Divergence Theorem not equaling 0

In summary, the conversation discusses why the integral of a function, f(r), over a volume, V, can result in a non-zero value even if f(r) is equal to zero. This is because the integral is a summation of the function multiplied by small differences in x, and even if f(r) is zero at every point in space, the sum can still result in a non-zero value. The conversation also mentions that the divergence of f(r) is the integral of a vector field over small surface areas, which can vary with location in space.
  • #1
xoxomae
23
1
Why is it possible that
∫∫∫ V f(r) dV ≠ 0 even if f(r) =0
 
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  • #2
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).
 
  • #3
Sorry, I forgot to mention that f(r) = div (grad (1/r))
 

1. What is the Divergence Theorem and why does it not always equal 0?

The Divergence Theorem is a mathematical concept that relates the flux (flow) of a vector field through a closed surface to the divergence (convergence or spreading out) of the vector field within the enclosed volume. It states that the net flux through a closed surface is equal to the volume integral of the divergence over the enclosed volume. However, the Divergence Theorem does not always equal zero because it depends on the properties of the vector field and the shape and orientation of the closed surface.

2. When does the Divergence Theorem not equal 0?

The Divergence Theorem does not equal 0 when the vector field is not divergence-free, meaning there is a net flow into or out of the enclosed volume. It also does not equal 0 when the closed surface is not a true "closed" surface, such as when there is a hole or opening in the surface.

3. Why is it important to understand when the Divergence Theorem does not equal 0?

Understanding when the Divergence Theorem does not equal 0 is important in various fields of science and engineering where vector fields are used. It allows for accurate calculations of flux and divergence, which have practical applications in fluid mechanics, electromagnetism, and other areas of physics and engineering.

4. Can the Divergence Theorem be used to solve real-world problems?

Yes, the Divergence Theorem can be used to solve real-world problems, particularly in the fields of fluid mechanics and electromagnetism. It allows for the calculation of important quantities such as fluid flow rate and electric flux, which are crucial in many engineering and scientific applications.

5. Are there any limitations to using the Divergence Theorem?

Yes, there are limitations to using the Divergence Theorem. It is only applicable to vector fields that are continuous and differentiable, and the closed surface must also be smooth and enclosed. Additionally, the Divergence Theorem is only valid in three-dimensional space, and cannot be applied to higher dimensions.

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