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Is it possible to replace $\textbf{F}$ in the divergence theorem by a real valued function ? Another question , is there any result which is similar to fundamental theorem of calculus II ( reduction of order of derivative) for $f:\mathbb{R^3}\to \mathbb{R}$ ?

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You need to more specific about what you're trying to accomplish. For the first question, when you say "replace F in the divergence theorem by a real-valued function", what kind of result are you looking for? Divergence only applies to vector fields. For the second question, you should state the "reduction of order of derivative" result as that isn't a standard name. Please edit these changes in your question rather than respond in comments. – Ted Dec 28 '12 at 20:14
up vote 2 down vote accepted

The fundamental theorem of calculus does not apply to scalar functions, even to functions from $\mathbb R$ to $\mathbb R$. It applies to integrals of differential forms, such as $f(x)\,dx$. Since in one dimension all differential forms are written as "a function times $dx$", it is possible to forget (or not to know) the language of differential forms and talk about integrating a scalar function. But this does not change the essence of the matter: we need a differential form to transform an integral in FTC-style, the operation succintly formalized in general Stokes' theorem.

So, the answer is no: there is nothing like the divergence theorem for scalar functions. But there are formulas (special cases of the aforementioned Stokes' theorem) which "reduce the order of derivative", e.g., $$\int_\gamma F_x\,dx+F_y\,dy+F_z\,dz = F(b)-F(a)$$ where $\gamma$ is any smooth curve going from point $a$ to point $b$.

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