Prove using divergence theorem

In summary, the divergence theorem can be used to show that the surface integral of the cross product of a vector field and the outward unit normal vector is equal to the triple integral of the dot product of the gradient of the vector field. This can be proven by using the hint provided, where the constant vector is replaced with the cross product of the unit normal vector and the vector field.
  • #1
grissom
2
0
Use the divergence theorem to show that [tex]\oint\oint[/tex]s (nXF)dS = [tex]\int\int\int[/tex]R ([tex]\nabla[/tex]XF)dV.

The divergence theorem states: [tex]\oint\oint[/tex]s (n.F)dS = [tex]\int\int\int[/tex]R ([tex]\nabla[/tex].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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  • #2
Hint: For any constant (position independent) vector [itex]\textbf{c}[/itex], the following is true (It's worthwhile if you prove this to yourself by looking at individual components)

[tex]\textbf{c}\cdot\int\int_{\mathcal{S}}\textbf{A}dS=\int\int_{\mathcal{S}}(\textbf{c}\cdot\textbf{A})dS[/tex]

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

Related to Prove using divergence theorem

1. What is the divergence theorem?

The divergence theorem, also known as Gauss's theorem, is a mathematical concept that relates the flux of a vector field through a closed surface to the divergence of the vector field within the enclosed volume. In other words, it states that the total outward flux of a vector field through a closed surface is equal to the volume integral of the divergence of the vector field within that surface.

2. How is the divergence theorem used in science?

The divergence theorem has many applications in various fields of science, such as fluid dynamics, electromagnetism, and thermodynamics. It allows us to simplify complex surface integrals into easier volume integrals, making it a useful tool in solving problems involving vector fields.

3. Can you provide an example of how the divergence theorem is used?

Sure. Let's say we have a vector field representing the velocity of air particles in a room. We want to calculate the total amount of air leaving the room through a window. Instead of calculating the flux through the window directly, which can be complicated, we can use the divergence theorem to convert it into a volume integral, making the calculation much simpler.

4. What are the conditions for the divergence theorem to be applicable?

The divergence theorem is only applicable to a closed surface, meaning that it completely encloses a volume. Additionally, the vector field must be continuous and differentiable within the volume enclosed by the surface.

5. How is the divergence theorem related to other theorems in mathematics?

The divergence theorem is closely related to other theorems in mathematics, such as Green's theorem and Stokes' theorem. These theorems all fall under the umbrella of the fundamental theorem of calculus, which relates the integral of a function over an interval to the values of that function at the endpoints of the interval.

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