By John C. Neu

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Although we haven’t presented a formula for σ, we can still figure out ∂v σ ∂p from a modified energy budget: This time the work done by the gas is ∂v σ payed for by a drop in its energy e per molecule, so de dv = −(rate of work) = −p . 6) Here, t is time and the time dependences are assumed to be sufficiently slow that this process is also “reversible”. 6), we have τ dv 3 dτ =− 2 dt v dt 36 Chapter 2. Differential calculus in many dimensions or 2 1 dv 1 dτ + = 0. τ dt 3 v dt It follows that 2 τ v 3 = c(σ) where c(σ) is a time independent constant that presumably depends on the similarly constant value of σ.

5) The determinant of this linear system for α, β is a2 +b2 = 0, since z = √a+ib = 0 means “at least one of a or b non-zero”. Geometrically, |z| := a2 + b2 , called the modulus of z, is the length of displacement from (0, 0) to (a, b) in 52 Chapter 3. Complex variables the complex plane. 5) for α, β is α = z −1 = a − bi z = 2. 7) is called the complex conjugate or conjugate of z. Geometrically, conjugation of z is reflection about the real axis. 2 ings of modulus and conjugate. As in real arithmetic, “z1 divided by z2 = 0 means (z1 )(z2−1 ) and we’ll denote it zz12 just like in real arithmetic.

Z1m z2N −m N ! (N − m)! m=0 N =0 ∞ 1 (z1 + z2 )N = ez1 +z1 . N! 17): Sum over m, n so m + n = N , and then sum over N . The third equality is the binomial expansion. 23). Chapter 3. Complex variables 65 Relatives of the exponential function in the complex plane Replacing x by z = x + iy in the real Taylor series for cos x and sin x gives the extensions of cosine and sine into the complex plane. For instance, cos z = 1 − z2 z4 + − ... 2! 24) and similarly for sin z. 24) and applying the binomial expansion to (x + iy)n is awkward.