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I was looking for examples of conservative and non conservative vector fields and stumbled on the question Is magnetic force non-conservative? and in the answer section, it is stated the magnetic force cannot be a vector field so it cannot possibly be considered conservative. As I am learning the maths for divergence and curl naturally I wanted to find some examples to help my intuition and that is what leads me to ask for an explanation of the explanation I found on Stack Exchange.

Emilio Pisanty
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The magnetic force is not a vector field because vector fields are functions $\mathbf F:\mathbf r \mapsto \mathbf F(\mathbf r)$ that take a single vector value at each position, and the magnetic force $\mathbf F=q\mathbf v\times \mathbf B(\mathbf r)$ also depends on the velocity of the particle that's experiencing the force.

You could ask, instead, for the magnetic force on a particle with a given velocity, which itself may or may not depend on the position, i.e. $\mathbf v=\mathbf v(\mathbf r)$ (where that dependence may just be a constant), in which case you'll get a map $$ \mathbf F:\mathbf r\mapsto \mathbf F(\mathbf r)=q\mathbf v(\mathbf r)\times \mathbf B(\mathbf r) $$ that does have a unique value at each position and which therefore does define a vector field, of which you can ask e.g. whether it's conservative or not. However, until you actually define what velocity dependence $\mathbf v(\mathbf r)$ you want to take, none of those terms are applicable.

Emilio Pisanty
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  • If it's not a function then that would explain it. I accept the answer . Although not my original question you have peaked my curiosity. Are you allowed to say a little more as to why the magnetic force also depends on the velocity of the particle that's experiencing the force? Is q charge times velocity crossed by a position function? A guess on my part. Couldn't we say this about any force field and exclude them from being functions or is there something special about the magnetic field ? vs. say the electric or gravitational field. Thank you –  Oct 17 '17 at 16:02
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    @Sedumjoy note that in the examples you mentioned (electric, gravitational), the "force" fields must likewise not be functions as they depend on the electric charge and object mass respectively. Again, for a given object, these fields will be fully determined independently of the velocity.

    Note however that the underlying potential fields (which in the case of electric and gravitational are scalars, and magnetic, vector) are uniquely functions of position and so therefore are fields proper.

     – tusky_mcmammoth Oct 17 '17 at 21:49
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It is indeed a vector field in the sense that it associates a vector to every point in the space, except that the vector is in $T_p^*M\otimes T_pM$ rather than $T_pM$, where commonly used vectors live.

Sean D
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    In that case it would be more accurate to say that magnetic force is a tensor field. Saying that it is a vector field might technically be correct, but is misleading. Moreover, most of the time a vector field is defined to be a section of $TM$, which would actually make that explanation incorrect. – silvascientist Oct 17 '17 at 18:20
  • I feel like you're moving the goalposts, but fine, if you want this to be a old-school vector (section of the tangent bundle) just take the phase space to be the underlying space and it still works. – Sean D Oct 17 '17 at 22:29