Divergence of the curl problem question

In summary: Essentially, you will be doing the same thing here, but with a vector field instead of a single vector.
  • #1
galactic
30
1

Homework Statement


if a vector can be written as the curl of another vector, its divergence vanishes. Can you justify the statement: "any vector field whose divergence vanishes identically can be written as the curl of some other vector"?

Homework Equations



Prove this by construction. Let [tex]{\nabla}{\cdot}{\vec{V}}=0[/tex] and try to find any vector, U, for which [tex]{\vec{V}}={\nabla}{\cdot}{\vec{U}}[/tex]
This amounts to showing that you have enough freedom to pick as components of \vec{U}, functions which satisfy some simple differetial equations. Simplify somewhat by trying to find a vector, [itex]{\vec{U}}[/itex], for which, say, [itex]{\vec{U_z}}=0[/itex]

The Attempt at a Solution



I know that for [tex]{\nabla}{\cdot}{\vec{V}}=0[/tex] we have to have for example, a vector field such as: [tex]{\vec{V}}={yz}\hat{x}+{xz}\hat{y}+{xy}\hat{z}[/tex] so that when we do [tex]{\nabla}{\cdot}{\vec{V}}=0[/tex]. I'm confused if the problem is just asking to prove the divergence of the curl is equal to 0 which I have already done a few homework problems ago or if its asking for something different here...because I'm confused by the hint that it's giving me (in relevant equations).
 
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  • #2
galactic said:

Homework Statement


if a vector can be written as the curl of another vector, its divergence vanishes. Can you justify the statement: "any vector field whose divergence vanishes identically can be written as the curl of some other vector"?

Homework Equations



Prove this by construction. Let [tex]{\nabla}{\cdot}{\vec{V}}=0[/tex] and try to find any vector, U, for which [tex]{\vec{V}}={\nabla}{\cdot}{\vec{U}}[/tex]
This amounts to showing that you have enough freedom to pick as components of \vec{U}, functions which satisfy some simple differetial equations. Simplify somewhat by trying to find a vector, [itex]{\vec{U}}[/itex], for which, say, [itex]{\vec{U_z}}=0[/itex]

The Attempt at a Solution



I know that for [tex]{\nabla}{\cdot}{\vec{V}}=0[/tex] we have to have for example, a vector field such as: [tex]{\vec{V}}={yz}\hat{x}+{xz}\hat{y}+{xy}\hat{z}[/tex] so that when we do [tex]{\nabla}{\cdot}{\vec{V}}=0[/tex]. I'm confused if the problem is just asking to prove the divergence of the curl is equal to 0 which I have already done a few homework problems ago or if its asking for something different here...because I'm confused by the hint that it's giving me (in relevant equations).

I believe there is a typo in your relevant equations. They want you to show that if
[tex]\nabla \cdot \vec{V} = 0[/tex]
for any particular vector field [itex]\vec{V}[/itex], then you can always find a vector field [itex]\vec{U}[/itex] such that
[tex]\nabla \times \vec{U} = \vec{V}[/tex]
They want you to construct a vector field U by x, y and z components that satisfies the second equation, using the equation for the divergence of V being 0 as the only given fact.
This is actually similar to a proof you may have done earlier: in 3-dimensional space, if [itex]\vec{A}\cdot\vec{B}=0[/itex], then there is a vector [itex]\vec{C}[/itex] such that [itex]\vec{A}\times\vec{C}=\vec{B}[/itex].
 
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FAQ: Divergence of the curl problem question

What is the divergence of the curl problem?

The divergence of the curl problem is a mathematical concept that describes the relationship between the divergence and curl of a vector field. It is based on the fundamental theorem of calculus for line integrals and the Green's theorem.

Why is the divergence of the curl problem important?

The divergence of the curl problem is important because it allows us to understand the behavior of vector fields and their sources and sinks. It also has many applications in physics and engineering, such as in fluid dynamics and electromagnetism.

How is the divergence of the curl problem calculated?

The divergence of the curl problem is calculated using the vector calculus operators of divergence and curl. The divergence of a vector field is computed by taking the dot product of the vector field with the del operator (∇). The result is a scalar function. The curl of a vector field is calculated by taking the cross product of the vector field with the del operator (∇). The result is a vector function. The divergence of the curl of a vector field is then calculated by taking the dot product of the curl with the del operator (∇).

Can the divergence of the curl problem be zero?

Yes, the divergence of the curl problem can be zero. This occurs when the vector field has no sources or sinks, and the curl is also zero. In other words, the vector field is both irrotational and incompressible.

What are some real-world applications of the divergence of the curl problem?

The divergence of the curl problem has many real-world applications. In fluid dynamics, it is used to study the flow of fluids and understand vortices and eddies. In electromagnetism, it helps us understand the behavior of electric and magnetic fields and their interactions. It also has applications in other fields such as meteorology, geophysics, and computer graphics.

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