Exercise. Below we have plotted a discrete "sampling" of a vector field: 40 -2 2 Let C be a circl...
Exercise. Below we have plotted a discrete "sampling of a vector field: -2 2 4 Let C be a circle of radins 3 centered at the origin drawn in a counterclockwise fashion. What concusions seem to be true? This is a gradieut field This is not a gradient field. This field has positive cur This field bas negative curl. c F.dp X Try again Note that the raclias of the circle is irreverent. Exercise. Below we have plotted a discrete...
Exercise. Below we have plotted a discrete sampling" of a vector field -4 -2 2 Let C bea circle of radis3centered at the origin n drawn in a counterclockwise fashion. What conelasions seem to he true? This is a gradient field. This is not a uruient Gelkl. This field has positive curl. This field has negative e curl. oF.dp- k.P.dp > ถ ? Check work Exercise. Below we have plotted a discrete sampling" of a vector field -4 -2 2...
PLEASE ANSWER ALL PARTS AND SHOW WORK. THANK YOU! If F is a continuous vector field on an oriented surface S with unit normal vector n, then llo F.JS = : Finds Select one: True False Let S be the bottom half of the unit sphere, oriented upward. Let C be the boundary of S, the unit circle in the zy-plane, oriented counterclockwise as viewed from above. Then for any vector field F with continuous first-order partial derivatives, SP.d -...
(1) Let G(,y, z) = (x,y, z). Show that there exists no vector field A : R3 -> R3 such that curl(A) Hint: compute its divergence G. (2) Let H R3 -> R3 be given as H(x,y, z) = (1,2,3). Find a vector potential A : R3 -> R3 such that curl(A) smooth function = H. Show that if A is a vector potential for H, then so is A+ Vf, for any f : R5 -> R (3) Let...
3. Consider the vector field F(x, y) + 2y F dr, where C is the circle (r-2)2 +y2 = 1, oriented counterclock (a) Compute wise (Hint: use the FT of line integrals. We could not use it for the circle centered at the origin, but we can use the theorem for this circle. Why?) (b) Let 0 be the angle in polar coordinates for a point (x, y). Check that 0 is a potential function for F 3. Consider the...
Exercise 9. Let n 2 2 be a positive integer. Let a -(ri,...,^n) ER". For any a,y E R" sphere of radius 1 centered at the origin. Let x E Sn-be fixed. Let v be a random vector that is uniformly distributed in S"1. Prove: 10Vn
Let C be the counter-clockwise planar circle with center at the origin and radius r o. VWithout computing them, determine for the following vector fields F whether the line integrals F. dr are positive, negative, or zero and type P, N, or Z as A. F the radial vector field-t1 + 30 B. F the circulating vector field -yi + xj C. F the circulating vector field -yi - zj D. F the constant vector field-i+j Let C be the...
Consider the vector field: f (x, y)= «M(x, y), N(x, y)= v promet Let C be any simple, positively oriented, closed curve that encloses the origin. Show that: F. do 21. We will solve this problem by completing the following steps: STEP 1 Let C be a positively oriented circle of radius r with the center at the origin. Letr be so small that the circle Člies within the region enclosed by the curve C(see figure below) Compute the integral...
exercise 4.18(2) proves that every longitude and every latitude is a line of curvature of a surface if revolution EXERCISE 4.23. Let S be the torus obtained by revolving about the axis the circle in the xz-plane with radius 1 centered at (2,0,0). This torus is illustrated in Fig. 4.8. Colored red (respectively green) is the region where 2y4 (respectively r2 +y > 4). Let N be the outward-pointing unit 2- normal field on S. (1) Verify that the unit...
2. Consider a charged particle in a magnetic field. Let -e = -1.60 x 10-191C) be an electron, under B=0.1T) of a uniform magnetic field directed out of the page direction (o-direction). At t = 0[sec] the particle is at the origin, and the initial velocity is toward positive y direction. (a) Sketch the initial configuration of this electron (-e at the origin) into (x,y)-plane. Applying the right hand rule, indicate the magnetic force on this moving electron. Note that...