{"id":5,"date":"2017-08-14T14:20:05","date_gmt":"2017-08-14T14:20:05","guid":{"rendered":"https:\/\/zuugs.ch\/showcase\/formeln\/chapter\/chapter-1\/"},"modified":"2019-05-31T14:03:09","modified_gmt":"2019-05-31T14:03:09","slug":"formulas","status":"web-only","type":"chapter","link":"https:\/\/zuugs.ch\/showcase\/formeln\/chapter\/formulas\/","title":{"raw":"Formulas","rendered":"Formulas"},"content":{"raw":"Here a few formulas:\r\n\r\n[latex]e=mc^2[\/latex]\r\n\r\nSeien [latex](V,\\langle \\cdot ,\\cdot \\rangle )[\/latex] ein endlichdimensionaler Euklidischer Vektorraum, sei [latex]T\\in \\mathrm {End}(V)[\/latex] orthogonal. Dann existiert eine ONB [latex]\\mathcal {B}[\/latex] von [latex]V[\/latex], sodass\r\n
[latex]\r\n\\begin{aligned}[][T]_{\\mathcal {B}}=\\left (\\begin{matrix}D_{1} & & 0 \\\\ & \\ddots & \\\\ 0 & & D_{m}\\end{matrix}\\right ),\\end{aligned}\r\n[\/latex]<\/div>\r\nwobei [latex]D_{k}=\\pm 1\\in M_{1\\times 1}(\\mathbb {R})[\/latex] oder von der Form [latex]D_{k}=(\\begin{smallmatrix}a_{k} & b_{k}\\\\ -b_{k} & a_{k}\\end{smallmatrix})[\/latex] mit [latex]a_{k},b_{k}\\in \\mathbb {R}[\/latex] und [latex]a_{k}^{2}+b_{k}^{2}=1[\/latex] ist. Falls [latex]\\det T=1[\/latex], dann kann man alle [latex]1\\times 1[\/latex] Blockdiagonaleintr\u00e4ge gleich [latex]1[\/latex] w\u00e4hlen.\r\n\r\n \r\n
\r\n

How to<\/h3>\r\nThe formulas are added with the latex-shortcode in Zuugs,\r\n\r\n\"\"\r\nand there are more advanced functionalities for the initiated.\r\n\r\n<\/div>\r\n \r\n\r\n[latexpage]\r\nAt first, we sample $f(x)$ in the $N$ ($N$ is odd) equidistant points around $x^*$:\r\n\\[\r\nf_k = f(x_k),\\: x_k = x^*+kh,\\: k=-\\frac{N-1}{2},\\dots,\\frac{N-1}{2}\r\n\\]\r\nwhere $h$ is some step.\r\nThen we interpolate points $\\{(x_k,f_k)\\}$ by polynomial\r\n\\begin{equation}\u00a0\\label{eq:poly}\r\nP_{N-1}(x)=\\sum_{j=0}^{N-1}{a_jx^j}\r\n\\end{equation}\r\nIts coefficients $\\{a_j\\}$ are found as a solution of system of linear equations:\r\n\\begin{equation}\u00a0\\label{eq:sys}\r\n\\left\\{ P_{N-1}(x_k) = f_k\\right\\},\\quad k=-\\frac{N-1}{2},\\dots,\\frac{N-1}{2}\r\n\\end{equation}\r\nReferences to existing equations are below.\r\n\r\n \r\n\r\n \r\n\r\n \r\n\r\n \r\n\r\n \r\n\r\n \r\n\r\n \r\n\r\n \r\n\r\n \r\n\r\n \r\n\r\n \r\n\r\n \r\n\r\n \r\n\r\nHere are references to existing equations: (\\ref{eq:poly}), (\\ref{eq:sys}).\r\n\r\n \r\n\r\n \r\n\r\n \r\n\r\n \r\n\r\n \r\n\r\n \r\n\r\n \r\n\r\n \r\n\r\n \r\n\r\n \r\n\r\n \r\n\r\n \r\n\r\n \r\n\r\n \r\n\r\n \r\n\r\n \r\n\r\n \r\n\r\n \r\n\r\n \r\n\r\n \r\n\r\n \r\n\r\n \r\n\r\n \r\n\r\n \r\n\r\n \r\n\r\n \r\n\r\n \r\n\r\n \r\n\r\n \r\n\r\n \r\n\r\n \r\n\r\n \r\n\r\n \r\n\r\n \r\n\r\n \r\n\r\n \r\n\r\n ","rendered":"

Here a few formulas:<\/p>\n

\"e=mc^2\"<\/p>\n

Seien \"(V,\langle \cdot ,\cdot \rangle )\" ein endlichdimensionaler Euklidischer Vektorraum, sei \"T\in \mathrm {End}(V)\" orthogonal. Dann existiert eine ONB \"\mathcal {B}\" von \"V\", sodass<\/p>\n

\"\begin{aligned}[][T]_{\mathcal {B}}=\left (\begin{matrix}D_{1} & & 0 \\ & \ddots & \\ 0 & & D_{m}\end{matrix}\right ),\end{aligned}\"<\/div>\n

wobei \"D_{k}=\pm 1\in M_{1\times 1}(\mathbb {R})\" oder von der Form \"D_{k}=(\begin{smallmatrix}a_{k} & b_{k}\\ -b_{k} & a_{k}\end{smallmatrix})\" mit \"a_{k},b_{k}\in \mathbb {R}\" und \"a_{k}^{2}+b_{k}^{2}=1\" ist. Falls \"\det T=1\", dann kann man alle \"1\times 1\" Blockdiagonaleintr\u00e4ge gleich \"1\" w\u00e4hlen.<\/p>\n

 <\/p>\n

\n

How to<\/h3>\n

The formulas are added with the latex-shortcode in Zuugs,<\/p>\n

\"\"
\nand there are more advanced functionalities for the initiated.<\/p>\n<\/div>\n

 <\/p>\n

At first, we sample \"f(x)\" in the \"N\" (\"N\" is odd) equidistant points around \"x^*\":<\/p>\n

  <\/span>   <\/span>\"\[<\/p>\n

where \"h\" is some step.
\nThen we interpolate points \"\{(x_k,f_k)\}\" by polynomial
\n<\/a><\/p>\n

(1) <\/span>   <\/span>\"\begin{equation*} <\/p>\n

Its coefficients \"\{a_j\}\" are found as a solution of system of linear equations:
\n<\/a><\/p>\n

(2) <\/span>   <\/span>\"\begin{equation*} <\/p>\n

References to existing equations are below.<\/p>\n

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Here are references to existing equations: (1<\/a>), (2<\/a>).<\/p>\n

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