The value of cos(4M) where M is the magnitude of the vector field with potential ƒ = e² sin(лy) cos(π²) at x = 1, y = 1/4, z = 1/3 is 0.602 -0.323 0.712 -0.816 0.781 0.102 0.075 0.013

Verify if the vector field F = (x² cos(3y), -³ sin(3y)) is conservative and evaluate the line integral Je where C' is the curve parameterized by R= for 0 < t < 1. cos(6) 3 A. I = B. I O c. I - OD. I = 0 E. I sin(6) 4 23 cos(3) 2 F. dR

Use the equation giving the flux of the vector field across the curve to calculate the flux of F(x, y) = x+1 y ( (x + 1)² + y²' (x + 1)² + y² across C, the segment 7≤ y ≤ 9 along the y-axis, oriented upwards. (Use symbolic notation and fractions where needed.) F. dr = 2 (1.9085 - 1.6987)

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Question A1 Consider a vector field F= 2x° yị +x'j., and a curve C defined by the equation y = 2x². Calculate the line integral (F•dr along the curve C starting at x = 0 and finishing at x = 1. 2 1.8 1.4 12 1 0.8 F 0.6 0.4 0.2 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 X-axis

Use the equation giving the flux of the vector field across the curve to calculate the flux of x+1 y (₁. (x + 1)² + y²' (x + 1)² + y² F(x, y): across C, the segment 7 ≤ y ≤ 9 along the y-axis, oriented upwards. (Use symbolic notation and fractions where needed.) [F C = Incorrect F. dr = 0.1074

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Let A be the vector potential and B the magnetic field of the infinite solenoid of radius R. Then 0 ={8/ B(r) = where r = √x² + y2 is the distance to the z-axis and B is a constant that depends on the current strength I and the spacing of the turns of wire. The vector potential for B is -{} J A(r) = if r> R Bk if r R r< R S

Compute the divergence and curl of the vector field cos? (x2 ) F = sin (²y=) 2. 10 yz4 – ryz , at the point (1, 1, c). a. Divergence = ab sin (a) f Ω b. Curl = sin (a) f Ω a 00

3. Let F = (x - y, xy) and C be arc of the circle x? + y? = 4 traversed counter-clockwise from (2,0) to (-2,0). A sketch of the vector field a curve are below. (a) Using the image above, determine whether F- dr is positive or negative. (b) Evaluate | F . dĩ. You may use that sin?(t) = /2. (Hint: the answer is 16/3 + 27)