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

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Green's, Stokes', and the divergence theorem asked 2021-01-13
How do Green’s Theorem, Stokes’ Theorem, and the Divergence Theorem relate to the Fundamental Theorem of Calculus for ordi-nary single integral?

## Answers (1) 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 asked 2021-01-13
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. asked 2021-01-02
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Write the integral(s), but do not evaluate. asked 2020-10-20
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$$\displaystyle\oint_{{C}}{F}{8}{d}{r}$$, where $$\displaystyle{F}{\left({x},{y}\right)}={\left\langle{x}^{{2}},{x}^{{2}}\right\rangle}$$ and C consists of the arcs $$\displaystyle{y}={x}^{{2}}{\quad\text{and}\quad}{y}={8}{x}{f}{\quad\text{or}\quad}{0}\le{x}\le{8}$$ asked 2021-02-09
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b. Evaluate both integrals in Green’s Theorem and check for consistency.
$$\displaystyle{F}={\left\langle{x},{y}\right\rangle},{R}={\left\lbrace{\left({x},{y}\right)}:{x}^{{2}}+{y}^{{2}}\le{4}\right\rbrace}$$ asked 2021-01-04
Suppose S is a region in the xy-plane with a boundary oriented counterclockwise. What is the normal to S? Explain why Stokes’ Theorem becomes the circulation form of Green’s Theorem. asked 2020-10-28
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 $$\displaystyle\frac{{x}^{{2}}}{{a}^{{2}}}+\frac{{y}^{{2}}}{{b}^{{2}}}={1},{z}={0}$$. asked 2020-10-27
Explain why or why not Determine whether the following statements are true and give an explanation or counterexample.
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b. If a vector field has zero divergence throughout a region (on which the conditions of Green’s Theorem are met), then the circulation on the boundary of that region is zero.
c. If the two-dimensional curl of a vector field is positive throughout a region (on which the conditions of Green’s Theorem are met), then the circulation on the boundary of that region is positive (assuming counterclockwise orientation). asked 2021-03-04
Use Green's Theorem in the form of this equation to prove Green's first identity, where D and C satisfy the hypothesis of Green's Theorem and the appropriate partial derivatives of f and g exist and are continuous. (The quantity grad g · n = Dng occurs in the line integral. This is the directional derivative in the direction of the normal vector n and is called the normal derivative of g.)
$$\displaystyle\oint_{{c}}{F}\cdot{n}{d}{s}=\int\int_{{D}}\div{F}{\left({x},{y}\right)}{d}{A}$$
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