How Does the Divergence Theorem Apply to a Vector Field on a Unit Sphere?

In summary, the divergence theorem is a mathematical theorem that relates the surface integral of a vector field over a closed surface to the triple integral of the divergence of the vector field in the enclosed volume. It is important because it allows for the calculation of flux through a closed surface and is applied in various real-world problems, such as fluid mechanics, electromagnetism, and heat distribution. However, it has limitations, such as only being applicable to continuously differentiable vector fields and three-dimensional spaces, and relies on certain assumptions.
  • #1
jlmac2001
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I need help evaluating both sides of the divergence theorem if V=xi+yj+zk and the surface S is the sphere x^2+y^2+z^2=1, and so verify the divergence theorem for this case.

Is the divergence theorem the triple integral over V (div V) dxdydz= the double integral over S (V dot normal)dS? If so I would I evaluate it for the above problem?
 
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  • #2
This was posted under the "calculus" area and I answered there. Please do not double post.
 
  • #3


Yes, the divergence theorem states that the triple integral over V of the divergence of a vector field V is equal to the double integral over the surface S of the dot product of V and the unit outward normal vector dS. In this case, we have V=xi+yj+zk and S is the sphere x^2+y^2+z^2=1.

To evaluate the left side of the theorem, we first need to find the divergence of V. The divergence of a vector field V = (V1, V2, V3) is given by div V = ∂V1/∂x + ∂V2/∂y + ∂V3/∂z. In this case, we have V1=x, V2=y, and V3=z, so the divergence of V is simply 1+1+1=3.

Next, we need to evaluate the triple integral over V of the divergence of V. Since V is a constant vector field, we can pull it out of the integral and evaluate only the divergence. This gives us ∫∫∫Vdiv V dxdydz = ∫∫∫3 dxdydz = 3∫∫∫dxdydz. Since V is defined over the entire volume V, this integral is equivalent to the volume of V, which in this case is the volume of the unit sphere. Therefore, the left side of the divergence theorem is equal to the volume of the unit sphere.

To evaluate the right side of the theorem, we need to find the dot product of V and the unit outward normal vector dS. The unit outward normal vector for a sphere is given by n = (x, y, z)/√(x^2+y^2+z^2), which in this case is simply n = (x, y, z). Therefore, the dot product V dot n is equal to x^2+y^2+z^2, which is equal to 1 on the surface S.

Next, we need to evaluate the double integral over S of V dot n dS. Since V dot n is a constant (equal to 1) on the surface S, we can pull it out of the integral and evaluate only the surface area dS. Since S is a sphere with radius 1, the surface area dS is given by dS = r^2sinθd
 

Related to How Does the Divergence Theorem Apply to a Vector Field on a Unit Sphere?

1. What is the divergence theorem?

The divergence theorem, also known as Gauss's theorem or Ostrogradsky's theorem, is a mathematical theorem that relates the surface integral of a vector field over a closed surface to the triple integral of the divergence of the vector field in the enclosed volume.

2. Why is the divergence theorem important?

The divergence theorem is important because it allows us to calculate the flux of a vector field through a closed surface by only knowing the values of the vector field on the boundary of the surface. This makes it a useful tool in many areas of physics and engineering, such as fluid mechanics, electromagnetism, and thermodynamics.

3. How is the divergence theorem applied in real-world problems?

The divergence theorem is applied in various real-world problems, such as calculating fluid flow through a pipe, determining electric flux through a closed surface, and analyzing the distribution of heat in a solid object. It is also used in solving differential equations and in numerical analysis.

4. What are the assumptions for using the divergence theorem?

The divergence theorem can only be applied to vector fields that are continuously differentiable and have a well-defined divergence throughout the enclosed volume. It also requires that the surface being integrated over is closed and has a finite boundary.

5. Are there any limitations to the divergence theorem?

While the divergence theorem is a powerful tool, it does have some limitations. It cannot be applied to vector fields that have singularities within the closed surface, and it only applies to three-dimensional spaces. Additionally, it relies on the assumptions mentioned in the previous question and may not be applicable in all situations.

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