MHB Exact Differential: Show f(z)dz is Exact

[ssh]

Q. Show that f(z)dz defined in a region is exact if and only if f(z) has a primitive.

 

[chisigma]

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 $\chi$ $\sigma$