# Why is the force of a magnetic field or a magnetic field not a vector field?

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.

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.
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.
• 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. – Reinstate Monica Oct 17 '17 at 18:20