Mathematical Formulas

Algebra
1 2
a x 2 + b x + c = 0 x = b ± b 2 4 a c 2 a
| a + b | | a | + | b | a n b n ( a n 2 ) ½ ( b n 2 ) ½
Trigonometry and Hyperbolics
1 2 3 4
Given a right triangle with sides a, b, and c and opposite angles A, B, and C, C being the right angle. Triangle Diagram
A + B + C = π
C = π 2
a = opp
b = adj
c = hyp
c 2 = a 2 + b 2
sin A = a c = opp hyp sin x = sin x
(i.e. sin is odd)
sin ( u + v ) = sin u cos v + cos u sin v
sin ( u v ) = sin u cos v cos u sin v
sin u + sin v = 2 sin ½ ( u + v ) cos ½ ( u v )
sin u sin v = 2 sin ½ ( u v ) cos ½ ( u + v )
cos A = b c = adj hyp cos x = cos x
(i.e. cos is even)
cos ( u + v ) = cos u cos v sin u sin v
cos ( u v ) = cos u cos v + sin u sin v
cos u + cos v = 2 cos ½ ( u + v ) cos ½ ( u v )
cos u cos v = 2 sin ½ ( u + v ) sin ½ ( u v )
sin 2 x + cos 2 x = 1 sin u sin v = ½ cos ( u v ) ½ cos ( u + v )
cos u cos v = ½ cos ( u v ) + ½ cos ( u + v )
sin u cos v = ½ sin ( u + v ) + ½ sin ( u v )
cos u sin v = ½ sin ( u + v ) ½ sin ( u v )
sin ( u + π 2 ) = cos u sin ( u π 2 ) = cos u cos ( u + π 2 ) = sin u cos ( u π 2 ) = sin u
tan A = a b = opp adj = sin A cos A
csc A = c a = 1 sin A sin π 6 = 1 2 = 0.5 sin π 4 = 2 2 = 0.707107 sin π 3 = 3 2 = 0.866025
sec A = c b = 1 cos A cos π 6 = 3 2 = 0.866025 cos π 4 = 2 2 = 0.707107 cos π 3 = 1 2 = 0.5
cot A = b a = 1 tan A = cos A sin A tan π 3 = 3 = 1.73205 tan π 6 = 3 3 = 0.577350 tan π 4 = 1
sinh x = x x 2 = ± ( cosh 2 x 1 ) ½ sinh x = sinh x
(i.e. sinh is odd)
sinh ( u + v ) = sinh u cosh v + cosh u sinh v
sinh ( u v ) = sinh u cosh v cosh u sinh v
sinh u + sinh v = 2 sinh ½ ( u + v ) cosh ½ ( u v )
sinh u sinh v = 2 cosh ½ ( u + v ) sinh ½ ( u v )
cosh x = x + x 2 = ( sinh 2 x + 1 ) ½ cosh x = cosh x
(i.e. cosh is even)
cosh ( u + v ) = cosh u cosh v + sinh u sinh v
cosh ( u v ) = cosh u cosh v sinh u sinh v
cosh u + cosh v = 2 cosh ½ ( u + v ) cosh ½ ( u v )
cosh u cosh v = 2 sinh ½ ( u + v ) sinh ½ ( u v )
cosh 2 x sinh 2 x = 1 sinh x + cosh x = x
sinh x cosh x = x
sinh u sinh v = ½ cosh ( u + v ) ½ cosh ( u v )
cosh u cosh v = ½ cosh ( u + v ) + ½ cosh ( u v )
sinh u cosh v = ½ sinh ( u + v ) + ½ sinh ( u v )
cosh u sinh v = ½ sinh ( u + v ) ½ sinh ( u v )
tanh x = sinh x cosh x = x x x + x = 2 x 1 2 x + 1 tanh x = tanh x
(i.e. tanh is odd)
csch x = 1 sinh x
sech x = 1 cosh x
coth x = 1 tanh x = cosh x sinh x
x = cos x + sin x sin x = x x 2 cos x = x + x 2
sinh x = sin x cosh x = cos x
Coordinate transformations
Spherical coordinates Cylindrical coordinates
x = r sin θ cos φ r = ( x 2 + y 2 + z 2 ) ½ x = r cos θ r = ( x 2 + y 2 ) ½
y = r sin θ sin φ θ = atan ( ( x 2 + y 2 ) ½ z ) y = r sin θ θ = atan y x
z = r sin θ φ = atan y z z = z z = z
Integration and Differentiation
1 2 3 4
x f ( g ( x ) ) = f ( g ( x ) ) g ( x ) q p q f ( x ) x = f ( q ) [p constant] p p q f ( x ) x = f ( p ) [q constant] a p q f ( x , a ) x = p q a [ f ( x , a ) ] x + f ( q , a ) q a f ( p , a ) p a
x ( u v ) = u v x + v u x x ( u v ) = 1 u u x u v 2 v x x ( u v w ) = u v w x + v w u x + u w v x
x ( u n v m ) = u n 1 v m 1 ( n v u x + m u v x ) x ( u n v m ) = u n 1 v m + 1 ( n v u x m u v x )
x ( x n ) = n x n 1 x n x = 1 n + 1 x n + 1 1 x x = ln x a x x = a x ln a
x ln x = 1 x ln x x = x ln x x x a x = a a x a x x = 1 a a x
x sin x = cos x sin x x = cos x x sinh x = cosh x sinh x x = cosh x
x cos x = sin x cos x x = sin x x cosh x = sinh x cosh x x = sinh x
x tan x = sec 2 x = 1 cos 2 x tan x x = ln sec x x tanh x = sech 2 x = 1 cosh 2 x tanh x x = ln cosh x
x asin a x = a ( 1 a 2 x 2 ) ½ asin a x x = x asin a x + 1 a ( 1 a 2 x 2 ) ½ x asinh a x = a ( a 2 x 2 + 1 ) ½ asinh a x x = x asinh a x 1 a ( a 2 x 2 + 1 ) ½
x acos a x = a ( 1 a 2 x 2 ) ½ acos a x x = x acos a x 1 a ( 1 a 2 x 2 ) ½ x acosh a x = a ( a 2 x 2 1 ) ½ acosh a x x = x acosh a x 1 a ( a 2 x 2 1 ) ½
x atan a x = a ( 1 + a 2 x 2 ) atan a x x = x atan a x ln ( 1 + a 2 x 2 ) 2 a x atanh a x = a ( 1 a 2 x 2 ) atanh a x x = x atanh a x + ln ( 1 a 2 x 2 ) 2 a
f : U V U n V m g : V D f ( x ) = ( f 1 x 1 ( x ) f 1 x 2 ( x ) f 1 x n ( x ) f 2 x 1 ( x ) f 2 x 2 ( x ) f 2 x n ( x ) f m x 1 ( x ) f m x 2 ( x ) f m x n ( x ) ) V g = U g f | det ( D f ) |
u v t = u v v u
Transforms
1 2 3
Fourier sine f ( x ) = n = 1 A n sin n π x l A n = 2 l 0 l f ( x ) sin n π x l x
Fourier cosine f ( x ) = 1 2 A 0 + n = 1 A n cos n π x l A n = 2 l 0 l f ( x ) cos n π x l x
Fourier f ( x ) = 1 2 A 0 + n = 1 ( A n cos n π x l + B n sin n π x l ) A n = 2 l l l f ( x ) cos n π x l x B n = 2 l l l f ( x ) sin n π x l x
Laplace F ( s ) = 0 s t f ( t ) t f ( t ) = 1 2 π 0 s t F ( s ) s
Series
1 2 3
i = 1 n i = n ( n + 1 ) 2 i = 1 n i 2 = n ( n + 1 ) ( 2 n + 1 ) 6 i = 1 n i 3 = n 2 ( n + 1 ) 2 4
i = 0 n x i = x n + 1 1 x 1 i = 0 x i = 1 1 x i = 1 x i = x 1 x
Linear Algebra
A 1 B 1 = ( B A ) 1 A T B T = ( B A ) T ( A 1 ) T = ( A T ) 1 Tr ( A B ) = Tr ( B A )
det ( A B ) = det ( A ) det ( B ) det ( A 1 ) = 1 det ( A ) det ( a 1 , 1 a 1 , 2 a 1 , n a 2 , 1 a 2 , 2 a 2 , n a n , 1 a n , 2 a n , n ) = ( 1 ) σ a 1 , i 1 a 2 , i 2 a n , i n
for all permutations i 1 , i 2 , i n of 1 , 2 , , n where σ is the degree of permutation
Characteristic equation det ( A λ I ) = 0
QM Linear Algebra
A = A1* A2* An* B = B1 B2 Bn B A = B1A1* B1A2* B1An* B2A1* B2A2* B2An* BnA1* BnA2* BnAn*
A B = A1* B1 + A2* B2 + + AN* BN A = A A = A φ ψ * = ψ φ
φ A ψ * = ψ A φ φ AB ψ * = ψ BA φ
For self-adjoint operators (Hermitian matrices) and lie rings:
AB = ABBA AA = 0 AB = BA ABC = ABC + BAC
ABC + BCA + CAB = 0 A = A ABC = ABC + ACB
n-dimensional spheres
Dimension (n) Shape (n-1)-Content
(Boundary)
n-Content
(Volume)
2 Circle Circumference = 2 π R Area = π R 2
3 Sphere Surface Area = 4 π R 2 Volume = 4 3 π R 3
4 4-Sphere 2 π 2 R 3 1 2 π 2 R 4
5 5-Sphere 8 3 π 2 R 4 8 15 π 2 R 5
6 6-Sphere π 3 R 5 1 6 π 3 R 6
even n-Sphere n ( n / 2 ) ! π n / 2 R n 1 1 ( n / 2 ) ! π n / 2 R n
odd n-Sphere 2 ( n + 1 ) / 2 ( n 2 ) ( n 4 ) 3 π ( n 1 ) / 2 R n 1 2 ( n + 1 ) / 2 n ( n 2 ) ( n 4 ) 3 π ( n 1 ) / 2 R n
volume
The volume of a k-dimensional parallelopiped in n specified by the vectors x 1 , x 2 , , x k
is given by v ( x 1 , x 2 , , x k ) = det ( X T X ) ½
where X is the n × k matrix with x 1 , x 2 , , x k as its columns
Let A be a n × n matrix. Let h : n n be the linear transformation h ( x ) = A x .
Let S be a rectifiable set in n , and let T = h ( S ) . Then v ( T ) = | det ( A ) | v ( S )

Earl Killian <earl at killian.com>
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