# Stokes theorem questions and answers Recent questions in Green's, Stokes', and the divergence theorem
Green's, Stokes', and the divergence theorem
ANSWERED ### Use Green's Theorem to evaluate the line integral $$\displaystyle\int_{{C}}{\left({y}+{e}^{{x}}\right)}{\left.{d}{x}\right.}+{\left({6}{x}+{\cos{{y}}}\right)}{\left.{d}{y}\right.}$$ where C is triangle with vertices (0,0),(0,2)and(2,2) oriented counterclockwise. a)6 b)10 c)14 d)4 e)8 f)12

Green's, Stokes', and the divergence theorem
ANSWERED ### Use the divergence theorem to evaluate $$\displaystyle\int\int_{{S}}{F}\cdot{N}{d}{S}$$, where $$\displaystyle{F}{\left({x},{y},{z}\right)}={y}^{{2}}{z}{i}+{y}^{{3}}{j}+{x}{z}{k}$$ and S is the boundary of the cube defined by $$\displaystyle-{5}\le{x},={5},-{5}\le{y}\le{5},{\quad\text{and}\quad}{0}\le{z}\le{10}$$.

Green's, Stokes', and the divergence theorem
ANSWERED ### Use Stokes' Theorem to evaluate $$\displaystyle\int\int_{{S}}{C}{U}{R}{L}{f}\cdot{d}{S}$$. $$\displaystyle{F}{\left({x},{y},{z}\right)}={x}^{{2}}{y}^{{3}}{z}{i}+{\sin{{\left({x}{y}{z}\right)}}}{j}+{x}{y}{z}{k}$$, S is the part of the cone $$\displaystyle{y}^{{2}}={x}^{{2}}+{z}^{{2}}$$ that lies between the planes y = 0 and y = 2, oriented in the direction of the positive y-axis.

Green's, Stokes', and the divergence theorem
ANSWERED ### z = x Let be the curve of intersection of the cylinder $$\displaystyle{x}^{{2}}+{y}^{{2}}={1}$$ and the plane , oriented positively when viewed from above . Let S be the inside of this curve , oriented with upward -pointing normal . Use Stokes ' Theorem to evaluate $$\displaystyle\int{S}{c}{u}{r}{l}{F}\cdot{d}{S}{\quad\text{if}\quad}{F}={y}{i}+{z}{j}+{2}{x}{k}$$.

Green's, Stokes', and the divergence theorem
ANSWERED ### Apply Green’s theorem to find the outward flux for the field $$\displaystyle{F}{\left({x},{y}\right)}={{\tan}^{{−{1}}}{\left(\frac{{y}}{{x}}\right)}}{i}+{\ln{{\left({x}^{{2}}+{y}^{{2}}\right)}}}{j}$$

Green's, Stokes', and the divergence theorem
ANSWERED ### 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 $$g \times n = Dng$$ 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.) $$\displaystyle\oint_{{c}}{F}\cdot{n}{d}{s}=\int\int_{{D}}\div{F}{\left({x},{y}\right)}{d}{A}$$

Green's, Stokes', and the divergence theorem
ANSWERED ### Use Green's Theorem to evaluate $$\displaystyle\int_{{C}}\vec{{{F}}}\cdot{d}\vec{{{r}}}$$ where $$\displaystyle\vec{{{F}}}{\left({x},{y}\right)}={x}{y}^{{2}}{i}+{\left({1}-{x}{y}^{{3}}\right)}{j}$$ and C is the parallelogram with vertices (-1,2), (-1,-1),(1,1)and(1,4). The orientation of C is counterclockwise.

Green's, Stokes', and the divergence theorem
ANSWERED ### Use Green's Theorem to evaluate $$\displaystyle\int_{{C}}{\left({e}^{{x}}+{y}^{{2}}\right)}{\left.{d}{x}\right.}+{\left({e}^{{y}}+{x}^{{2}}\right)}{\left.{d}{y}\right.}$$ where C is the boundary of the region(traversed counterclockwise) in the first quadrant bounded by $$\displaystyle{y}={x}^{{2}}{\quad\text{and}\quad}{y}={4}$$.

Green's, Stokes', and the divergence theorem
ANSWERED ### Let $$\displaystyle{F}{\left({x},{y}\right)}={\left\langle{4}{\cos{{\left({y}\right)}}},{2}{\sin{{\left({y}\right)}}}\right\rangle}$$. Compute the flux $$\displaystyle\oint{F}\cdot{n}{d}{s}$$ of F across the boundary of the rectangle $$\displaystyle{0}\le{x}\le{5},{0}\le{y}\le\frac{\pi}{{2}}$$ using the vector form of Green's Theorem. $$\displaystyle\oint{F}\cdot{n}{d}{s}=$$?

Green's, Stokes', and the divergence theorem
ANSWERED ### Use Stokes' Theorem to evaluate $$\displaystyle\int_{{C}}{F}\cdot{d}{r}$$ where C is oriented counterclockwise as viewed from above. $$\displaystyle{F}{\left({x},{y},{z}\right)}={\left({x}+{y}^{{2}}\right)}{i}+{\left({y}+{z}^{{2}}\right)}{j}+{\left({z}+{x}^{{2}}\right)}{k}$$, C is the triangle with vertices (3,0,0),(0,3,0), and (0,0,3).

Green's, Stokes', and the divergence theorem
ANSWERED ### Evaluate $$\displaystyle\int_{{C}}{x}^{{2}}{y}^{{2}}{\left.{d}{x}\right.}+{4}{x}{y}^{{3}}{\left.{d}{y}\right.}$$ where C is the triangle with vertices(0,0),(1,3), and (0,3). (a)Use the Green's Theorem. (b)Do not use the Green's Theorem.

Green's, Stokes', and the divergence theorem
ANSWERED ### Use Green's Theorem to evaluate $$\displaystyle\oint_{{C}}{\left({x}^{{2}}+{y}\right)}{\left.{d}{x}\right.}-{\left({3}{x}+{y}^{{3}}\right)}{\left.{d}{y}\right.}$$ Where c is the ellipse $$\displaystyle{x}^{{2}}+{4}{y}^{{2}}={4}$$

Green's, Stokes', and the divergence theorem
ANSWERED ### If E(t,x,y,z) and B(t,x,y,z)represent the electric and magnetic fields at point (x,y,z) at time t, a basic principle of electromagnetic theory says that $$\displaystyle\nabla\times{E}=\frac{{-\partial{B}}}{{\partial{t}}}$$. In this expression $$\displaystyle\nabla\times{E}$$ is computed with t held fixed and $$\displaystyle\frac{{\partial{B}}}{{\partial{t}}}$$ is calculated with (x,y,z) fixed. Use Stokes' Theorem to derive Faraday's law, $$\displaystyle\oint_{{C}}{E}\cdot{d}{r}=-\frac{\partial}{{\partial{t}}}\int\int_{{S}}{B}\cdot{n}{d}\sigma$$,

Green's, Stokes', and the divergence theorem
ANSWERED ### Let $$\displaystyle{F}={\left[{x}^{{2}},{0},{z}^{{2}}\right]}$$, and S the surface of the box $$\displaystyle{\left|{{x}}\right|}\le{1},{\left|{{y}}\right|}\le{3},{0}\le{z}\le{2}$$. Evaluate the surface integral $$\displaystyle\int\int_{{S}}{F}\cdot{n}{d}{A}$$ by the divergence theorem.

Green's, Stokes', and the divergence theorem
ANSWERED ### Set up the integral for the divergence theorem both ways. Then find the flux. $$\displaystyle{F}{\left({x},{y},{z}\right)}={3}{x}\hat{{{i}}}+{x}{y}\hat{{{j}}}+{2}{x}{z}\hat{{{k}}}$$ E is the cube bounded by the planes $$x = 0, x = 3, y = 0, y = 3,\ and\ z = 0, z = 3$$.

Green's, Stokes', and the divergence theorem
ANSWERED ### Explain carefully why Green's Theorem is a special case of Stokes' Theorem.

Green's, Stokes', and the divergence theorem
ANSWERED ### A particle moves along line segments from the origin to the points(3,0,0),(3,3,1),(0,3,1), and back to the origin under the influence of the force field $$\displaystyle{F}{\left({x},{y},{z}\right)}={z}^{{2}}{i}+{3}{x}{y}{j}+{4}{y}^{{2}}{k}$$. Use Stokes' Theorem to find the work done.

Green's, Stokes', and the divergence theorem
ANSWERED ### What is the Divergence Theorem? Explain how it generalizes Green’s Theorem to three dimensions.

Green's, Stokes', and the divergence theorem
ANSWERED ### Use Stokes' Theorem to compute $$\displaystyle\oint_{{C}}\frac{{1}}{{2}}{z}^{{2}}{\left.{d}{x}\right.}+{\left({x}{y}\right)}{\left.{d}{y}\right.}+{2020}{\left.{d}{z}\right.}$$, where C is the triangle with vertices at(1,0,0),(0,2,0), and (0,0,2) traversed in the order.
ANSWERED 