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Exact Differential

ssh

New member
Jun 30, 2012
17
Q. Show that f(z)dz defined in a region is exact if and only if f(z) has a primitive.
 

chisigma

Well-known member
Feb 13, 2012
1,704
Re: exact differential

Q. Show that f(z)dz defined in a region is exact if and only if f(z) has a primitive.
Usually thye concept of 'exact differential' refers to a multivariable function. In case of two variables x and y, an expression like...


$\displaystyle A(x.y)\ dx + B(x,y)\ dy\ (1)$


... where A(*,*) and B(*,*) are defined in a field D, is called exact differential if it exist an F(x,y) differentiable in D for which is...


$\displaystyle dF = A(x,y)\ dx + B(x,y)\ dy\ (2)$

The expression (1) is an exact differential if and only if $A(x,y)$, $B(x,y)$, $\displaystyle \frac{\partial A}{\partial y}$ and $\displaystyle \frac{\partial B}{\partial x}$ are continuos and is...


$\displaystyle \frac{\partial A}{\partial y}= \frac{\partial B}{\partial x}\ (3)$


Kind regards


$\chi$ $\sigma$