Example One

Topology Algebra Differential Equations Integrals Complex example

Topology

Multiplication by an axis.

Let us note the following formulas:

$\underset{\mathrm{xy}}{\overset{}{E}}\varphi \left(y\right)=X\times \underset{\mathrm{xy}}{\overset{}{E}}\varphi \left(y\right)$

Algebra

Example

$f\left(x\right)=(x+y{)}^3$

=$\sum _{n=0}^3\left(\begin{array}{c}3\\ n\end{array}\right)x^{3-n}{y}^n$ $\begin{array}{c}\\ \end{array}$

by the Binomial theorem =${x^3+3x^{2}y}^{}+3x{y}^2+y^3$

Example

Let $A=\left(\begin{array}{ccc}1& 2& 3\\ 4& 5& 6\\ 7& 8& 9\end{array}\right).$

The characteristic polynomial is equivalent to the determinant of the matrix $B=A-\lambda I$ .

We can compute $detB$ by using cofactors:

$(1-\lambda )\Vert \begin{array}{cc}5-\lambda & 6\\ 8& 9-\lambda \end{array}\Vert -2\Vert \begin{array}{cc}4& 6\\ 7& 9-\lambda \end{array}\Vert +3\Vert \begin{array}{cc}4& 5-\lambda \\ 7& 8\end{array}\Vert =0\begin{array}{}\end{array}$

The roots of this cubic polynomial are thethree eigenvalues of A: ${\lambda }_1^{}=0,{\lambda }_2^{}=\frac{3}{2}(5-\sqrt{33}),{\lambda }_3^{}=\frac{3}{2}(5+\sqrt{33})$

Differential Equations

Suppose $v\ge 2\mathrm{and}{{\zeta }^1,⃛,{\zeta }^{v-1}{}^{}\in C^2\left(G\right).}^{}\mathrm{Then}$

$\sum _{\alpha =1}^v(-1{)}^{\alpha }\frac{\partial }{\partial x^{\alpha }}$ [$\frac{\partial ({\zeta }^1,⃛,{\zeta }^{\alpha -1},{\zeta }^{\alpha },⃛,{\zeta }^{v-1})}{\partial (x^1,⃛,x^{\alpha -1}{}^{},x^{\alpha +1}{,}^{}⃛,x^{v-1}{}^{})}]=0.$

Integration

We are concerned with the equation

$\frac{d^{2}x}{{}^{d{t}^2}}+(a+\mathrm{ϵcos\; t})x=0$

We shall carry out a complete study only for $a$ =${\omega }^2>0$ The equation is then equivalent to $\frac{\mathrm{dX}}{\mathrm{dt}}$ =AX+εP(t)X where

A=$\left(\begin{array}{cc}0& 1\\ -a{\omega }^2& 0\end{array}\right)\mathrm{and\; P}\left(\mathrm{t}\right)=\left(\begin{array}{cc}0& 0\\ -\mathrm{cos\; t}& 0\end{array}\right)$

We shall carry calculate an approximation to${\Phi }_{ϵ}(t,0)\mathrm{by\; the\; method\; of\; section}1.1.\mathrm{Thus\; let}$

${\Phi }_{ϵ}(t,0$ )=$\exp$ (At)+ε${\Phi }_1$ (t)+${ϵ}^2{\Phi }_2$ (t)+…

We shall calculate exp (A2Π)+ε${\Phi }_1$ (2Π)+${ϵ}^2{\Phi }_2$ (2Π), the beginning of the expansion of ${\Phi }_{ϵ}(\mathrm{2\Pi },0$ ), and then an approximation to the second order in ε of $\mathrm{tr}{\Phi }_{ϵ}(2\Pi ,0$ ). We know that

exp$\text{[A(t-s)]=}$ $\left(\begin{array}{cccc}\mathrm{cos\omega }(t-s)& & \frac{1}{\omega }\mathrm{sin\omega }(t-s)& \\ -\mathrm{\omega sin\omega }(t-s)& & \text{cosw(t-s)}& \end{array}\right)$

and

tr${\mathrm{exp\Phi }}_{ϵ}(A2\mathrm{\Pi )=2cos\; 2\omega \Pi }$

Let us calculate

${\Phi }_1\text{(t)=}$ ${\int }_0^t\mathrm{exp\; [A(t-s)]P(s)exp(As)ds}$

$={\int }_0^t\left(\begin{array}{cccc}-\frac{1}{\omega }\mathrm{sin\omega }(t-s)\mathrm{cos\omega s\; coss}& & -\frac{1}{{\omega }^2}\mathrm{sin\omega }(t-s)\mathrm{sin\omega s\; coss}& \\ -\mathrm{cos\omega }(t-s)\mathrm{cos\omega s\; coss}& & -\frac{1}{\omega }\mathrm{cos\omega }(t-s)\mathrm{sin\omega s\; coss}{}^{}& \end{array}\right)ds$

Thus $\mathrm{tr}{\Phi }_1$ (2Π)=$-\frac{1}{\omega }{\int }_0^{2\Pi }\sin 2\mathrm{\Pi \omega coss\; ds}$

Complex Example

Exampl1

$\left(\sum _{i\in n}{a}_{l_i}{x}_i\right)\det K(t=1,x_1,⃛,x_n;l\Vert l)$

$=\left(\underset{i\in n}{\Π}\stackrel{⩓}{x_i}\right)\sum _{I\subseteq n-\left\{l\right\}}(-1{)}^{\Vert I\Vert }{A}^{\left(\lambda \right)}(I\Vert I\left)\det A^{\left(\lambda \right)}\right(\stackrel{\_}{I}\bigcup \left\{l\right\}\Vert \stackrel{\_}{I}\bigcup \{l$ }).

Example2

$\frac{d^{2}P}{d{\Omega }_{s}d{\omega }_s}=r_e^2{\int }_V<S_i>d^{3}r\int \Vert \stackrel{⩓}{e}.\overleftrightarrow{\Pi }.\stackrel{⩓}{e}{\Vert }^2{k}^2\mathrm{f\delta }(k.v-\omega )d^{3}v$

$=r_e^2{\int }_V<S_i>d^{3}r\int \Vert 1-\frac{(1-\stackrel{⩓}{s}.\stackrel{⩓}{i})}{(1-{\beta }_i)(1-{\beta }_s)}{\beta }_e^2{\Vert }^2\Vert \frac{1-{\beta }_i}{1-{\beta }_s}{\Vert }^2{}^{}$

$\times (1-{\beta }^2)\mathrm{f\delta (}k.v-\omega )d^{3}v$