Circulation of two-dimensional flows Let C be the unit circle with counterclockwise orientation. Find the circulation on C of the following vector fields. a. The radial vector field F = <> b. The rotation vector field F = << -y, x>>

Circulation of two-dimensional flows Let C be the unit circle with counterclockwise orientation. Find the circulation on C of the following vector fields. a. The radial vector field F = <<x, y>> b. The rotation vector field F = << -y, x>>

Question
Analytic geometry
asked 2021-02-05
Circulation of two-dimensional flows Let C be the unit circle with counterclockwise orientation. Find the circulation on C of the following vector fields.
a. The radial vector field \(\displaystyle{F}={\left\langle{x},{y}\right\rangle}\)
b. The rotation vector field \(\displaystyle{F}={\left\langle-{y},{x}\right\rangle}\)

Answers (1)

2021-02-06
Similar to circle sphere is a two dimensional space where the set of points that are at the same distance r from a given point in a three dimensional space. In analytical geometry with a center and radius is the locus of all points is called sphere.
a) \(\displaystyle{F}{\left({x},{y}\right)}={\left\langle{x},{y}\right\rangle}\)
Here the curve C is the unit circle \(\displaystyle{x}^{{1}}+{y}^{{2}}={1}\)
The parametric equations are
\(\displaystyle{x}={\cos{{t}}},{y}={\sin{{t}}}\)
\(\displaystyle{\left.{d}{x}\right.}=−{\sin{{t}}}{\left.{d}{t}\right.},{\left.{d}{y}\right.}={\cos{{t}}}{\left.{d}{t}\right.}\)
the flux of \(\displaystyle\vec{{F}}\) across the curve C is
\(\displaystyle\int_{{C}}\vec{{F}}\hat{{n}}{d}{s}=\int_{{C}}{p}{\left.{d}{y}\right.}-{Q}{\left.{d}{x}\right.}\)
\(\displaystyle\int_{{C}}\vec{{F}}\hat{{n}}{d}{s}=\int_{{C}}{x}{\left.{d}{y}\right.}-{y}{\left.{d}{x}\right.}\)
\(\displaystyle\int_{{C}}\vec{{F}}\hat{{n}}{d}{s}={\int_{{0}}^{{1}}}{\left({\cos{{t}}}\right)}{\cos{{t}}}{\left.{d}{t}\right.}-{\sin{{\left(-{\sin{{t}}}\right)}}}{\left.{d}{t}\right.}\)
\(\displaystyle\int_{{C}}\vec{{F}}\hat{{n}}{d}{s}={\int_{{0}}^{{1}}}{1}{\left.{d}{t}\right.}={1}\)
b) \(\displaystyle\vec{{F}}={\left\langle-{y},{x}\right\rangle}\)
Here
p=-y, Q=x
\(\displaystyle\int_{{C}}\vec{{F}}{n}{d}{s}=\int_{{C}}-{y}{\left.{d}{y}\right.}-{x}{\left.{d}{x}\right.}\)
\(\displaystyle\int_{{C}}\vec{{F}}{n}{d}{s}=\in{i}{{t}_{{0}}^{{1}}}-{\sin{{t}}}{\cos{{t}}}{\left.{d}{t}\right.}-{\cos{{t}}}{\left(-{\sin{{t}}}{t}\right)}{\left.{d}{t}\right.}\)
\(\displaystyle\int_{{C}}\vec{{F}}{n}{d}{s}={\int_{{0}}^{{10}}}{\left.{d}{t}\right.}={0}\)
0

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 2021-05-05
The bulk density of soil is defined as the mass of dry solidsper unit bulk volume. A high bulk density implies a compact soilwith few pores. Bulk density is an important factor in influencing root development, seedling emergence, and aeration. Let X denotethe bulk density of Pima clay loam. Studies show that X is normally distributed with \(\displaystyle\mu={1.5}\) and \(\displaystyle\sigma={0.2}\frac{{g}}{{c}}{m}^{{3}}\).
(a) What is thedensity for X? Sketch a graph of the density function. Indicate onthis graph the probability that X lies between 1.1 and 1.9. Findthis probability.
(b) Find the probability that arandomly selected sample of Pima clay loam will have bulk densityless than \(\displaystyle{0.9}\frac{{g}}{{c}}{m}^{{3}}\).
(c) Would you be surprised if a randomly selected sample of this type of soil has a bulkdensity in excess of \(\displaystyle{2.0}\frac{{g}}{{c}}{m}^{{3}}\)? Explain, based on theprobability of this occurring.
(d) What point has the property that only 10% of the soil samples have bulk density this high orhigher?
(e) What is the moment generating function for X?
asked 2021-02-08

Consider the vector field \(\displaystyle{F}{\left({x},{y}\right)}=⟨{y},{5}⟩,\) and let C be the curve of the ellipse having equation \(\displaystyle{\frac{{{x}^{{{2}}}}}{{{4}}}}+{\frac{{{y}^{{{2}}}}}{{{9}}}}={1}.\) Compute the line integral \( \int CF\) dr, where r(t) is a parametrization of C going once around counterclockwise.

asked 2021-05-28
Find the area of the parallelogram with vertices A(-3, 0), B(-1 , 3), C(5, 2), and D(3, -1).
asked 2021-04-30
Two oppositely charged but otherwise identical conducting plates of area 2.50 square centimeters are separated by a dielectric 1.80 millimeters thick, with a dielectric constant of K=3.60. The resultant electric field in the dielectric is \(\displaystyle{1.20}\times{10}^{{6}}\) volts per meter.
Compute the magnitude of the charge per unit area \(\displaystyle\sigma\) on the conducting plate.
\(\displaystyle\sigma={\frac{{{c}}}{{{m}^{{2}}}}}\)
Compute the magnitude of the charge per unit area \(\displaystyle\sigma_{{1}}\) on the surfaces of the dielectric.
\(\displaystyle\sigma_{{1}}={\frac{{{c}}}{{{m}^{{2}}}}}\)
Find the total electric-field energy U stored in the capacitor.
u=J
asked 2021-05-20
Assume that a ball of charged particles has a uniformly distributednegative charge density except for a narrow radial tunnel throughits center, from the surface on one side to the surface on the opposite side. Also assume that we can position a proton any where along the tunnel or outside the ball. Let \(\displaystyle{F}_{{R}}\) be the magnitude of the electrostatic force on the proton when it islocated at the ball's surface, at radius R. As a multiple ofR, how far from the surface is there a point where the forcemagnitude is 0.44FR if we move the proton(a) away from the ball and (b) into the tunnel?
asked 2021-05-04
Determine whether the lines L1 and L2 are parallel, skew, or intersecting. If they intersect, find the point of intersection.
\(L_1: \frac{x-2}{1}=\frac{y-3}{-2}=\frac{z-1}{-3}\)
asked 2021-04-25
The unstable nucleus uranium-236 can be regarded as auniformly charged sphere of charge Q=+92e and radius \(\displaystyle{R}={7.4}\times{10}^{{-{15}}}\) m. In nuclear fission, this can divide into twosmaller nuclei, each of 1/2 the charge and 1/2 the voume of theoriginal uranium-236 nucleus. This is one of the reactionsthat occurred n the nuclear weapon that exploded over Hiroshima, Japan in August 1945.
A. Find the radii of the two "daughter" nuclei of charge+46e.
B. In a simple model for the fission process, immediatelyafter the uranium-236 nucleus has undergone fission the "daughter"nuclei are at rest and just touching. Calculate the kineticenergy that each of the "daughter" nuclei will have when they arevery far apart.
C. In this model the sum of the kinetic energies of the two"daughter" nuclei is the energy released by the fission of oneuranium-236 nucleus. Calculate the energy released by thefission of 10.0 kg of uranium-236. The atomic mass ofuranium-236 is 236 u, where 1 u = 1 atomic mass unit \(\displaystyle={1.66}\times{10}^{{-{27}}}\) kg. Express your answer both in joules and in kilotonsof TNT (1 kiloton of TNT releases 4.18 x 10^12 J when itexplodes).
asked 2021-05-16
Consider the curves in the first quadrant that have equationsy=Aexp(7x), where A is a positive constant. Different valuesof A give different curves. The curves form a family,F. Let P=(6,6). Let C be the number of the family Fthat goes through P.
A. Let y=f(x) be the equation of C. Find f(x).
B. Find the slope at P of the tangent to C.
C. A curve D is a perpendicular to C at P. What is the slope of thetangent to D at the point P?
D. Give a formula g(y) for the slope at (x,y) of the member of Fthat goes through (x,y). The formula should not involve A orx.
E. A curve which at each of its points is perpendicular to themember of the family F that goes through that point is called anorthogonal trajectory of F. Each orthogonal trajectory to Fsatisfies the differential equation dy/dx = -1/g(y), where g(y) isthe answer to part D.
Find a function of h(y) such that x=h(y) is the equation of theorthogonal trajectory to F that passes through the point P.
asked 2021-06-06
Let X and Y be independent, continuous random variables with the same maginal probability density function, defined as
\(f_{X}(t)=f_{Y}(t)=\begin{cases}\frac{2}{t^{2}},\ t>2\\0,\ otherwise \end{cases}\)
(a)What is the joint probability density function f(x,y)?
(b)Find the probability density of W=XY. Hind: Determine the cdf of Z.
...