Discovering a Vector Field for ∇xF with Properties of Curl

[icesalmon]

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.

 

[Dick]

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?

 

[icesalmon]

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.

 

[Dick]

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.

 

[icesalmon]

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.

 

[Dick]

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.

 

[LCKurtz]

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.

 

[icesalmon]

yeah I didn't notice that myself in the question, my apologies. Thank you for your help!

 

[icesalmon]

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

 

[Dick]

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.

 

[HallsofIvy]

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".