44. The vector field F has F 7 everywhere and C is the circle of radius 1 centered at the origin....
(a) Suppose F is a field such that divF 14 everywhere and S is the closed surface of a cylinder with height 4, radius 3, and oriented inward. Find FdA (b) Suppose G is a smooth field everywhere and S is a sphere centered at the origin with radius 9, oriented outward. Find curl G dA Explain your answer. Why is it true? What conditions were necessary?
(a) Suppose F is a field such that divF 14 everywhere and S...
c. Evaluate ,f(z) dz with า the circle of radius 1 centered at the origin and traveled once counterclockwise ˊ们: (1-2 For real twith-1 < t < 1 and +12)-1 Explain why f(:)) has an expansion of the form in C , let f(z) be defined by fG)- a. b. Compute Uo(t), Ui(t), and Uz(t) in terms of t. c. Recalling that t is a real number smaller than 1 in absolute value, find the radius of convergence of this...
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...
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...
9. 9a) A vector field A has divergence ? ⋅ ? = 0 everywhere. Represent (draw) how A could look like (one possible vector field whose divergence is null everywhere). 9b) A vector field A has divergence ? ⋅ ? > 0 in the origin of the axis. Draw how A could look like (one possible vector field whose divergence is null everywhere).
Calculate • (-3x, 2y). Nds where C is circle of radius 5 centered at the origin and N is an outward pointing unit normal vector of C. Answer: WA 2
Suppose F is a radial force field, S1 is a sphere of radius 4 centered at the origin, and the flux integral ??S1 F ·dS = 3.Let S2 be a sphere of radius 12 centered at the origin, and consider the flux integral ??S2 F ·dS.
1) Suppose a dart board is a circle of radius one centered at the origin. A player throws darts many times and hits the board every time, recording where the dart hits. The player then finds the probability density function of where they hit on the dartboard is given by f(r,y) r2 +y -(r2 y)) a) Based on your knowledge of probabilities determine the value of C given that the player never misses the dart board.
1) Suppose a dart...
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...
(1 point) Consider the vector field F(x, y, z) = (2z + 3y)i + (2z + 3x)j + (2y + 2x)k. a) Find a function f such that F = Vf and f(0,0,0) = 0. f(x, y, z) = b) Suppose C is any curve from (0,0,0) to (1,1,1). Use part a) to compute the line integral / F. dr. (1 point) Verify that F = V and evaluate the line integral of F over the given path: F =...