Mathematical Review

Algebra

Quadratic equation:

The solutions ofax^2 +bx+c= 0

arex=−b±

√b (^2) − 4 ac
2 a.
Logarithms and exponentials:
ln(ab) = lna+ lnb
ea+b=eaeb
lnex=elnx=x
ln(ab) =blna
Geometry, area, and volume
area of a triangle of baseband heighth =^12 bh
circumference of a circle of radiusr = 2πr
area of a circle of radiusr =πr^2
surface area of a sphere of radiusr = 4πr^2
volume of a sphere of radiusr =^43 πr^3
Trigonometry with a right triangle
sinθ=o/h cosθ=a/h tanθ=o/a
Pythagorean theorem:h^2 =a^2 +o^2
Trigonometry with any triangle
Law of Sines:
sinα
A

sinβ
B

sinγ
C
Law of Cosines:
C^2 =A^2 +B^2 − 2 ABcosγ
Properties of the derivative and integral
Letfandgbe functions ofx, and letcbe a con-
stant.
Linearity of the derivative:
d
dx
(cf) =cdf
dx
d
dx(f+g) =
df
dx+
dg
dx
The chain rule:
d
dx
f(g(x)) =f′(g(x))g′(x)
Derivatives of products and quotients:
d
dx
(fg) =
df
dx
g+
dg
dx
f
d
dx
(
f
g
)

f′
g
−
fg′
g^2
Some derivatives:
d
dxx
m=mxm− (^1) , except form= 0
d
dxsinx= cosx
d
d dxcosx=−sinx
dxe
x=ex d
dxlnx=
1
x
Linearity of the integral:
∫
cf(x) dx=c
∫
f(x) dx
∫
[f(x) +g(x)] =
∫
f(x) dx+
∫
g(x) dx
The fundamental theorem of calculus:
The derivative and the integral undo each other,
in the following sense:
∫b
a
f′(x) dx=f(b)−f(a)
1018 Chapter 14 Additional Topics in Quantum Physics