Divergence Theorm example for 28 Nov 12:00

So, now we have \iiint \nabla \cdot (x \vec{r}) dV = 4 \iiint x dV. But what is \nabla \cdot (x \vec{r}) ?In summary, using the Divergence Theorem, we can show that the integral of x times the unit outward normal over a smooth surface enclosing a volume is equal to 4 times the triple integral of x over the enclosed volume. This can be achieved by calculating the divergence of the given vector function and applying the Divergence Theorem.
  • #1
debian
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Homework Statement



Let S be a smooth surface enclosing the volume V, and let [itex]\vec{n}[/itex] to be the unit outward normal. Using the Divergence Theorm show that:


∫∫ x [itex]\vec{r}[/itex] ° [itex]\vec{n}[/itex] dS = 4 * ∫∫∫ x dV,

where [itex]\vec{r}[/itex]=(x,y,z)

Homework Equations



Divergence theorm

http://www.math.oregonstate.edu/home...rg/diverg.html [Broken]

The Attempt at a Solution



I tried to change the form of the those two equations to the form stated in divergence theorm and then to compare the u (or F as in link above), but the u (F) on the left hand side is never equal to this on the right.
 
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  • #2
Your link is broken. But let's suppose the divergence theorem says [tex]\iint (\vec{F} \cdot \vec{n}) dS = \iiint \nabla \cdot \vec{F} dV. [/tex] Now you are given [itex] \vec{F}= x \vec{r} [/itex]. Can you calculate [itex] \nabla \cdot \vec{F} [/itex]?
 
  • #3
[itex]\nabla[/itex] [itex]\cdot[/itex] [itex]\vec{F}[/itex] = (d/dx, d/dy, d/dz) [itex]\cdot[/itex] (x^2,xy,xz) = 2x+x+x=4x
 
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  • #4
debian said:
[itex]\nabla[/itex] [itex]\cdot[/itex] [itex]\vec{F}[/itex] = (d/dx, d/dy, d/dz) [itex]\cdot[/itex] (x^2,xy,xz) = 2x+x+x=4x

Good job. :)
 

What is the Divergence Theorem?

The Divergence Theorem is a mathematical theorem that relates the flow of a vector field through a closed surface to the behavior of the vector field inside the surface.

What does the Divergence Theorem state?

The Divergence Theorem states that the outward flux of a vector field through a closed surface is equal to the volume integral of the divergence of the vector field throughout the region enclosed by the surface.

Can you provide an example of the Divergence Theorem?

Sure, for example, if we have a vector field F(x,y,z) = (x^2, y^2, z^2) and a closed surface S that encloses a region R, the Divergence Theorem states that the flux of F through S is equal to the volume integral of the divergence of F over R.

What is the significance of the Divergence Theorem?

The Divergence Theorem is important in the field of vector calculus as it allows us to relate the behavior of a vector field throughout a region to its behavior on the boundaries of that region. This is especially useful in applications such as fluid dynamics and electromagnetism.

How is the Divergence Theorem used in real-world applications?

The Divergence Theorem is used in various real-world applications such as fluid flow calculations, electrical field calculations, and even in computer graphics for simulating particle systems. It allows for a more efficient and accurate calculation of flux and volume integrals in these applications.

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