Prove using divergence theorem

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SUMMARY

The discussion focuses on applying the divergence theorem to demonstrate the equation \(\oint\oint (n \times F) dS = \int\int\int_R (\nabla \times F) dV\). The divergence theorem is correctly stated as \(\oint\oint (n \cdot F) dS = \int\int\int_R (\nabla \cdot F) dV\), highlighting the transition from dot product to cross product. A key hint provided is to consider the vector \(\textbf{A} = \textbf{n} \times \textbf{F}\) and utilize the triple scalar product rule for further insights.

PREREQUISITES
  • Understanding of vector calculus, specifically the divergence theorem.
  • Familiarity with cross products and dot products in vector operations.
  • Knowledge of triple scalar products and their applications.
  • Basic proficiency in mathematical proofs and manipulation of vector fields.
NEXT STEPS
  • Study the divergence theorem in detail, focusing on its applications in vector calculus.
  • Learn about the properties of cross products and how they relate to vector fields.
  • Explore the triple scalar product rule and its implications in vector analysis.
  • Practice solving problems involving the divergence theorem and cross products to solidify understanding.
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Students and professionals in mathematics, physics, and engineering who are looking to deepen their understanding of vector calculus and the divergence theorem.

grissom
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Use the divergence theorem to show that \oint\oints (nXF)dS = \int\int\intR (\nablaXF)dV.

The divergence theorem states: \oint\oints (n.F)dS = \int\int\intR (\nabla.F)dV.

The difference is switching from dot product to cross product. I have no idea how to start. Can someone please point me in the right direction. Any help is appreciated.
 
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Hint: For any constant (position independent) vector \textbf{c}, the following is true (It's worthwhile if you prove this to yourself by looking at individual components)

\textbf{c}\cdot\int\int_{\mathcal{S}}\textbf{A}dS=\int\int_{\mathcal{S}}(\textbf{c}\cdot\textbf{A})dS

What happens if you let \textbf{A}=\textbf{n}\times\textbf{F} and apply the triple scalar product rule?:wink:
 

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