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Unformatted text preview: Math 18.02 (Spring 2009): Lecture 14 Non-independent variables. Partial differential equations March 6 Reading Material: From Simmons : 19.6. From Course Notes : N. Last time: Lagrange Multipliers Today: Non-independent variables. Partial differential equations. 2 Non-independent variables If you go to OCW and look at Lecture 21 of 3.00: Thermodynamics of Materials, you will see that the title of this class is ”Mathematics of Thermodynamics”. In the middle of the lecture notes relative to this lecture you will find an equation involving differentials that looks like this: dU = TdS- PdV + C X i =1 μ i dN i , (2.1) where all the functions are measuring the following quantities relative to a gas: T = Temperature U = Internal energy S = Entropy V = Volume N i = Numbers of molecules of type i . The relationship among these quantities described by (2.1) is a combination of the First and Second Laws of Thermodynamics that I am not going to state here. The point though is that if one thinks about T, U, S, V and N i , for i = 1 , ..., C as variables, then (2.1) is a way to describe how they are related. From the definition of differential we can deduce the partial derivatives of a function. In fact recall that for a function of two variables z = f ( x, y ) , where x and y are the independent variables, we have the differential dz = f x dx + f y dy....
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