§A1.1 偏导数：固定其余变量§A1.2 全微分：线性近似§A1.3 链式法则自测题 正文从第1章 起就大量使用带下标的偏导数，例如 ( ∂ S / ∂ U ) V , N (\partial S/\partial U)_{V,N} ( ∂ S / ∂ U ) V , N
§A1.1 偏导数：固定其余变量 设 f f f x , y x,y x , y 偏导数 是只让一个自变量变化、其余自变量保持不变时的变化率：
( ∂ f ∂ x ) y = lim h → 0 f ( x + h , y ) − f ( x , y ) h , ( ∂ f ∂ y ) x = lim k → 0 f ( x , y + k ) − f ( x , y ) k (A1.1) \left(\frac{\partial f}{\partial x}\right)_y = \lim_{h\to0}\frac{f(x+h,y) - f(x,y)}{h},\qquad \left(\frac{\partial f}{\partial y}\right)_x = \lim_{k\to0}\frac{f(x,y+k) - f(x,y)}{k} \tag{A1.1} ( ∂ x ∂ f ) y = h → 0 lim h f ( x + h , y ) − f ( x , y ) , ( ∂ y ∂ f ) x = k → 0 lim k f ( x , y + k ) − f ( x , y ) ( A1.1 ) 计算时把其余变量当作常数，按一元函数求导即可。例如 f = x 2 y f = x^2y f = x 2 y ( ∂ f / ∂ x ) y = 2 x y (\partial f/\partial x)_y = 2xy ( ∂ f / ∂ x ) y = 2 x y ( ∂ f / ∂ y ) x = x 2 (\partial f/\partial y)_x = x^2 ( ∂ f / ∂ y ) x = x 2 f x f_x f x f y f_y f y ∂ x f \partial_xf ∂ x f ∂ y f \partial_yf ∂ y f f x x f_{xx} f xx f x y = ∂ y ( ∂ x f ) f_{xy} = \partial_y(\partial_xf) f x y = ∂ y ( ∂ x f )
下标不能省略的原因。 数学书里自变量是固定的一组，下标往往省去；热力学里同一个量可以用不同的变量组来描述（S S S T , V T,V T , V T , p T,p T , p T T T f = x 2 y f = x^2y f = x 2 y x x x u = x y u = xy u = x y f = x u f = xu f = xu
( ∂ f ∂ x ) u = u = x y ≠ ( ∂ f ∂ x ) y = 2 x y \left(\frac{\partial f}{\partial x}\right)_u = u = xy\ \ne\ \left(\frac{\partial f}{\partial x}\right)_y = 2xy ( ∂ x ∂ f ) u = u = x y = ( ∂ x ∂ f ) y = 2 x y 两者都是"f f f x x x x x x y y y x y xy x y ( ∂ S / ∂ T ) V = C V / T (\partial S/\partial T)_V = C_V/T ( ∂ S / ∂ T ) V = C V / T ( ∂ S / ∂ T ) p = C p / T (\partial S/\partial T)_p = C_p/T ( ∂ S / ∂ T ) p = C p / T (1.18) ），二者之差乘以 T T T (1.20) 。两种偏导数之间的换算公式见 §A4.4 的 (A4.4) 。
§A1.2 全微分：线性近似 若 f f f
f ( x + d x , y + d y ) − f ( x , y ) = ( ∂ f ∂ x ) y d x + ( ∂ f ∂ y ) x d y + ( 比 d x 、 d y 高阶的小量 ) (A1.2) f(x+dx,\,y+dy) - f(x,y) = \left(\frac{\partial f}{\partial x}\right)_ydx + \left(\frac{\partial f}{\partial y}\right)_xdy + (\text{比 }dx\text{、}dy\text{ 高阶的小量}) \tag{A1.2} f ( x + d x , y + d y ) − f ( x , y ) = ( ∂ x ∂ f ) y d x + ( ∂ y ∂ f ) x d y + ( 比 d x 、 d y 高阶的小量 ) ( A1.2 ) 右边的线性部分称为 f f f 全微分 d f df df z = f ( x , y ) z = f(x,y) z = f ( x , y ) n n n d f = ∑ i ( ∂ f / ∂ x i ) d x i df = \sum_i(\partial f/\partial x_i)\,dx_i df = ∑ i ( ∂ f / ∂ x i ) d x i
热力学基本方程 (1.6) 、(1.7) 就是全微分。反过来，如果已知 d f = A d x + B d y df = A\,dx + B\,dy df = A d x + B d y A = ( ∂ f / ∂ x ) y A = (\partial f/\partial x)_y A = ( ∂ f / ∂ x ) y B = ( ∂ f / ∂ y ) x B = (\partial f/\partial y)_x B = ( ∂ f / ∂ y ) x (1.8) 和 §1.6 中从 d F dF d F S S S p p p μ \mu μ 附录 A2 的内容。
§A1.3 链式法则 沿一条曲线。 若 x = x ( t ) x = x(t) x = x ( t ) y = y ( t ) y = y(t) y = y ( t ) g ( t ) = f ( x ( t ) , y ( t ) ) g(t) = f(x(t),y(t)) g ( t ) = f ( x ( t ) , y ( t ))
d g d t = ∂ f ∂ x d x d t + ∂ f ∂ y d y d t (A1.3) \frac{dg}{dt} = \frac{\partial f}{\partial x}\frac{dx}{dt} + \frac{\partial f}{\partial y}\frac{dy}{dt} \tag{A1.3} d t d g = ∂ x ∂ f d t d x + ∂ y ∂ f d t d y ( A1.3 ) 记忆方法：把 (A1.2) 两边"除以 d t dt d t
换一组变量。 若 x = x ( u , v ) x = x(u,v) x = x ( u , v ) y = y ( u , v ) y = y(u,v) y = y ( u , v )
( ∂ f ∂ u ) v = ∂ f ∂ x ( ∂ x ∂ u ) v + ∂ f ∂ y ( ∂ y ∂ u ) v (A1.4) \left(\frac{\partial f}{\partial u}\right)_v = \frac{\partial f}{\partial x}\left(\frac{\partial x}{\partial u}\right)_v + \frac{\partial f}{\partial y}\left(\frac{\partial y}{\partial u}\right)_v \tag{A1.4} ( ∂ u ∂ f ) v = ∂ x ∂ f ( ∂ u ∂ x ) v + ∂ y ∂ f ( ∂ u ∂ y ) v ( A1.4 ) （固定 v v v u u u u u u (A1.3) 。）
正文中的几个例子。
§1.5 ：总熵 S = S 1 ( U 1 ) + S 2 ( U − U 1 ) S = S_1(U_1) + S_2(U - U_1) S = S 1 ( U 1 ) + S 2 ( U − U 1 ) U 1 U_1 U 1 S 2 S_2 S 2 U 2 = U − U 1 U_2 = U - U_1 U 2 = U − U 1 d U 2 / d U 1 = − 1 dU_2/dU_1 = -1 d U 2 / d U 1 = − 1 d S / d U 1 = ∂ S 1 / ∂ U 1 − ∂ S 2 / ∂ U 2 dS/dU_1 = \partial S_1/\partial U_1 - \partial S_2/\partial U_2 d S / d U 1 = ∂ S 1 / ∂ U 1 − ∂ S 2 / ∂ U 2 §1.6 ：换用 β = 1 / ( k B T ) \beta = 1/(k_{\mathrm B}T) β = 1/ ( k B T ) ∂ / ∂ β = ( d T / d β ) ∂ / ∂ T = − k B T 2 ∂ / ∂ T \partial/\partial\beta = (dT/d\beta)\,\partial/\partial T = -k_{\mathrm B}T^2\,\partial/\partial T ∂ / ∂ β = ( d T / d β ) ∂ / ∂ T = − k B T 2 ∂ / ∂ T (1.15) 。§1.9 的欧拉定理：对 f ( λ x 1 , … , λ x n ) f(\lambda x_1,\dots,\lambda x_n) f ( λ x 1 , … , λ x n ) λ \lambda λ λ x i \lambda x_i λ x i λ \lambda λ x i x_i x i (A1.3) 得 ∑ i x i ∂ f / ∂ ( λ x i ) \sum_ix_i\,\partial f/\partial(\lambda x_i) ∑ i x i ∂ f / ∂ ( λ x i ) 附录 A9 ）。§27.3 麦克斯韦 1860 年的推导：Ψ ( v x 2 + v y 2 + v z 2 ) \Psi(v_x^2+v_y^2+v_z^2) Ψ ( v x 2 + v y 2 + v z 2 ) v x v_x v x v 2 v^2 v 2 v x v_x v x 2 v x 2v_x 2 v x 2 v x Ψ ′ ( v 2 ) 2v_x\Psi'(v^2) 2 v x Ψ ′ ( v 2 )
常见错误。 把"f f f x , y x,y x , y x x x f f f x x x f ( x , y ) f(x,y) f ( x , y ) y = y ( x ) y = y(x) y = y ( x ) d f / d x = ∂ f / ∂ x + ( ∂ f / ∂ y ) d y / d x df/dx = \partial f/\partial x + (\partial f/\partial y)\,dy/dx df / d x = ∂ f / ∂ x + ( ∂ f / ∂ y ) d y / d x
自测题
设 f = x 2 + y 2 f = x^2 + y^2 f = x 2 + y 2 x = r cos θ x = r\cos\theta x = r cos θ y = r sin θ y = r\sin\theta y = r sin θ (A1.4) 求 ( ∂ f / ∂ r ) θ (\partial f/\partial r)_\theta ( ∂ f / ∂ r ) θ ( ∂ f / ∂ θ ) r (\partial f/\partial\theta)_r ( ∂ f / ∂ θ ) r ［答：2 r 2r 2 r 0 0 0 f = r 2 f = r^2 f = r 2
理想气体 p V = N k B T pV = Nk_{\mathrm B}T p V = N k B T ( ∂ p / ∂ T ) V (\partial p/\partial T)_V ( ∂ p / ∂ T ) V ( ∂ p / ∂ T ) ρ (\partial p/\partial T)_{\rho} ( ∂ p / ∂ T ) ρ ρ = N / V \rho = N/V ρ = N / V ( ∂ p / ∂ T ) V / T (\partial p/\partial T)_{V/T} ( ∂ p / ∂ T ) V / T ［答：N k B / V Nk_{\mathrm B}/V N k B / V ρ k B \rho k_{\mathrm B} ρ k B p = N k B / ( V / T ) p = Nk_{\mathrm B}/(V/T) p = N k B / ( V / T ) V / T V/T V / T T T T 0 0 0
设 g ( t ) = f ( t , t 2 ) g(t) = f(t,t^2) g ( t ) = f ( t , t 2 ) f ( x , y ) = x y f(x,y) = xy f ( x , y ) = x y (A1.3) 求 g ′ ( t ) g'(t) g ′ ( t ) ［答：y + x ⋅ 2 t = 3 t 2 y + x\cdot2t = 3t^2 y + x ⋅ 2 t = 3 t 2
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