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Unformatted text preview: Multivariable Calculus The world is not one-dimensional, and calculus doesn’t stop with a single independent variable. The ideas of partial derivatives and multiple integrals are not too different from their single-variable coun- terparts, but some of the details about manipulating them are not so obvious. Some are downright tricky. 8.1 Partial Derivatives The basic idea of derivatives and of integrals in two, three, or more dimensions follows the same pattern as for one dimension. They’re just more complicated. The derivative of a function of one variable is defined as df ( x ) dx = lim Δ x → f ( x + Δ x )- f ( x ) Δ x (8 . 1) You would think that the definition of a derivative of a function of x and y would then be defined as ∂f ( x,y ) ∂x = lim Δ x → f ( x + Δ x,y )- f ( x,y ) Δ x (8 . 2) and more-or-less it is. The ∂ notation instead of d is a reminder that there are other coordinates floating around that are temporarily being treated as constants. In order to see why I used the phrase “more-or-less,” take a very simple example: f ( x,y ) = y . Use the preceding definition, and because y is being held constant, the derivative ∂f/∂x = 0 . What could be easier? I don’t like these variables so I’ll switch to a different set of coordinates, x and y : y = x + y and x = x What is ∂f/∂x now? f ( x,y ) = y = y- x = y- x Now the derivative of f with respect to x is- 1 , because I’m keeping the other coordinate fixed. Or is the derivative still zero because x = x and I’m taking ∂f/∂x and why should that change just because I’m using a different coordinate system? The problem is that the notation is ambiguous. When you see ∂f/∂x it doesn’t tell you what to hold constant. Is it to be y or y or yet something else? In some contexts the answer is clear and you won’t have any difficulty deciding, but you’ve already encountered cases for which the distinction is crucial. In thermodynamics, when you add heat to a gas to raise its temperature does this happen at constant pressure or at constant volume or with some other constraint? The specific heat at constant pressure is not the same as the specific heat at constant volume; it is necessarily bigger because during an expansion some of the energy has to go into the work of changing the volume. This sort of derivative depends on type of process that you’re using, and for a classical ideal gas the difference between the two molar specific heats obeys the equation c p- c v = R If the gas isn’t ideal, this equation is replaced by a more complicated and general one, but the same observation applies, that the two derivatives dQ/dT aren’t the same. James Nearing, University of Miami 1 8—Multivariable Calculus 2 In thermodynamics there are so many variables in use that there is a standard notation for a partial derivative, indicating exactly which other variables are to be held constant....
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This note was uploaded on 01/30/2012 for the course PHYS 315 taught by Professor Nearing during the Fall '08 term at University of Miami.
- Fall '08