(http://web.nchu.edu.tw/pweb/users/wtluh)
Some mathematical relations [Appendix C/p. 1015]
Derivatives:
(d/dx)(yz) = y(dz/dx) + z(dy/dx)
df(y)/dx = [df(y)/dy](dy/dx)
d(sin x)/dx = cos x
dey/dx = ey(dy/dx)
d(ln y)/dx = (1/y)(dy/dx)
Integrals: [definite integrals/p. 43]
 xn dx = xn+1/(n + 1) + I
 eax dx = (1/a) eax + I
 (ln ax) dx = x(ln ax) - x + I
Partial derivatives:
f(x,y) --- a function of more than one variable
df = (f/x)ydx + (f/y)xdy
If x, y and z are mutually dependent, like P, V, and
T, then we have
(a) (f/x)z = (f/x)y + (f/y)x(y/x)z
(b) (x/y)z = 1/[(y/x)z]
(c) (x/y)z = -(x/z)y(z/y)x
Eulers chain relation: (x/y)z(y/z)x(z/x)y = -1
(d) df = g(x,y)dx + h(x,y)dy is an exact differential if
(g/y)x = (h/x)y
If df is exact, then its integral between specified
limits is independent of the path.
Thermodynamical concepts, U, H, A, and G follow
this behavior.
Differential equations
b(dy/dx) + cy = 0: a first-order differential equation
a(d2y/dx2) + b(dy/dx) + c y =0 :
a second-order differential equation
a, b, and c may be functions of x
Solution: y = f(x)
a general solution
a particular solution: which follows initial or
boundary conditions
--1--a(dx/x) + bydy = 0, a and b are constants
 a(dx/x) +  bydy = 0
a ln x + b(y2/2) = I
--2--dy/dx = axy
rearranged: dy/y = axdx
integrated: ln y = (1/2)ax2 + A : a general solution
initial condition: y = y0 when x = 0 --> A = ln y0
we have a particular solution:
ln y = (1/2)ax2 + ln y0 or y = y0exp[ax2/2]
--3--dy/dt + ay = b; y(t=0) = 0
Use an integrating factor: exp[adt] = exp[at]
(dy/dt)exp[at] + ay exp[at] = b exp[at]
d{y exp[at]}/dt = b exp[at]
y exp[at] = (b/a) {exp[at] - 1}
y = (b/a) {1 - exp[-at]}
useful for describing the kinetics of consecutive
reactions
--4--2y/t2 = a(2y/x2): a partial differential equation
solved through a separation of variables procedure by
setting y = T(t)X(x)