 u_{n + 2} + 3 u_{n + 1} - u_{n}$

$∑_{k = 1}^{\infty} (1 + \frac{1}{2} + ... + \frac{1}{k}) k^{− 2} = \frac{17 π^{4}}{360}$

${(a + b)}^{n} = ∑_{k = 0}^{n} {C_{n}}^{k} a^{k} b^{n - k}$

${(a + b)}^{n} = ∑_{k = 0}^{n} C_{n}^{k} a^{k} b^{n - k}$

$A_{n}^{k}$

$m. \vec{a_{x}} = \vec{P_{x}} + \vec{F_{ski}} + \vec{F_{air}}$

$a_{x} = g. sin(α) - \frac{F_{ski}}{m} - \frac{F_{air}}{m}$

$\bar{z} = 2 - 3 i$

$E = h ν$

$P = \frac{h}{λ}$

$E = \frac{p^{2}}{2} m + V(r)$

$E = h ν$

$P = \frac{h}{λ}$

$E = \frac{p^{2}}{2} m + V(r)$

$\vec{p}$

$\hat{σ} = \sqrt{\frac{1}{n - 1} ∑_{i = 1}^{n} {(x - \bar{X})}^{2}}$

$\overset{︵}{AB}$

$f : \begin{matrix} ℝ \to ℝ \\ x \mapsto 2 x + 3 \end{matrix}$

$f : \begin{matrix} ℝ \to ℝ \\ x \mapsto f(x) \end{matrix}$

$C(n; k) = C_{n}^{k} = {nC}_{k} = (\begin{matrix} n \\ k \end{matrix}) = \frac{n!}{k! (n - k)!}$

$erf(z) = \frac{2}{\sqrt{π}} ∑_{n = 0}^{+ \infty} {(− 1)}^{n} \frac{z^{2 n + 1}}{n! (2 n + 1)}$

$⋃_{i = 1}^{N} f$

$⋂_{i = 1}^{N} f$

$\int_{1}^{N} f d i$

$∬_{i = 1}^{N} f$

$∭_{i = 1}^{N} f$

$∫_{i = 1}^{N} f$

$\underset{Ω}{∬} f$

$∭_{i = 1}^{N} f$

$∮_{i = 1}^{N} f$

$∯_{i = 1}^{N} f$

$∰_{i = 1}^{N} f$

$\underset{Ω}{∰} f$

$⊕_{i = 1}^{N} f$

$⊗_{i = 1}^{N} f$

$⊙_{i = 1}^{N} f$

$∑_{i = 1}^{N} f$

${\begin{matrix} x + y = 2 \\ 2 x + 3 y = 32 \end{matrix}$

$\begin{matrix} x + y = 2 \\ 2 x + 3 y = 32 \end{matrix}} \Leftrightarrow x \in {− 26; 28}$

${\begin{matrix} x + y = 2 \\ 2 x + 3 y = 32 \end{matrix} \Leftrightarrow x \in {− 26; 28}$

${\begin{matrix} x + y = 2 \\ 2 x + 3 y = 32 \end{matrix}} \Leftrightarrow x \in {− 26; 28}$

$∣ x ∣ = {\begin{matrix} x si x \geq 0 \\ − x si x < 0 \end{matrix}$

$f(x) = {\begin{matrix} \begin{matrix} x si x \in] − \infty; − 1] \\ − 2 x si x \in] 1; 2] \end{matrix} \\ − 2 x + \frac{1}{2} si x \in] 2; + \infty [ \end{matrix}$

$a \oplus b$

$a \otimes b$

$a ⊙ b$

$a \oplus b$

$\vec{z} \land (\vec{a} \land \vec{b})$

$\vec{z} . (\vec{a} \land \vec{b})$

$\vec{a} . \vec{b}$

$\vec{a} . (\vec{b} + \vec{c}) = \vec{a} . \vec{b} + \vec{a} . \vec{c}$

${\begin{matrix} x \equiv 2 \mod 3 \\ x \equiv 5 \mod 7 \end{matrix}$

$(f \circ g) (x) = f(g(x))$

$(f \circ g)'(x) = f'(g(x)) \times g'(x)$

$(f \times g)' = f' \times g + g' \times f$

$(f \times g)'(x) = f'(x) \times g(x) + g'(x) \times f(x)$

${(\frac{f}{g})}^{'} = \frac{f' \times g - g' \times f}{g^{2}}$

$σ(X)$

$Physic$

$〈 a | b 〉$

$〈 a |$

$| b 〉$

$H_{2} {SO}_{4}$

${SO}_{4}^{2 -}$

${_{92}}^{238} U$

${CH}_{4} + 2 O_{2} \to {CO}_{2} + 2 H_{2} O$

$\frac{{\hat{\vec{p}}}^{2}}{2 m} ∣ Ψ(t) + V(\hat{\vec{r}}, t) ∣ Ψ(t) 〉$

$| tyo 〉$

$f(p_{i}, q_{i}, t)$

$is given by$

$\frac{df}{dt} = {f, H} + \frac{\partial f}{\partial t}$

${f; g} = ∑_{i = 1}^{N} (\frac{\partial f}{\partial q_{i}} \frac{\partial g}{\partial p_{i}} - \frac{\partial f}{\partial p_{i}} \frac{\partial g}{\partial q_{i}})$

$[A; BC] = [A; B] C + B [A; C]$

$\frac{d}{ds} BC = \frac{dA}{ds} C + B \frac{dC}{ds}$

$grad T = \frac{\partial T}{\partial x}$

$\nabla T(x) = T'(x) = \frac{dT}{dx} (x)$

$f : ℝ^{2} \mapsto ℝ$

$grad f = \nabla f$

$\frac{\partial^{2}}{\partial t^2} (f)$

$\frac{\partial}{\partial t} (f)$

$\frac{\partial^{2}}{\partial t \partial u} (f)$

$\frac{\partial^{2} f}{\partial t \partial u}$

$E = ℏ ν$

$\hat{H} | Ψ 〉 = i ℏ \frac{d}{dt} | Ψ 〉$

$J^{2} = j (j + 1) ℏ^{2} with : j = 0, 1, 2, 3, 4, \dots$

$J_{z} = m ℏ$

$Δ x Δ p \geq \frac{1}{2} ℏ$

$Gradient et variété riemanniene$

$Développement limité$

$Soit une application:$

$avec ( n In {2;3} for example)$

$est monotone (resp. strictement monotone), alors f est convexe (resp. strictement convexe).$

$C'est-à-dire, en utilisant la caractérisation par les cordes :$

$Relations vectorielles$

$M(x; y; z) \in (ABC) \Leftrightarrow x + 3 y - 5 z + 5 = 0$

$(\begin{matrix} x1 \\ y1 \\ z1 \end{matrix}) \land (\begin{matrix} x2 \\ y2 \\ z2 \end{matrix}) = (\begin{matrix} (− y1) z2 - y2 z1 \\ −(((x1 z2 - x2 z1))) \\ x1 y2 - x2 y1 \end{matrix})$

$\vec{n} = \vec{AB} \land \vec{AC} = (− 2; − 6; 10)$

$\vec{n1} = − 1 / 2 \vec{n}$

$\vec{grad} f = \vec{\nabla} f$

$\frac{\partial}{\partial t} (\nabla f) = \nabla \frac{\partialf}{\partial t}$

$div(\vec{grad} f) = Δ f$

$\vec{grad} (div f) = \vec{rot} (\vec{rot} f) + \vec{Δ} \vec{f}$

$Changement of Base$

$\vec{\nabla} S . \vec{dOM} = (y \vec{i} + x \vec{j}) . (dx \vec{i} + dy \vec{j}) = (\frac{∂(xy)}{\partial x} \vec{i} + \frac{∂(xy)}{\partial y} \vec{j}) . (dx \vec{i} + dy \vec{j})$

${\bar{12345}}^{two} = 11000000111001$

$N_{p} = {\begin{matrix} p - 1 si p \equiv + 1 (\mod 4) \\ p + 1 si p \equiv − 1 (\mod 4) \end{matrix}$

$2 \equiv + 1$

$t = 0, 1, \dots, p - 1, \infty$

$− 1 = i^{2}$

$∏_{p} \frac{p}{N_{p}} = ∏_{p} {(1 - \frac{a_{p}}{p})}^{− 1}$

$= (∏_{(p \equiv 1 (4))} {(1 - \frac{1}{p})}^{− 1}) . (∏_{(p \equiv − 1 (4))} {(1 - \frac{1}{p})}^{− 1})$

$= (∏_{(p \equiv 1 (4))} 1 + \frac{1}{p} + \frac{1}{p^{2}} + \frac{1}{p^{3}} + ...) . (∏_{(p \equiv − 1 (4))} 1 - \frac{1}{p} + \frac{1}{p^{2}} - \frac{1}{p^{3}} + ...)$

$= 1 - \frac{1}{3} + \frac{1}{5} - \frac{1}{7} + \frac{1}{9} - \frac{1}{11} + ....$

$= \frac{π}{4}$