Fhd expression bothering me for weeks

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Suppose the magnetization of some substance depends on the field H and temperature, i.e. M(H,T) and we have the mathematical identity

[tex]\nabla \int _0 ^H M dH = M \nabla H + \int _0 ^H \nabla M dH[/tex]

then it is derived in Ferrohydrodynamics (fhd) by Rosensweig that

[tex]\nabla \int _0 ^H M dH = M \nabla H + \int _0 ^H \frac{\partial M}{\partial T} dH[/tex]

but what happend with the dependence of M on H?! Shouldn't the above include a term

[tex] \int _0 ^H \frac{\partial M}{\partial H} \nabla H dH[/tex]

or is there some reason this should vanish? Can someone help me out of my misery?
 

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