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

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The discussion centers on the condition for the exactness of the differential form f(z)dz in a given region, asserting that it is exact if and only if f(z) possesses a primitive. The concept of exact differentials typically applies to multivariable functions, specifically in the context of two variables, x and y. An expression is deemed an exact differential if there exists a differentiable function F(x,y) such that its differential equals the given expression. The criteria for exactness include the continuity of the functions involved and the equality of mixed partial derivatives. Understanding these principles is crucial for analyzing the behavior of multivariable functions in calculus.
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Q. Show that f(z)dz defined in a region is exact if and only if f(z) has a primitive.
 
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Re: exact differential

ssh said:
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 $\chi$ $\sigma$
 

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