Question

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.

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.

Answers (1)

2021-01-05
Step 1
Consider the surface in xy-plane as z = g(x,y).
Step 2
Obtain the value of \(\displaystyle{z}_{{x}}{\quad\text{and}\quad}{z}_{{y}}{a}{s}{z}_{{x}}={g}_{{x}}{\quad\text{and}\quad}{z}_{{y}}={g}_{{y}}\) respectively
Thus, the parametric form becomes \(\displaystyle{\left\langle{z}_{{x}},{z}_{{y}},{1}\right\rangle}={\left\langle{g}_{{x}},{g}_{{y}},{1}\right\rangle}\)
If the vector product of two vectors is obtained, the direction of the vector product is perpendicular to both the vectors.
The components \(\displaystyle{z}_{{x}},{\quad\text{and}\quad}{z}_{{y}}\) lie on the xy- plane, hence the vector product \(\displaystyle{z}_{{x}}\times{z}_{{y}}\), lie perpendicular to the xy- plane.
Therefore, the normal to the surface is the same as that of xy- plane.
Here, surface is just a plane.
Hence, there is only one normal vector at every point of the plane.
Thus, the normal vector in Stokes’ theorem becomes the unit vector that results in circulation form of Green’s theorem.
0
 
Best answer

expert advice

Need a better answer?

Relevant Questions

asked 2020-10-27
Explain why or why not Determine whether the following statements are true and give an explanation or counterexample.
a. The work required to move an object around a closed curve C in the presence of a vector force field is the circulation of the force field on the curve.
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 2020-11-02
Suppose that the plane region D, its boundary curve C, and the functions P and Q satisfy the hypothesis of Green's Theorem. Considering the vector field F = Pi+Qj, prove the vector form of Green's Theorem \(\displaystyle\oint_{{C}}{F}\cdot{n}{d}{s}=\int\int_{{D}}\div{F}{\left({x},{y}\right)}{d}{A}\)
where n(t) is the outward unit normal vector to C.
asked 2021-01-06

Use Stokes' Theorem to evaluate \(\int_C F \cdot dr\) where C is oriented counterclockwise as viewed above.
\(\displaystyle{F}{\left({x},{y},{z}\right)}={x}{y}{i}+{3}{z}{j}+{5}{y}{k}\), C is the curve of intersection of the plane \(x+z=10\) and the cylinder \(\displaystyle{x}^{{2}}+{y}^{{2}}={9}\).

asked 2020-12-02
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.
asked 2021-03-08
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 \(\displaystyle\int{S}{c}{u}{r}{l}{F}\cdot{d}{S}{\quad\text{if}\quad}{F}={y}{i}+{z}{j}+{2}{x}{k}\).
asked 2021-02-25
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 \(\displaystyle{y}={x}^{{2}}{\quad\text{and}\quad}{y}={4}\).
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 2020-12-16
Let \(\displaystyle{f}={\left[{x}^{{2}}{y}^{{2}},-\frac{{x}}{{y}^{{2}}}\right]}\) and \(\displaystyle{R}:{1}\le{x}^{{2}}+{y}^{{2}},+{4},{x}\ge{0},{y}\ge{x}\). Evaluate \(\displaystyle\int_{{C}}{F}{\left({r}\right)}\cdot{d}{r}\) counterclockwise around the boundary C of the region R by Green's theorem.
asked 2021-02-25
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).
asked 2021-03-12
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
...