Differences

This shows you the differences between two versions of the page.

Link to this comparison view

Both sides previous revision Previous revision
Next revision
Previous revision
Last revision Both sides next revision
determinant_differentiation [2016/07/18 23:15]
nikolaj
determinant_differentiation [2016/07/21 00:59]
nikolaj
Line 19: Line 19:
 This comes from [[http://​en.wikipedia.org/​wiki/​Jacobi%27s_formula|Jacobi'​s formula]]: This comes from [[http://​en.wikipedia.org/​wiki/​Jacobi%27s_formula|Jacobi'​s formula]]:
  
-${\mathrm d} \log\left(\det (F(t))\right) = \mathrm{det}(F(t))^{-1}{\mathrm d} \det (F(t)) = \mathrm{tr} (F(t)^{-1} {\mathrm d}F(t))$+${\mathrm d} \det (F(t)) = \det (F(t)) \mathrm{tr} (F(t)^{-1} {\mathrm d}F(t))$
  
 where $F(t)$ is a parameter dependent matrix where $F(t)$ is a parameter dependent matrix
Line 25: Line 25:
 This is a special case of the product rule and generalizes ​ This is a special case of the product rule and generalizes ​
  
-${\mathrm d}\left(u\cdot v\right) = u\,{\mathrm d}v+v\,{\mathrm d}u\cdot v\left(\dfrac{1}{u}{\mathrm d}u+\dfrac{1}{v}{\mathrm d}v\right)$.+${\mathrm d}\left(a\cdot b\right) = a\,{\mathrm d}b+b\,{\mathrm d}a\cdot b\left(\dfrac{1}{a}{\mathrm d}a+\dfrac{1}{b}{\mathrm d}b\right)$.
  
 which you get for which you get for
  
-$F(t) := \mathrm{diag}(u(t),v(t))$+$F(t) := \mathrm{diag}(a(t),b(t))$
  
-The expression $\dfrac{1}{u}{\mathrm d}u$ is the so called logarithmic derivative of $u$.+which can be seen to represent the growing area of a rectangle. 
 + 
 +The expression $\dfrac{1}{a}{\mathrm d}a$ is the so called logarithmic derivative of $aand scale invariant.
  
 == Perspective == == Perspective ==
Link to graph
Log In
Improvements of the human condition