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MATHEMATICS REVISION NOTES:
1 Circles and Spheres
Circumference of a circle of radius R
C = 2 R
Area of a circle of radius R A = R^2
Surface Area of a sphere of radius R
A = 4 R^2
Volume of a sphere of radius R
V =
R^3
2 Trigonometry
sin = 2 sin
cos
cos = cos^2
sin^2
= 2 cos 2
1 = 1 2 sin 2
d d tan = sec^2 =
cos^2 d d cot = csc^2 =
sin^2
3 Integration
Z (^) a 0
x sin xdx = sin a a cos a
4 Spherical Polar Co ordinates
A p oint whose cartesian co ordinates are (x; y ; z ) can b e describ ed in terms of spherical p olar co ordinates (r; ; ) where x = r sin cos y = r sin sin z = r cos ;
or inverting, we have
r =
q x^2 + y 2 + z 2
= tan ^1
p x^2 + y 2 z
= tan ^1
(^) y x
5 Volume Integrals
In Cartesian Co ordinates (x; y ; z )
dV = dx dy dz
In spherical p olar co ordinates (r; ; )
dV = r 2 dr sin d d :
6 Di erential Equations
dy dx = ay
Solution y = y (0) exp( ax)
dy dx = ce bx^ ay
Particular solution: Search for a solution of the form
y = Ae bx^ :
x dy dx = by 2 We rewrite this as dy y 2 = b dx x Integrating b oth sides and adding the constant of integration, we get 1 y = b ln(x) + C
Taking the inverse y =
(C + b ln(x)) C is determined from the b oundary condition, y = y 0 when x = x 0 - we can write the solution as y =
1 =y 0 ln(x 0 ) + b ln(x)
This may b e rewritten in a simpler form as
y = y 0 1 + by 0 ln(x=x 0 )
7 Complex Numb ers
z = x + i y = jz j ei^ ;
where jz j =
q x^2 + y 2 ; tan = y x w =
z
x + i y
x i y x^2 + y 2 = jw j e i^ ;
where jw j =
p x^2 + y 2
