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PRACTICE EXERCISE 4.12 Given that E = (3x2 + y)a, + xay kV/m, find the work...
Challenge Yourself PRACTICE EXERCISE 4.12 Given that E = (3x² + v) a, + xa, kV/m, find the work done in moving a -2 uc charge from (0.5.0) to (2, -1.0) by taking the path (a) (0,5, 0) -> (2, 5,0) -→ (2.-1.0) (b) y = 5 -- 3x Answer: (a) 12 m), (b) 12 m).
Problem 4.14 (b) Show that R) (e) What is )? hs. 4.12. Compute fqu'l dz 4.13. Show, that, zas-0 for any closed piecewise smooth γ and any integer "t-1. (if" is negative, assume that γ does not pass through the origin, since otherwise the integral is not defined.) 4·1 4, Exercise 4.13 excluded " =-1 for a good reason: Exercise 4.4 gives a coun- terexample. Generalizing these, if m is any integer, find a closed path y so that 4.15....
Find the work done by the force field F on a particle moving along the given path. F(x, y) = xi + 4yj C: x = t, y = 13 from (0, 0) to (2,8)
nc = 13 1. Find the charge in the volume defined by 1<r<2m, in the spherical coordinates if pv = (No cos?0)/r* (uC/mº). 2. Given that D = 7r2 a, + Nc sin 0 ag in spherical coordinates, find the charge density. 3. Find the work done in moving a point charge Q = - 20 uC from (4,2,0)m to the origin in the field E = (x/2 + 2y) ax + Nc xay (V/m). 1
Given the potential V=x^yz2 [V] find the electric field E at (x 1,y=2,z=1 (i) (ii) calculate the work done in moving a 2 uC charge from A-(1,1,1) to B-(4,-1,1)
(b) Find the work done in moving a particle along the path x-cos y, z 0 from y-0 to y 2m, in the field F(x, y, z)-c" cosy i-xe® sínyi + 2xe2: cos y k. (10 Marks) EvaluatelFdA for surface S: x-z2,0 F(x, y, z)--Зугі + zer cosyj + 3xz2k. (c) y 2,-1 251and (7 Marks) (b) Find the work done in moving a particle along the path x-cos y, z 0 from y-0 to y 2m, in the field...
show work please a) Find the second order derivatives of y=x - 3x2 + 3x-1 b) When x >1, is the above function concave up or concave down? And why?
Question 1. (15 pts) Given f(x, y) = 3x 2 + y 3 . (a) Find the gradient of f. (b) Find the directional derivative of f at P0 = (3, 2) in the direction of u = (5/13)i + (12/13)j. Question 1. (15 pts) Given f(L,y) = 3x2 +y?. (a) Find the gradient of f. (b) Find the directional derivative off at P =(3,2) in the direction of u=(5/13)i + (12/13)j.
EXERCISE 4.12. For fixed øo E (0, 7), let y be a unit-speed parametriza- tion of the o-latitudinal curve in the sphere S2 (the 0-parameter curve in Example 3.24 on page 129 with = ¢o). With respect to the outward- pointing orientation of S2, show that the geodesic curvature of y is constant cot (φο). at Kg e next USurface patches EXAMPLE 3.24 (The Sphere). Recall that a its spherical coordinates (0,6), illustrated in Fig. 3.11 (left). The conver- point...
13. Find the work done by the force field F on an object moving along the specified path. The specified path: Counterclockwise along the semicircle y=V4-n- from (2,0) (-2,0) to 13. Find the work done by the force field F on an object moving along the specified path. The specified path: Counterclockwise along the semicircle y=V4-n- from (2,0) (-2,0) to