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Stroke's Theorem and Direct Evaluation

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(a) Let F(x,y,z)=(x^2+y-4)i + (3xy)j + (2xy+z^2)k. Evaluate the double integral over S of (curl(F). dS) where S is the surface x^2 + y^2 + z^2 = 16, z >=0

(I) Using Stroke's theorem
(II)By direct evaluation

(b) Find the flux of the vector field
F(x,y,z) = (y-x)i + (x+y)j + y k across the side of the triangle with vertices at (1,0,0), (0,1,0) and (0,0,1)

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Solution Summary

This does several things with a function: evaluates a double integral using Streok's theorem and direct evaluation, and then finds the flux of a given vector field

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(a) Let F(x,y,z)=(x^2+y-4)i + (3xy)j + (2xy+z^2)k. Evaluate the double integral over S of (curl(F). dS) where S is the surface x^2 + y^2 + z^2 = 16, z >=0

(I) Using Stroke's theorem
(II)By direct evaluation

Solution. (I) I don't know how to use Stroke's theorem to calculate the surface integral with respect to coordinate elements. I am ...

Solution provided by:
Changping Wang, MA
Education
  • 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."
  • "excellent work"
  • "Thank you so much for all of your help!!! I will be posting another assignment. Please let me know (once posted), if the credits I'm offering is enough or you ! Thanks again!"
  • "Thank you"
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