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- The divergence theorem states that the integral of the divergence of a vector field over a volume is equal to the integral of the vector field over the boundary surface of the volume. A proof of the divergence theorem can be given by12:
- Assuming that the surface is closed and any line parallel to the coordinate axes cuts the surface in two points.
- Dividing the volume into small cubes and approximating the divergence by finite sums and difference quotients.
- Summing over all the cubes and canceling the terms that appear twice on opposite faces of the cubes.
- Taking the limit as the cubes become infinitesimal and using the definition of the integrals.
Learn more:✕This summary was generated using AI based on multiple online sources. To view the original source information, use the "Learn more" links.Divergence Theorem Proof The divergence theorem-proof is given as follows: Assume that “S” be a closed surface and any line drawn parallel to coordinate axes cut S in almost two points. Let S 1 and S 2 be the surface at the top and bottom of S. These are represented by z=f (x,y)and z=ϕ (x,y) respectively.byjus.com/maths/divergence-theorem/Proof We can approximate the integral of the divergence over the volume by the finite sum by dividing densely the space inside the volume into small cubes with the edges and the corners as well as approximating three of the coordinate derivatives by their difference quotients.en.wikiversity.org/wiki/Divergence_theorem/Proof - People also ask
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WEBThis theorem is used to solve many tough integral problems. It compares the surface integral with the volume integral. It means that it gives the …
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WEBNov 10, 2020 · Proof: Let Σ be a closed surface which bounds a solid S. The flux of ∇ × f through Σ is. ∬ Σ ( ∇ × f) · dσ = ∭ S ∇ · ( ∇ × f)dV (by the Divergence Theorem) = ∭ S 0dV (by Theorem 4.17) = 0. There is …
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