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Unformatted text preview: Total differential for y = f ( x ) For y=f(x), we have y = dy dx = df dx We can treat dx = x as an independent variable In the limit x 0, then dy dx = lim x y x If x finite, then dy is not exactly y Patrick K. Schelling Introduction to Theoretical Methods Total differential for z = f ( x , y ) and for many independent variables For a function of two variables, z = f ( x , y ), we can define the total differential dz = z x dx + y dy We can have dx and dy independent variables Then dz is the change in z along the tangent plane at x , y As with the previous example, dz is not equal to z for finite dx and dy For a function of many variables u = f ( x 1 , x 2 ,..., x N ), we define the total differential du = N X n =1 u x n dx n Patrick K. Schelling Introduction to Theoretical Methods Thermodynamics In thermodynamics, we have quantities that might pressure p , volume V , temperature T , entropy S , particle number N , and chemical potential . These are not all independent, so if we know p then V is determined, hence we describe quantities in terms of some subset of all the possible variables (In fact, p and V are conjugate pairs, as are T and S , and also N and .) The total energy...
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 Spring '03
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