# How do Green’s Theorem, Stokes’ Theorem, and the Divergence Theorem relate to the Fundamental Theorem of Calculus for ordi-nary single integral?

Question
How do Green’s Theorem, Stokes’ Theorem, and the Divergence Theorem relate to the Fundamental Theorem of Calculus for ordi-nary single integral?

2021-01-14
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.

### Relevant Questions

Stokes' Theorem is a generalization of the
(a)fundamental theorem of line integrals.
(b)flux form of Green's Theorem.
(c)circulation form of Green's Theorem.
Consider the vector field $$\displaystyle{F}={\left\langle{5}{z},{x},{5}{y}\right\rangle}$$ and the surface which is the part of the elliptic paraboloid $$\displaystyle{z}={x}^{{2}}+{5}{y}^{{2}}$$ that lies below the plane z = 5. Calculate curl(F) and then apply Stokes' Theorem to compute the exact magnitude of the flux of curl(F) through the surface using line integral. You do not need to cinfirm your answer by evaluating the double integral of curl(F) over the surface(the right-hand side of Stokes' Theorem).
Evaluate the line integral $$\displaystyle\oint_{{C}}{x}{y}{\left.{d}{x}\right.}+{x}^{{2}}{\left.{d}{y}\right.}$$, where C is the path going counterclockwise around the boundary of the rectangle with corners (0,0),(2,0),(2,3), and (0,3). You can evaluate directly or use Green's theorem.
Write the integral(s), but do not evaluate.
Use Green's Theorem to evaluate the line integral. Orient the curve counerclockwise.
$$\displaystyle\oint_{{C}}{F}{8}{d}{r}$$, where $$\displaystyle{F}{\left({x},{y}\right)}={\left\langle{x}^{{2}},{x}^{{2}}\right\rangle}$$ and C consists of the arcs $$\displaystyle{y}={x}^{{2}}{\quad\text{and}\quad}{y}={8}{x}{f}{\quad\text{or}\quad}{0}\le{x}\le{8}$$
What is the Divergence Theorem? Explain how it generalizes Green’s Theorem to three dimensions.
Green’s Theorem, flux form Consider the following regions R and vector fields F.
a. Compute the two-dimensional divergence of the vector field.
b. Evaluate both integrals in Green’s Theorem and check for consistency.
$$\displaystyle{F}={\left\langle{x},{y}\right\rangle},{R}={\left\lbrace{\left({x},{y}\right)}:{x}^{{2}}+{y}^{{2}}\le{4}\right\rbrace}$$
Use Stokes' theorem to evaluate the line integral $$\displaystyle\oint_{{C}}{F}\cdot{d}{r}$$ where A = -yi + xj and C is the boundary of the ellipse $$\displaystyle\frac{{x}^{{2}}}{{a}^{{2}}}+\frac{{y}^{{2}}}{{b}^{{2}}}={1},{z}={0}$$.
$$\displaystyle\oint_{{c}}{F}\cdot{n}{d}{s}=\int\int_{{D}}\div{F}{\left({x},{y}\right)}{d}{A}$$