Cramer's Rule

$Ax\; =\; c$

$x\_i\; =\; \{\; \backslash det(A\_i)\; \backslash over\; \backslash det(A)\}$

Example

$ax+by\; =\; e$ and

$cx+dy\; =\; f$,

$x\; =\; \{\; ed\; -\; bf\; \backslash over\; ad\; -\; bc\}$

$y\; =\; \{\; af\; -\; ec\; \backslash over\; ad\; -\; bc\}$

Applications to Differential Geometry

$dF\; =\; \backslash frac\{\backslash partial\; F\}\{\backslash partial\; x\}\; dx\; +\; \backslash frac\{\backslash partial\; F\}\{\backslash partial\; y\}\; dy\; +\backslash frac\{\backslash partial\; F\}\{\backslash partial\; u\}\; du\; +\backslash frac\{\backslash partial\; F\}\{\backslash partial\; v\}\; dv\; =\; 0$

$dG\; =\; \backslash frac\{\backslash partial\; G\}\{\backslash partial\; x\}\; dx\; +\; \backslash frac\{\backslash partial\; G\}\{\backslash partial\; y\}\; dy\; +\backslash frac\{\backslash partial\; G\}\{\backslash partial\; u\}\; du\; +\backslash frac\{\backslash partial\; G\}\{\backslash partial\; v\}\; dv\; =\; 0$

$dx\; =\; \backslash frac\{\backslash partial\; X\}\{\backslash partial\; u\}\; du\; +\; \backslash frac\{\backslash partial\; X\}\{\backslash partial\; v\}\; dv$

$dy\; =\; \backslash frac\{\backslash partial\; Y\}\{\backslash partial\; u\}\; du\; +\; \backslash frac\{\backslash partial\; Y\}\{\backslash partial\; v\}\; dv$

$dF\; =\; \backslash left(\backslash frac\{\backslash partial\; F\}\{\backslash partial\; x\}\; \backslash frac\{\backslash partial\; x\}\{\backslash partial\; u\}\; +\backslash frac\{\backslash partial\; F\}\{\backslash partial\; y\}\; \backslash frac\{\backslash partial\; y\}\{\backslash partial\; u\}\; +\backslash frac\{\backslash partial\; F\}\{\backslash partial\; u\}\; \backslash right)\; du\; +\; \backslash left(\backslash frac\{\backslash partial\; F\}\{\backslash partial\; x\}\; \backslash frac\{\backslash partial\; x\}\{\backslash partial\; v\}\; +\backslash frac\{\backslash partial\; F\}\{\backslash partial\; y\}\; \backslash frac\{\backslash partial\; y\}\{\backslash partial\; v\}\; +\backslash frac\{\backslash partial\; F\}\{\backslash partial\; v\}\; \backslash right)\; dv\; =\; 0$

$dG\; =\; \backslash left(\backslash frac\{\backslash partial\; G\}\{\backslash partial\; x\}\; \backslash frac\{\backslash partial\; x\}\{\backslash partial\; u\}\; +\backslash frac\{\backslash partial\; G\}\{\backslash partial\; y\}\; \backslash frac\{\backslash partial\; y\}\{\backslash partial\; u\}\; +\backslash frac\{\backslash partial\; G\}\{\backslash partial\; u\}\; \backslash right)\; du\; +\; \backslash left(\backslash frac\{\backslash partial\; G\}\{\backslash partial\; x\}\; \backslash frac\{\backslash partial\; x\}\{\backslash partial\; v\}\; +\backslash frac\{\backslash partial\; G\}\{\backslash partial\; y\}\; \backslash frac\{\backslash partial\; y\}\{\backslash partial\; v\}\; +\backslash frac\{\backslash partial\; G\}\{\backslash partial\; v\}\; \backslash right)\; dv\; =\; 0$

$\backslash frac\{\backslash partial\; F\}\{\backslash partial\; x\}\; \backslash frac\{\backslash partial\; x\}\{\backslash partial\; u\}\; +\backslash frac\{\backslash partial\; F\}\{\backslash partial\; y\}\; \backslash frac\{\backslash partial\; y\}\{\backslash partial\; u\}\; =\; -\backslash frac\{\backslash partial\; F\}\{\backslash partial\; u\}$

$\backslash frac\{\backslash partial\; G\}\{\backslash partial\; x\}\; \backslash frac\{\backslash partial\; x\}\{\backslash partial\; u\}\; +\backslash frac\{\backslash partial\; G\}\{\backslash partial\; y\}\; \backslash frac\{\backslash partial\; y\}\{\backslash partial\; u\}\; =\; -\backslash frac\{\backslash partial\; G\}\{\backslash partial\; u\}$

$\backslash frac\{\backslash partial\; F\}\{\backslash partial\; x\}\; \backslash frac\{\backslash partial\; x\}\{\backslash partial\; v\}\; +\backslash frac\{\backslash partial\; F\}\{\backslash partial\; y\}\; \backslash frac\{\backslash partial\; y\}\{\backslash partial\; v\}\; =\; -\backslash frac\{\backslash partial\; F\}\{\backslash partial\; v\}$

$\backslash frac\{\backslash partial\; G\}\{\backslash partial\; x\}\; \backslash frac\{\backslash partial\; x\}\{\backslash partial\; v\}\; +\backslash frac\{\backslash partial\; G\}\{\backslash partial\; y\}\; \backslash frac\{\backslash partial\; y\}\{\backslash partial\; v\}\; =\; -\backslash frac\{\backslash partial\; G\}\{\backslash partial\; v\}$

$\backslash frac\{\backslash partial\; x\}\{\backslash partial\; u\}\; =\; -\; \backslash frac\{\backslash left(\backslash frac\{\backslash partial\backslash left(F,\; G\backslash right)\}\{\backslash partial\backslash left(y,\; u\backslash right)\}\backslash right)\}\{\backslash left(\backslash frac\{\backslash partial\backslash left(F,\; G\backslash right)\}\{\backslash partial\backslash left(x,\; y\backslash right)\}\backslash right)\}$

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