The Fundamental theorem of Calculus says that if f(x) is differentiable on (a,b) and continuous on \(\displaystyle{\left[{a},{b}\right]}\), then the integral of the differential function \(\displaystyle\frac{{{d}{f}}}{{{\left.{d}{x}\right.}}}\) is

\(\displaystyle{\int_{{a}}^{{b}}}\frac{{{d}{f}}}{{{\left.{d}{x}\right.}}}{\left.{d}{x}\right.}={f{{\left({b}\right)}}}-{f{{\left({a}\right)}}}\).

If F=f(x)i throughout \(\displaystyle{\left[{a},{b}\right]}\), then \(\displaystyle\frac{{{d}{f}}}{{{\left.{d}{x}\right.}}}=\nabla\cdot{F}\). If the unit vector field n normal to the boundary of \(\displaystyle{\left[{a},{b}\right]}\) to be i at b and -i at a, then.

\(\displaystyle{f{{\left({b}\right)}}}-{f{{\left({a}\right)}}}={f{{\left({b}\right)}}}{i}\cdot{\left({i}\right)}-{f{{\left({a}\right)}}}{i}\cdot{\left(-{i}\right)}\)

\(\displaystyle{F}{\left({b}\right)}\cdot{n}+{F}{\left({a}\right)}\cdot{n}\)

=total outward flux of F across the boundary of \(\displaystyle{\left[{a},{b}\right]}\)

The Fundamental theorem expressed as follows:

\(\displaystyle{F}{\left({b}\right)}\cdot{n}+{F}{\left({a}\right)}\cdot{n}=\int_{{\begin{array}{cc} {a}&{b}\end{array}}}\nabla\cdot{F}{\left.{d}{x}\right.}\)

The sum of the normal field components over the boundary enclosing the region is equals to the integral of the differential operator grad. operating on a field F over a region. It is obtained from Fundamental Theorem of Calculus, the normal form of Green’s theorem, and The Divergence theorem.

The sum of the field components over the boundary of the surface is equal to the surface integral of the differential operator grad xx operating on a field. It is obtained from Stokes’ theorem and the tangential form of Green’s theorem

A unifying Fundamental theorem of Vector integral Calculus:

The integral of a differential operator acting on a field over a region equals the sum of the field components suitable to the operator over the boundary of the region.

\(\displaystyle{\int_{{a}}^{{b}}}\frac{{{d}{f}}}{{{\left.{d}{x}\right.}}}{\left.{d}{x}\right.}={f{{\left({b}\right)}}}-{f{{\left({a}\right)}}}\).

If F=f(x)i throughout \(\displaystyle{\left[{a},{b}\right]}\), then \(\displaystyle\frac{{{d}{f}}}{{{\left.{d}{x}\right.}}}=\nabla\cdot{F}\). If the unit vector field n normal to the boundary of \(\displaystyle{\left[{a},{b}\right]}\) to be i at b and -i at a, then.

\(\displaystyle{f{{\left({b}\right)}}}-{f{{\left({a}\right)}}}={f{{\left({b}\right)}}}{i}\cdot{\left({i}\right)}-{f{{\left({a}\right)}}}{i}\cdot{\left(-{i}\right)}\)

\(\displaystyle{F}{\left({b}\right)}\cdot{n}+{F}{\left({a}\right)}\cdot{n}\)

=total outward flux of F across the boundary of \(\displaystyle{\left[{a},{b}\right]}\)

The Fundamental theorem expressed as follows:

\(\displaystyle{F}{\left({b}\right)}\cdot{n}+{F}{\left({a}\right)}\cdot{n}=\int_{{\begin{array}{cc} {a}&{b}\end{array}}}\nabla\cdot{F}{\left.{d}{x}\right.}\)

The sum of the normal field components over the boundary enclosing the region is equals to the integral of the differential operator grad. operating on a field F over a region. It is obtained from Fundamental Theorem of Calculus, the normal form of Green’s theorem, and The Divergence theorem.

The sum of the field components over the boundary of the surface is equal to the surface integral of the differential operator grad xx operating on a field. It is obtained from Stokes’ theorem and the tangential form of Green’s theorem

A unifying Fundamental theorem of Vector integral Calculus:

The integral of a differential operator acting on a field over a region equals the sum of the field components suitable to the operator over the boundary of the region.