# lec4-1 - Total differential for y = f x • For y=f(x we...

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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 = Δ x as an independent variable • In the limit Δ x → 0, then dy dx = lim Δ x → Δ y Δ x • If Δ x finite, then dy is not exactly Δ 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 Δ 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 μ . • 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 μ .) • The total energy...
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lec4-1 - Total differential for y = f x • For y=f(x we...

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