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Stokes and Gauss' Theorem

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1. A vector field v(x, y, z) is given by the formula

v(x, y, z) = xyˆx − yˆ2y.

Consider a square path in the xy plane which starts at (0,0,0) and moves along the corners (1,0,0), (1,1,0) and (0,1,0). Calculate the path integral of v, i.e. v · dr, and calculate the area integral of the divergence, R ∇ × v · da, and verify that Stokes' theorem holds. (Note: For the first leg of the path, dr = ˆxdx, and for the second leg of the path, dr = ˆydy. The area element here is da = dxdyˆz, integrated over the square.)

2. Given a vector t = −ˆxy + ˆyx, use Stokes' theorem to show that the integral around a closed curve of
arbitrary shape in the xy plane

(see attachment for formula)

where A is the area enclosed by the curve. (Hint: Use Stokes' theorem to write the integral in terms
of the curl of the vector. What does the integral now represent?)

3. Show that (see attachment for formula)

where V is the volume enclosed by the surface S, and r = xˆx + yˆy + zˆz. (Hint: This is similar to the
previous problem, but using Gauss' theorem instead of Stokes' theorem.)

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Solution provided by:
  • BSc , Wuhan Univ. China
  • MA, Shandong Univ.
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  • "Your solution, looks excellent. I recognize things from previous chapters. I have seen the standard deviation formula you used to get 5.154. I do understand the Central Limit Theorem needs the sample size (n) to be greater than 30, we have 100. I do understand the sample mean(s) of the population will follow a normal distribution, and that CLT states the sample mean of population is the population (mean), we have 143.74. But when and WHY do we use the standard deviation formula where you got 5.154. WHEN & Why use standard deviation of the sample mean. I don't understand, why don't we simply use the "100" I understand that standard deviation is the square root of variance. I do understand that the variance is the square of the differences of each sample data value minus the mean. But somehow, why not use 100, why use standard deviation of sample mean? Please help explain."
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