Question about variational derivative

[Hassan2]

I have a question as to what notation should I use for the following derivation ( I have no background on calculus of variation ):

In linear media , the stored energy in a given volume is given by

$W_{mag}=\int_{V}\frac{1}{2}H.Bdv$

Now intuitively

1) $H=\frac{d W_{mag}}{dv d B}$But this is not the way they write it, instead one of the following notations are used:2) $H=\frac{\partial W_{mag}}{\partial B}$
3) $H=\frac{\delta W_{mag}}{\delta B}$ ( dv is missing)

What is the right notation for equation 1 ?Thanks.

Edit: In 1) it is assumed that B inside the infinitesimal volume (dv) changes to B+dB while outside dv it remained constant.

 

[Valerie Park]

The correct notation for equation 1 is 2), H=\frac{\partial W_{mag}}{\partial B}. This notation indicates that the partial derivative of W_{mag} with respect to B is taken, while keeping other variables (in this case, v) constant.