I Divergence Theorem not equaling 0

AI Thread Summary
The discussion centers on the conditions under which the integral of a function f(r) over a volume V can be non-zero, even if f(r) is zero at certain points. It highlights that f(r) may not be zero everywhere, which is crucial for the integral's value. The integration process involves summing the product of f(r) and small volume elements dV, suggesting that if f(r) is zero at all points, the integral should also be zero. Additionally, the divergence theorem relates to the behavior of vector fields and their integrals over surfaces, emphasizing the importance of the function's spatial variability. The conversation underscores the nuances of mathematical integration and divergence in vector calculus.
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Why is it possible that
∫∫∫ V f(r) dV ≠ 0 even if f(r) =0
 
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I think something is missing here. What is missing is that whether f(r) is zero at every single point in space or just at a particular distance.
There is another thing:
When you are integrating you are essentially summing up a multiplication of the function by very small differences in x.
Simply put: f(r)*dV is your differential.
If f(r) is always zero, why would the sum of zeroes equal to something non-zero?
Moreover the divergence is the integral of f(r) vector field over small parts of surface areas with a vector pointing outwards multiplying it (is inside the integral if it changes with respect to location in space).
 
Sorry, I forgot to mention that f(r) = div (grad (1/r))
 

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