Hi, I'm having trouble with the following question.

Q. Let p be a real constant and [itex]\mathop F\limits^ \to = \left( {yz^p ,x^p z,xy^p } \right)[/itex] be a vector field. For what value of p is the line integral

[tex]\int\limits_{C_2 }^{} {\mathop F\limits^ \to \bullet d\mathop s\limits^ \to } = 0[/tex]

Where C_2 is any closed path in R^2.

Firstly, how can C_2 be a path in R^2 when the vector field is '3D'? That doesn't seem to make sense in the context of the line integral. Assuming that C_2 is any closed path in R^3 then it should be sufficient to find the values of p so that curl F = 0.

I found [itex]curl\mathop F\limits^ \to = \nabla \times \mathop F\limits^ \to [/itex]

[tex]

= \left( {x\left( {p - 1} \right)y^{p - 1} - x^p ,y\left( {p - 1} \right)z^{p - 1} - y^p ,z\left( {p - 1} \right)x^{p - 1} - z^p } \right)

[/tex]

The answer is p = 1 but substituting p = 1 to what I found doesn't give the result curl F = 0. I've checked over my calculation a few times but I still can't see what's wrong with the curlf that I've computed. Can someone help me out?

Q. Let p be a real constant and [itex]\mathop F\limits^ \to = \left( {yz^p ,x^p z,xy^p } \right)[/itex] be a vector field. For what value of p is the line integral

[tex]\int\limits_{C_2 }^{} {\mathop F\limits^ \to \bullet d\mathop s\limits^ \to } = 0[/tex]

Where C_2 is any closed path in R^2.

Firstly, how can C_2 be a path in R^2 when the vector field is '3D'? That doesn't seem to make sense in the context of the line integral. Assuming that C_2 is any closed path in R^3 then it should be sufficient to find the values of p so that curl F = 0.

I found [itex]curl\mathop F\limits^ \to = \nabla \times \mathop F\limits^ \to [/itex]

[tex]

= \left( {x\left( {p - 1} \right)y^{p - 1} - x^p ,y\left( {p - 1} \right)z^{p - 1} - y^p ,z\left( {p - 1} \right)x^{p - 1} - z^p } \right)

[/tex]

The answer is p = 1 but substituting p = 1 to what I found doesn't give the result curl F = 0. I've checked over my calculation a few times but I still can't see what's wrong with the curlf that I've computed. Can someone help me out?

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