Why does Faraday's law involve a partial vs an ordinary derivative?

[Nick R]

Faraday's law is often stated in SI units as

$$\nabla x E(x,y,z,t) = \frac{\partial B(x,y,z,t)}{\partial t}$$

But x, y, z (or some other set of coordinates) are variables not functions and thus have no "t dependence".

So it would seem that the "total derivative" of B is the same as the partial:

$$\frac{dB}{dt}=\frac{\partial B}{\partial t}+\frac{\partial B}{\partial x}\frac{\partial x}{\partial t}+\frac{\partial B}{\partial y}\frac{\partial y}{\partial t}+\frac{\partial B}{\partial z}\frac{\partial z}{\partial t} = \frac{\partial B}{\partial t} + 0 + 0 + 0$$

It seems that there isn't any difference between the partial of B wrt to t and the "full derivative" of B wrt t.

The reason I'm concerned is the issue where you CANNOT do the following (or maybe sometimes... like in this case you can?):

$$\int^a_b\frac{\partial f(x,y)}{\partial x}dx = f(b,y) - f(a,y)$$

 

[Bhut]

It's the change in flux that matters so B can be constant and area change.