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For any state function of n independant variables $F(x_1, \dots ,x_n)$, $$dF(x_1, \ldots ,x_n)=\frac{\partial F}{\partial x_1} dx_1 + \cdots + \frac{\partial F}{\partial x_n} dx_n$$ where each partial derivative are each taken by holding the other state variables constant. What do you think about this more compact notation with the del operator, $$dF(x_1, \ldots ,x_n)=\nabla F \cdot dx$$ where $x = (x_1, \ldots ,x_n)$ ?

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It's an abuse of notation, though one which would probably be understood. –  Zhen Lin Sep 23 '11 at 2:53
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Why do you want to introduce a new notation for $dF$? –  Mariano Suárez-Alvarez Sep 23 '11 at 3:31
    
I want to isolate dx and to use vector notations to get ride of the "..." or sum in all my calculations. –  ths1104 Sep 23 '11 at 3:49
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But you can use $dF$! There is nothing added in you $\nabla F.dx$ that's not in $dF$, and the latter has the nice property of being actually a meaningful notation and not an abuse thereof :) –  Mariano Suárez-Alvarez Sep 23 '11 at 4:00
    
My \$0.02: I like the idea behind "$dF = \nabla F \cdot dx$" because it has the same spirit as the $1$-dimenional analog: $dy = f'(x) dx$ (where of course $y = f(x)$. The lhs is a notation, while the rhs is a computation. –  Shaun Ault Sep 23 '11 at 4:04

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