Proving the Divergence Theorem for Bounded Domains and Differentiable Fields

In summary, the conversation discusses using the divergence theorem to prove an integral involving a bounded domain and differentiability conditions. The problem is related to Laplace's, Poisson's, and Greens Formulas, but the person is unsure of how to approach it. They eventually solve it using a vector identity, but then question whether a vector potential could be used instead.
  • #1
EngageEngage
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Homework Statement


Let the domain D be bounded by the surface S as in the divergence theorem, and assume that all fields satisfy the appropriate differentiability conditions.
Suppose that:
[tex]\nabla\cdot\vec{V}=0[/tex]
[tex]\vec{W}=\nabla\phi with \phi = 0 on S[/tex]
prove:
[tex]\int\int\int_{D}\vec{V}\cdot\vec{W}dV=0[/tex]

This problem is in the Laplace's, Poisson's and Greens Formulas section. Truthfully I'm not sure where to even get started here. If anyone could give me a push in the right direction I would appreciate it greatly.
 
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  • #2
nevermind; got it!
 
  • #3
I used a vector identity, but can someone please help me do this one without an identity. This is in a greens identities section, but none of the greens identities look like they would work. Is it right of me to say that
[tex] \vec{V} = curl \vec{G}[/tex]

where g i ssome vector potential? or would this not help me at all?
 

Related to Proving the Divergence Theorem for Bounded Domains and Differentiable Fields

1. What is a vector identity?

A vector identity is a mathematical equation that expresses a relationship between vectors and is true for all possible values of the vectors involved. It is often used in physics and engineering to prove relationships between physical quantities.

2. How do you prove a vector identity?

To prove a vector identity, you must start with one side of the equation and manipulate it using vector operations such as addition, subtraction, and scalar multiplication until it is equal to the other side. This process is similar to algebraic proofs, but with vectors instead of numbers.

3. What are some common techniques used to prove vector identities?

Some common techniques used to prove vector identities include using the properties of vector operations, converting vectors into their component form, and using special identities such as the triple product rule and the dot product rule.

4. Are there any tips for making vector identity proofs easier?

Yes, some tips for making vector identity proofs easier include breaking the problem into smaller, more manageable steps, using geometric interpretations of vector operations, and being familiar with common vector identities and their properties.

5. What are some real-world applications of vector identities?

Vector identities are commonly used in physics and engineering to model and solve problems related to motion, forces, and energy. They can also be used in computer graphics and animation to manipulate and transform objects in three-dimensional space.

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