Component of the velocity gradient part of Vlasov equation in cylindrical coordinate

I am stuck with a problem in expanding the velocity gradient part of the Vlasov equation in cylindrical basis. It is also applicable to the spatial gradient part as well. For example here I consider only the velocity gradient part, which could be written as,

$$vec{A}(vec{r},vec{v},t).nabla_{v}f$$

where, $A=(vec{E}+vec{v}timesvec{B})$, with $vec{E}$ is the electric field, $vec{v}$ is the velocity, $vec{B}$ is the magnetic field and $f$ is the distribution function. A naïve expansion of this term in cylindrical coordinate gives (see for reference),

$$vec{A}(vec{r},vec{v},t).nabla_{v}f=A_{perp}frac{partial f}{partial v_{perp}}+frac{A_{phi}}{v_{perp}}frac{partial f}{partial phi}+A_{||}frac{partial f}{partial v_{||}}$$

Also it is known that for Vlasov equation $vec{A}$ can be taken inside the velocity gradient making it a divergence equation as,

$$A(vec{r},vec{v},t).nabla_{v}f=nabla_{v}.(vec{A}f)$$

whose expansion in cylindrical coordinate gives (see for reference),

$$A(vec{r},vec{v},t).nabla_{v}f=nabla_{v}.(vec{A}f)=frac{1}{v_{perp}}frac{partial (v_{perp}A_{perp}f)}{partial v_{perp}}+frac{1}{v_{perp}}frac{partial (A_{phi}f)}{partial phi}+frac{partial (A_{||}f)}{partial v_{||}}$$

Now these two expansion are same. So,

$$A_{perp}frac{partial f}{partial v_{perp}}+frac{A_{phi}}{v_{perp}}frac{partial f}{partial phi}+A_{||}frac{partial f}{partial v_{||}}=frac{1}{v_{perp}}frac{partial (v_{perp}A_{perp}f)}{partial v_{perp}}+frac{1}{v_{perp}}frac{partial (A_{phi}f)}{partial phi}+frac{partial (A_{||}f)}{partial v_{||}}$$

The last term from both the sides are equal as $vec{E}$ does not depends on $vec{v}$ and $(vec{v}timesvec{B})_{||}$ will not depend on $v_{||}$. But for the second term in right hand side we have,

$$frac{1}{v_{perp}}frac{partial (A_{phi}f)}{partial phi}=frac{A_{phi}}{v_{perp}}frac{partial f}{partial phi}+frac{f}{v_{perp}}frac{partial A_{phi}}{partial phi}$$

where the first term resembles with the second term of the left hand side of the above equation. But the next term in $frac{1}{v_{perp}}frac{partial (A_{phi}f)}{partial phi}$ is a problem.

Now with the first term in the right hand side of the equation,

$$frac{1}{v_{perp}}frac{partial (v_{perp}A_{perp}f)}{partial v_{perp}}=A_{perp}frac{partial f}{partial v_{perp}}+ffrac{partial A_{perp}}{partial v_{perp}}+frac{A_{perp}f}{v_{perp}}$$

So my question is, whether $ffrac{partial A_{perp}}{partial v_{perp}}+frac{A_{perp}f}{v_{perp}}+frac{f}{v_{perp}}frac{partial A_{phi}}{partial phi}=0$?

Any help would be highly appreciated.