Divergence Theorem not equaling 0

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SUMMARY

The discussion centers on the Divergence Theorem and its implications when the function f(r) is zero at specific points in space. It emphasizes that the integral ∫∫∫ V f(r) dV can yield a non-zero result even if f(r) equals zero at certain locations. The key takeaway is that the divergence of a vector field, represented as f(r) = div(grad(1/r)), involves integrating over surface areas where the vector field may not be uniformly zero, leading to non-zero integrals despite pointwise zero values.

PREREQUISITES
  • Understanding of vector calculus concepts, particularly the Divergence Theorem.
  • Familiarity with the notation and operations of gradients and divergences.
  • Knowledge of multivariable integration techniques.
  • Basic comprehension of vector fields and their properties.
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  • Study the Divergence Theorem in detail, focusing on its mathematical proofs and applications.
  • Explore vector calculus textbooks that cover the implications of pointwise zero functions in integrals.
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  • Investigate examples of non-zero integrals in cases where functions are zero at specific points.
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Students and professionals in mathematics, physics, and engineering who are working with vector calculus and the Divergence Theorem, particularly those interested in understanding the nuances of integrals involving zero functions.

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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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