Chapter 4: Partial differentiation

Transcription

1 Chapter 4: Partial differentiation It is generally the case that derivatives are introduced in terms of functions of a single variable. For example, y = f (x), then dy dx = df dx = f. However, most of the time we are dealing with quantities that are functions of several variables. For example, we usually want physical quantities in three dimensional space. For example, the electric field at each point in space might depend on x, y, and z, E E(x, y, z). Or, it might be convenient in some cases to use spherical coordinates, and then E E(r, φ, θ). We hence have to think about partial differentiation in physics.

2 Chapter 4 goals By the end of this chapter, you should be able to: Work with power series in two or more variables

3 Chapter 4 goals By the end of this chapter, you should be able to: Work with power series in two or more variables Use total differentials

4 Chapter 4 goals By the end of this chapter, you should be able to: Work with power series in two or more variables Use total differentials Use total differentials for approximation

5 Chapter 4 goals By the end of this chapter, you should be able to: Work with power series in two or more variables Use total differentials Use total differentials for approximation Use the chain rule for differentiation of a function of a function

6 Chapter 4 goals By the end of this chapter, you should be able to: Work with power series in two or more variables Use total differentials Use total differentials for approximation Use the chain rule for differentiation of a function of a function Use partial differentiation in maximum/minimum problems

7 Chapter 4 goals By the end of this chapter, you should be able to: Work with power series in two or more variables Use total differentials Use total differentials for approximation Use the chain rule for differentiation of a function of a function Use partial differentiation in maximum/minimum problems Use Lagrange multipliers in maximum/minimum problems with constraints

8 Chapter 4 goals By the end of this chapter, you should be able to: Work with power series in two or more variables Use total differentials Use total differentials for approximation Use the chain rule for differentiation of a function of a function Use partial differentiation in maximum/minimum problems Use Lagrange multipliers in maximum/minimum problems with constraints Make changes of variables, including using spherical and cylindrical coordinate systems

9 Chapter 4 goals By the end of this chapter, you should be able to: Work with power series in two or more variables Use total differentials Use total differentials for approximation Use the chain rule for differentiation of a function of a function Use partial differentiation in maximum/minimum problems Use Lagrange multipliers in maximum/minimum problems with constraints Make changes of variables, including using spherical and cylindrical coordinate systems Take derivatives of integrals

10 Introduction and notation For example, say we have z = f (x, y), we then need partial derivatives z x = f x = f x The key is we take derivative with respect to x, while keeping y fixed For example, z = f (x, y) = x 2 cos y, then z x = 2x cos y Then we can take another derivative, 2 z y x Does the order matter? Notice that 2 z x y We see 2 z. This is most often true. x y = 2 z y x = 2x sin y = 2x sin y. Sometimes we explicitly note that one variable is fixed (for example, in thermodynamics) ( ) z = 2x cos y x y

11 Power series in two variables We can take a Taylor series expansion about the point x = a, y = b of the function f (x, y) The power series can be represented by f (x, y) = a 00 +a 10 (x a)+a 01 (y b)+a 20 (x a) 2 +a 02 (y b) 2 +a 11 (x a)(y We see that a 00 = f (a, b) We can take partial derivatives with respect to x and y of the power series f x = f x = a a 20 (x a) + a 11 (y b) +... f y = f y = a a 02 (y b) + a 11 (x a) +... We evaluate the derivatives at x = a, y = b, and obtain the infinite Taylor series

12 Power series continued a 10 = f x (a, b), a 01 = f y (a, b), a 20 = 1 2 f xx(a, b), a 02 = 1 2 f yy(a, b), a 11 = f xy (a, b) = f yx (a, b), etc. We can express with h = x a and k = y b, f (x, y) = n=0 Where we mean y f (a, b) = f y (a, b) ( 1 h n! x + k ) n f (a, b) y

13 Example: Section 2, Problem 2 Find the Mclaurin series (expansion about x = 0,y = 0) of f (x, y) = cos(x + y) We see f x = f y = sin(x + y), and f xx = f yy = f xy = f yx = cos(x + y), etc. sin 0 = 0 and cos 0 = 1, so with h = x and k = y, f (x, y) = cos(x + y) n=0 ( 1 h n! x + k ) n f (a, b) y cos(x+y) = 1 1 2! (x 2 +2xy+y 2 )+ 1 4! (x 4 +4x 3 y+2x 2 y 2 +4xy 3 +y 4 )+...

14 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 y x 0 x If x finite, then dy is not exactly y

15 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 n=1 u x n dx n

16 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 U(S, V, N), so du = U S We define T = U S, p = U V U U ds + dv + V N dn, and µ = U N du = TdS pdv + µdn

17 Legendre transformations Construct a new function F = U TS, then df = du TdS SdT = SdT pdv + µdn We see that F (T, V, N), different independent variables! This is an example of a Legendre transformation Consider another example, G = F + pv, so dg = df + pdv + Vdp = SdT + Vdp + µdn The thermodynamics function G(T, p, N) is quite convenient because in experiments it is easy to control and measure T and p, as opposed to S and V (entropy and volume)