Conservative force, prove.

Question

I've problem to understand the notation of this problem:

"Let x=xi+yj+zk; say if the force F=(x * k)x is conservative and find a potential function".

I do not understead how the vector multiplication has to be made.

Thanks.

Answer 1

In terms of $(x,y,z)$-coordinates one has ${\bf x}\cdot{\bf k}=z$. Interpreting the $*$ as scalar product the field ${\bf F}$ is therefore given by $${\bf F}(x,y,z):=(xz,yz,z^2)\ ,$$ and it is easy to see that ${\rm curl}\,{\bf F}$ is not identically ${\bf 0}$, whence ${\bf F}$ is not conservative.

Interpreting the $*$ as vector product makes ${\bf x}\times{\bf k}=(y,-x,0)$ a vector, whence $\bigl({\bf x}\times{\bf k}\bigr){\bf x}$ makes no sense as a vector, and neither does the triple product $\bigl({\bf x}\times{\bf k}\bigr)\cdot{\bf x}$.

Check the exact formulation of your problem!