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Jacobians

 
 

Jacobians

If ~f is the function of one variable, ~x, which is itself the function of another variable, ~t

~f _ = _ ~f ( ~x ) , _ _ ~x _ = _ ~x ( ~t )

Then we can form the composite function: _ _ ~f ( ~t ) #:= ~f ( ~x ( ~t ) ) , _ _ and

int{,,,} _ ~f ( ~x ) d~x _ = _ int{,,,} _ ~f ( ~t ) fract{d~x,d~t} d~t

We will now genralize this to a function of two variables:

~f _ = _ ~f ( ~x , ~y ) , _ _ ~x _ = _ ~x ( ~u , ~v ) , _ _ ~y _ = _ ~y ( ~u , ~v )

dblint{,,,~A,,} _ ~f ( ~x , ~y ) d~x d~y _ = _ dblint{,,,~A,,} _ ~f ( ~u , ~v ) | J | d~u d~v

where

J _ = _ fract{{&partial. ( ~x , ~y )},{&partial. ( ~u , ~v )}} _ #:= _ _ det{ fract{&partial.~x,&partial.~u} , fract{&partial.~x,&partial.~v} / _ / fract{&partial.~y,&partial.~u} , fract{&partial.~y,&partial.~v} } _ _ = _ fract{&partial.~x,&partial.~u}fract{&partial.~y,&partial.~v} _ - _ fract{&partial.~x,&partial.~v}fract{&partial.~y,&partial.~u}

Is called the #~{Jacobian} of the transformation