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1. ### Grand Canonical Partition function

There is something on Wikipedia, but not quite what you want: http://en.wikipedia.org/wiki/Table_of_thermodynamic_equations Are you familiar with partial and total derivatives? That would be more useful to get what you want as opposed to a table.
2. ### Grand Canonical Partition function

1) Yes, they are essentially the same. Calculations are done in the same fashion. The grand canonical ensemble allows for the particle number of a system to change as well as energy. The canonical ensemble keeps particle number constant. In the microcanonical ensemble both particle number and...
3. ### I don't understand gauss's law for sheet surfaces

First, it's a closed surface, not an open one. I'm guessing that the plane you describe is infinite and has a uniform charge density. In that case the x-components cancel everywhere and there exists only a y-component. In fact, you can use Gauss's law to show that the electric field is constant...
4. ### Is this use of the grad function correct?

I totally missed that -x part. Thanks! The fact that the gradient points in the direction of greatest increase is the reasons for the negative sign in your case. The temperature is decreasing as you go away from the source. Without this negative sign, the resulting vector field would be...
5. ### Is this use of the grad function correct?

The gradient points in the direction of greatest increase. Since temperature is decreasing, you will want to take the negative gradient of your function. You also differentiated incorrectly. The derivative of x^(-2) is not 2^(-1). It's -2^(-3). Remember: d/dx(x^n) = nx^(n-1).