Discovering a Vector Field for ∇xF with Properties of Curl

  • Thread starter icesalmon
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In summary: This means that if you take the divergence of the curl of a vector field, the result will always be zero.
  • #1
icesalmon
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Homework Statement


find a vector field that satisfies ∇xF = xi +yj +zk



The Attempt at a Solution


I was just kind of staring at this problem for a while, maybe I could use some properties of the curl? I can't really think of any intelligent way to do this problem other than just muscling through the formula and setting the i,j,k components equal to x, y, and z. Which means I don't understand something conceptually.
 
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  • #2
icesalmon said:

Homework Statement


find a vector field that satisfies ∇xF = xi +yj +zk



The Attempt at a Solution


I was just kind of staring at this problem for a while, maybe I could use some properties of the curl? I can't really think of any intelligent way to do this problem other than just muscling through the formula and setting the i,j,k components equal to x, y, and z. Which means I don't understand something conceptually.

Why do you think there is such a field? Isn't divergence of a curl zero?
 
  • #3
Well it exists in some form, and it was asked of me on some homework. The div(curl(f)) = 0 sure. Maybe I could try integrating the rhs. It won't hurt to try I suppose.
 
  • #4
icesalmon said:
Well it exists in some form, and it was asked of me on some homework. The div(curl(f)) = 0 sure. Maybe I could try integrating the rhs. It won't hurt to try I suppose.

You could try to find a solution, but it's pointless. Take the divergence of both sides of ∇xF = xi +yj +zk. Don't you see something wrong? There must be some error or misunderstanding in the homework.
 
  • #5
It asks me if I can find a vector field that satisfies the OQ. I get 0 = 3 when I take the divergence, the problem is: that doesn't make any sense. So there are no solutions to this expression.
 
Last edited:
  • #6
icesalmon said:
It asks me if I can find a vector field that satisfies the OQ. I get 0 = 3 when I take the divergence, the problem is: that doesn't make any sense. So there are no solutions to this expression.

That's it.
 
  • #7
icesalmon said:
It asks me if I can find a vector field that satisfies the OQ.

It would have been good for you to state the original question correctly, including the "if". Then since the divergence of your vector isn't zero, you could just say "no" and be done with it instead of causing a discussion about the statement of the problem.
 
  • #8
yeah I didn't notice that myself in the question, my apologies. Thank you for your help!
 
  • #9
why is taking the divergence of the RHS and LHS a valid method of proving the vector field doesn't exist? I'm sort of confused about this
 
  • #10
icesalmon said:
why is taking the divergence of the RHS and LHS a valid method of proving the vector field doesn't exist? I'm sort of confused about this

The curl of any vector field has divergence 0. The divergence of xi +yj +zk is 3. Therefore xi +yj +zk cannot be the curl of any vector field. I'm sort of confused about what you are confused about.
 
  • #11
You originally asked "maybe I could use some properties of the curl?"

Yes! The specific property you can use is "the divergence of a curl is 0".
 

FAQ: Discovering a Vector Field for ∇xF with Properties of Curl

What is curl(F)?

Curl(F) is a mathematical operator that describes the rotation or circulation of a vector field in three-dimensional space. It is also known as the curl of a vector field or the vector curl.

How is curl(F) calculated?

The curl of a vector field F is calculated using the partial derivative of its components with respect to each of the three spatial coordinates (x, y, and z). It can be represented by a three-dimensional vector, where each component corresponds to the rate of change of the field in a specific direction.

What is the physical significance of curl(F)?

Curl(F) is a fundamental concept in physics and has many important applications. It is used to describe the flow of fluids, the electromagnetic field, and many other phenomena in nature. It also has practical applications in engineering, such as in fluid dynamics and electromagnetism.

How is curl(F) related to divergence?

Curl(F) and divergence are two different mathematical operators that describe different aspects of a vector field. While curl(F) represents the rotation of a field, divergence represents its expansion or contraction. They are related through the fundamental theorem of vector calculus, known as the Kelvin-Stokes theorem.

What are some common uses of curl(F)?

Curl(F) is commonly used in many fields, including physics, engineering, and mathematics. It is used to describe the behavior of fluid flow, electromagnetism, and even weather patterns. It is also used in computer graphics and image processing to simulate natural phenomena and create realistic visual effects.

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