Is Multiplying Divergence by Area Correct in Divergence Theorem Problems?

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The discussion focuses on the application of the Divergence Theorem in a problem involving the vector field F = xi + yj + zk and the surface of a sphere defined by s = x^2 + y^2 + z^2. The divergence of F is calculated as 3, but the initial assumption to multiply this by the area of the sphere (4π) to find the answer is incorrect. Instead, the volume of the sphere should be used, which is calculated as 4/3π for a radius of 1. The correct final answer for the problem is 4π. Understanding the distinction between surface area and volume is crucial in applying the Divergence Theorem correctly.
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Homework Statement



F = xi + yj + zk, s = x^2 + y^2 + z^2

Homework Equations





The Attempt at a Solution



div F = 1+1+1=3

area of sphere = 4pi

i can just multiply them to get 12pi as an answer right?
 
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No. You want the volume of the sphere, not the area.
 
ooooops

V of sphere = 4/3 pi r^3 where r = 1, V = 4/3 pi

so i got 4pi
 
Correct.
 
Question: A clock's minute hand has length 4 and its hour hand has length 3. What is the distance between the tips at the moment when it is increasing most rapidly?(Putnam Exam Question) Answer: Making assumption that both the hands moves at constant angular velocities, the answer is ## \sqrt{7} .## But don't you think this assumption is somewhat doubtful and wrong?

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