Formulas: Flashcards
Complex Analysis: Basic Formula
Z=x+yi
Complex Analysis: Polar formula
z=rcos@+rsin@i
z=r<@
Complex Analysis: Euler’s
z=re^@i
Complex Analysis: r (modulus)
|z|=sqr root of (x^2 + y^2)
Complex Analysis: Theta @ (Argument)
@ = arctan (y/x)
Complex Analysis: De moivre’s
Caltech: CMPLX mode
|z|^n <(n x arg (z)) =
Number System: Binary, Bin
0 and 1
Number System: Decimal, Dec
0 to 9
Number System: Octagecimal, Oct
0 to 7
Number System: Hexadecimal, Hex
0 to 9 and A to F
Linear Equation
ax+b
Vieta’s Formula
If X1 and X2 are roots of quadratic equation
Vieta’s Formula: Sum of Roots
X1 + X2 = -B/A
Vieta’s Formula: Product of Roots
X1X2=C/A
Binomial Theorem: (x+y)^n
Formula for rth term
nCn-r+1(x^(n-r+1))(y^(r-1))
Binomial Theorem: (x+y)^n
Formula for Sum of coefficient
f(1,1)-f(0,0)
Binomial Theorem: (x+y)^n
Formula for Sum of Exponents
(SOEi)(n+1C2)
Binomial Theorem: (x+y)^n
Formula for Number of Terms
n+1
Binomial Theorem: (x+y)^n
Formula for middle term
r=(n/2) +1
Multinomial Theorem:
(x^alpha + y^beta + z^ gamma) ^n
Formula for rth term
[n!/(a! x b! x c!) ] (x^a y^b z ^c)
Multinomial Theorem:
(x^alpha + y^beta + z^ gamma) ^n
Formula for Sum of Exponents
(SOEi) (n+t-1Ct)
Multinomial Theorem:
(x^alpha + y^beta + z^ gamma) ^n
Formula for number of terms
n+t-1Ct-1 (t no. Of terms)
Arithmetic Sequence: Stat 2
An
An=A1+(n-1)d
Arithmetic Sequence: Stat 2
Sn
Sn=(n/2)(2A1+ (n-1)d)
Geometric Sequence: Stat 6
An
An=A1(r^(n-1))
Geometric Sequence: Stat 6
Sn
Sn=A1(1-r^n)/1-r
Fibonacci Sequence: add 2 numbers before it
Golden ratio, Phi Formula
Phi=(1+sqr root of 5)/2
Fibonacci Sequence: add 2 numbers before it
Silver ratio, Gamma Y Formula
Gamma, Y=(1- sqr root of 5)/2
Fibonacci Sequence: add 2 numbers before it
Rth Term formula
(Phi^n - Y^n)/sqr root of 5
Lucas Sequence, L
Rth term:
Phi^n +Y^n
i in matrix (Aij)
Row
j in matrix (Aij)
Column
Clock Problem: Stat 2 Caltech
x|y
0|T(-30deg)
60|T(-30deg) +330
Clock Problem: First
-Theta
Clock Problem: 2nd
Theta
Clock Problem: 3rd
- theta + 360
Clock Problem: 4th
Theta + 360
Clock Problem: 5th
-theta + 720
Clock Problem: 6th
Theta + 720
Clock Problem: Together/Coincide
Theta = 0
Clock Problem: Perpendicular
90deg
Clock Problem: Align
180 deg
Rowing Boat Problem: d=v x t
V is
Still water
Rowing Boat Problem: d=v x t
w is
Current speed
Rowing Boat Problem: d=v x t
Upstream
v-w
Rowing Boat Problem: d=v x t
Downstream
v + w
Plane Problem: d=v x t
Tail wind
Same direction of wind & Plane
Plane Problem: d=v x t
Downwind
Opposite direction of wind and plane
Trains and tunnels Problem:
La + Lb= (Va +- Vb) t
+ for opposite
- for same direction
Diophantine Equation
2 equation 3 unknowns
TRIGO: Arc length formula
s=r@
TRIGO: Angular Velocity Formula
v=wr
Csc@
1/sin@
Sec@
1/cos@
Cot@
1/tan@
Pythagorean:
sin2@ +cos2@ = 1
tan2@ + 1 = sec2@
Cot2@ + 1 = csc2 @
Double angle: sin2@
2sin@cos@
Double angle: cos2@
Cos2@ - sin2@
Half Angle: Sin (1/2 @)
Sqr root of (1/2)(1-cos@)
Half Angle: Cos (1/2 @)
Sqr root of (1/2)(1+cos@)
Half Angle: tan (1/2 @)
Sqr root of (1-cos@)/(1+cos@)
Negative angles: sin(-@)
-sin@
Negative angles: cos(-@)
Cos@
Negative angles: tan(-@)
-tan@
Sum & Difference: cos (theta +- Beta)
Cos@ cosB -+ sin@ sinB
Sum & Difference: sin (theta +- Beta)
Sin@ cos B +- cos@ Sin B
Sine Law: 2 angles are known
(a/sinA) = (b/sinB) = (c/sinC)
Cosine Law:
a^2 = b^2 + c^2 -2bc cos A
Heron’s formula: for oblique triangles
Semi perimeter formula
s = (a+b+c)/2
Heron’s formula: for oblique triangles
Area of Triangle formula
A= sqr root of s(s-a)(s-b)(s-c)
For Cyclic Quadrilateral: Brahmagupta’s equation
Semi Perimeter Formula
s = (a+b+c+d)/2
For Cyclic Quadrilateral: Brahmagupta’s equation
Area Formula
A = sqr root of (s-a)(s-b)(s-c)(s-d)
Spherical Triangle
A= piR^E/180
Epsilon, E = A + B + C - 180
Napier’s Triangle for Right Spherical
SIN-TAAD
Sin(B)= tanc tan a
Napier’s Triangle for Right Spherical
SIN-COOP
Sin(a) = cosc cosA
Cosine Law for Oblique Spherical Triangle
cosa = cosbcosc + sinbsinc cos A
Sine law for oblique spherical Triangle
(sina/sinA) = (sinb/sinB) = (sinc/sinC)
Terrestrial Sphere: horizontal line
Equator
Terrestrial Sphere: parallel to equator
Latitude
Terrestrial Sphere: perpendicular to equator
Longitude
1 deg is equal to blank nm
60 nautical miles
1 deg is equal to blank mins
4 mins
1 nautical miles is equal to blank meters
1852 meters
1 knots is equal to
nautical miles/hr
Which quadrant is sin cos tan positive?
1st quadrant
Which quadrant is sin positive?
2nd quadrant
Which quadrant is cos positive?
4th quadrant
Which quadrant is tan positive?
3rd quadrant
Angle less then 90 deg
Acute
Angle more than 90 deg
Obtuse
Sum of 90 deg
Complement angle
Sum of 180 deg
Supplement angle
Sum of all angles of triangle are equal to
180 deg
Vectors has
Both magnitude and direction
Components of vectors
A= axi +ayj+azk
Magnitude of vectors
|A| = sqr root of ax2 + ay2 +az2
Unit vector
a = component/magnitude
Dot product is
Scalar
Cross product is
Vector
Dot product A • B =
|A| |B| cos @
Cross product |A x B| =
|A| |B| sin @
Area by three coplanar points
A plane = 1/2 det (V1 x V2)
Area by three coplanar points if 3D
1/2 x Abs (Vct A x Vct B)
Distance formula
d= sqr root of (X2-X1)2 + (Y2-Y1)2
Distance formula Caltech
Cmplx mode
Abs((X1+Y1i)-(X2+Y2i))
Slope, m
m= Y2-Y1/X2-X1 = tan@
Inclination Caltech
CMPLX
arg ((X1+Y1i) - (X2+Y2i))
Angle Between 2 Lines Caltech
CMPLX
arg (X1+Y1i/X2+Y2i)
Answer must be acute!
Point to a line Caltech
Comp
Ax+By+C/sqr root of A2 + B2
Calc X and Y = Xo,Yo (Point)
Distance Between 2 Parallel lines
d= C2-C1/sqr root of A2 + B2
Forms of a line:
Standard form
Ax+By+C=0
Forms of a line:
Slope-intercept form
y=mx+b
Forms of a line:
Point-slope form
y-yo=m(x-xo)
Forms of a line:
Two-point form
y-y1 = (y2-y1/x2-x1)(x-x1)
Forms of a line:
Intercept form
x/a + y/b = 1
Conic Sections Formula
Ax2 + Bxy + Cy2 + Dx + Ey + F = 0
Locus of points that are equidistant from a fixed point/center
Circle
Circle General Form
Ax2 + Cy2 + Dx + Ey + F = 0
A and C should be equal
Circle (h,k) formula of h
h= -D/2A
Circle (h,k) formula of k
k= -E/2A
Circle Standard Form
(x-h)2 + (y-k)2 = r2
Formula of Circle’s Radius from general form
r= sqr root of (D2 + E2 -4AF)/4A2
Locus of points that are equidistant from a fixed point (focus) and directrix (line)
Parabola
Parabola Vertical Formula
4p(y-k)=(x-h)2
Parabola Horizontal Formula
4p(x-h)=(y-k)2
Latus rectum of Parabola
4f (f is focus to vertex)
Locus of points such that the sum of its distance from 2 fixed points (foci) is constant
Ellipse
Area of ellipse
A=pi(ab)
Vertical Ellipse Formula
(y-k)2/a2 + (x-h)2/b2 = 1
Horizontal Ellipse Formula
(x-h)2/a2 + (y-k)2/b2 = 1
Ellipse eccentricity
e = c/a < 1
Ellipse Latus Rectum
2b2/a
Ellipse Flatness
f=a-b/a
Ellipse apogee
a + c
Ellipse directrix
a/e
Locus points such that the difference of the distance between two fixed points (foci) is at constant 2a
Hyperbola
Ellipse general form/length
b2 + c2 = a2
Hyperbola General form
b2 = c2 -a2
Axis in hyperbola that passes two foci
Transverse axis
Axis in hyperbola that is perpendicular to transverse
Conjugate axis
Axis in hyperbola formed by transverse and conjugate
Auxiliary rectangle
Hyperbola Vertical Formula
(y-k)2/a2 - (x-h)2/b2 = 1
Hyperbola Horizontal Formula
(x-h)2/a2 - (y-k)2/b2 = 1
Hyperbola eccentricity
e= c/a
Hyperbola Latus Rectum
2a^2/b
Caltech for Parabola
Stat 3 (3 points, 1 point is mirrored)
Caltech of Ellipse/Hyperbola
Stat 2 (2 points but square the other)
Permutations: Circular w/ flip
p= (n-1)!/2
Permutations: Circular w/ no flip
P=(n-1)!
Permutation repetition
P=n!/(k1! x k2! x k3!…)
Permutation Alteration (equal number per group)
P= g! (a!)^g (g is number of groups)
Permutation Alteration (not equal number per group)
P=a!b!
Permutation proximity
P= (n+1) x (n!p!)
p- together
n- not together
Probability Formula
of Favorable outcomes/ total outcomes
Law of total probability
P(A U B) = P(A) + P(B) - P(AŪB)
If P(AŪB) = 0 what does it mean?
It means events are mutually exclusive
2 types Discreet Probability
Binomial Probability (n<20) and Poisson Probability (n>20)
Binomial Probability Formula
P(x) = nCx (P)^x (1-P)^(n-x)
Poisson Probability Formula
P(x) = (e^-np (np)^x)/x!
n= # of trials p= success rate x= interval
Geometric Mean
Xg= nth root of X1X2X3…Xn
Mean of two middle values
Median
Value that appears frequently
Mode
Perimeter
P=sn
Interior angle
Theta= 180(n-2)/n
Sum of interior angles
180(n-2)
Diagonals
D=n(n-3)/2
Area
(ns^2)/4tan(180/n)
Apothem, a
s/2tan(180/n)
Volume of Cylinder
V=pi r^2 h
Lateral Surface area of cylinder
2 pi r h
Volume of cone
1/3 pi r^2 h
Slant height of cone
Sqr root of (h^2 + r^2)
Lateral Surface area of cone
Pi r Sl
Total surface area of pyramid
TSA=Area of B + LSA
Lateral Surface area of Pyramid
1/2 (P)(SI)
Volume of Pyramid
1/3 (Area of Base) (h)
Volume of Sphere
4/3 pi r^3
Surface area of Sphere
4pir^2
For spherical segment of 1base
(Pih^2/3)(3r-h)
If two solids have the same height and cross sectional area at every level then they have the same volume
Cavalieri’s principle
Area of Polygon Inscribed in a Circle
(1/2)(nr^2 sin360/n)
Area of Polygon circumscribed on a Cirle
nr^2 tan180/n
Perimeter of a parallelogram
2a + 2b
Area of a parallelogram
Bh=absin@=1/2 (d1d2)(sin @)
2 diagonals of a parallelogram
p^2 = a^2 + b^2 -2abcos@ q^2 = a^2 + b^2 -2abcos (180-@)
Rhombus Perimeter
4b
Rhombus area
1/2 (d1d2)
Trapezoid Area
h(b1 + b2)/2
Area of a circle
Pi r ^2
Circumference of a circle
2 pi r
Sector area of a circle
1/2 (theta) (r^2)
Should be in rad
Segment (zone) area of a circle
1/2 r^2 (@-sin@)
Area of a triangle
1/2 bh
Incenter (circle on a triangle)
r=area of traingle/semi perimeter
Circumcenter (triangle on a circle)
r= abc/4(Area of a triangle)
Type of polyhedra that has two identical faces that are parallel to each other
Prisms
Rectangular parallelipiped, LSA
2lh + 2wh
Rectangular parallelipiped, TSA
2lh + 2wh +2lw
Rectangular parallelipiped, Volume
Lwh
Area of a quadrilateral
A= sqr root of (s-a)(s-b)(s-c)(s-d)-abcdcos^2 @
Lune Surface Area
(Pi r^2 theta)/90
Wedge Volume
Pi r ^3 theta / 270
Spherical sector
V=(pi/6)(h)(3a^2 +3b^2 +h)
Spheroids Prolate (rotate on major axis)
4/3 pi a b^2
Spheroids Oblate (rotate on minor axis)
4/3 pi a^2 b
Frustum Volume
1/3 h (B1 + B2 + sqr root of B1B2)
Truncated Cylinder Formula
Pi r^2 [(h1 + h2)/2]
Truncated Prism Formula
Area of base x [(h1+h2 +…+hn)/n]
Platonic Solids: Tetrahedron
(Sqr root of 2 /12) (a^3)
Platonic Solids: Hexahedron
a^3
Platonic Solids: octahedron
(Sqr root of 2 / 3) (a^3)
Platonic Solids: Dodecahedron
7.66a^3
Platonic Solids: Hexahedron (20)
2.18a^3
Paraboloid Volume
Pi/8 d^2 h
Paraboloid Area
2pi/3 d h
Torus Volume
2pi^2 R r^2
Torus Area
4pi^2 R r
Pappus’ No. 1
SA= (2 pi R) (C)
Pappus’ No. 2
V= 2piR (area)
Hydrostatic Pressure and Force Formula
P= rho g h / gc
Centroid Technique
F= gamma hc A
Derivative of a^u
a^u (ln a)
Derivative of log(a) u
1/ulna
Largest rectangle inscribed in a triangle
x=b/2 y=h/2
Sector of area but minimum perimeter
A sec= (1/2) theta r^2 (in rad)
s=r theta =2r
Theta =2 rad
r= sqr root of A
Rectangle to be fenced at 3 sides with given area at min. Perimeter
x=2y
P=4y
Area=2y^2
Largest square inscribed in a circle
a=r sqr root of 2
Largest rectangle inscribed in a semi circle
a=r sqr root of 2
Largest rectangle inscribed in a triangle w/ one side lying on the base of the triangle
x=b/2
y=h/2
Area of rectangle = 1/2 area of triangle
Triangle with 2 equal sides
Isosceles traingle
Max light on norman window
x=2y
h=x
P=x(2+pi/2)
Longest segment that can fit on a perpendicular intersection
L= (a^2/3 + b^2/3)^3/2
Sweet spot
x= sqr root of y1y2
Location of a stake on the ground to minimize wire length
x=dh1/h1+h2
L1 + L2 = (sqr root of h1^2 + x^2) + (sqr root of h2^2 + (d-x)^2)
Maximum Cylinder Inscribed in a Cone
h = (1/3)H
r=2/3 R
Cylinder on a Sphere
V of sphere = (4/ cube root of 27) (pi r^3)
V of cylinder = (4/ sqr root of 27) (pi r^3)
Integral of sin x
-cos x
Integral of cos x
Sin x
Integral of tan x
ln(secx)
Integral of cot x
ln(sinx)+c
Integral of sec x
ln|secx + tanx| + c
Integral of csc x
ln|cscx - cotx| + c
Integral of sec^2 x
tan x
Integral of csc^2 x
-cot x
Integral of tan x sec x
Sec x
Integral of cot x csc x
-csc x
What kind of polar graph is
r=a+-b sin theta or r= a +-b cos theta?
Limacons
What kind of polar graph is
r=a sin (n@) or r= a cos (n@) ?
Roses
What kind of polar graph is
r^2 =a^2 sin theta or r^2 = a^2 cos theta?
Lemniscate
Roses with even petals
Petals =2n
Roses with odd petals
petals = n
Guess what type of ODe (Ordinary Differential Equation):
dy/dx = f(x)g(y)
Separable ODe
Guess what type of ODe (Ordinary Differential Equation):
dy/dx + P(x)y=Q(x)
Linear ODe
Guess what type of ODe (Ordinary Differential Equation):
dy/dx = f(y/x)
Homogenous ODe
Guess what type of ODe (Ordinary Differential Equation):
dy/dx + P(x)y =Q(x)y^n
Bernoulli ODe
Guess what type of ODe (Ordinary Differential Equation):
M(dx) +N(dy) = 0
Exact Equation
Application DE (formula and caltech): For Nuclear Decay
Formula: m=mo e*-kt ln|m/mo|=-kt k=ln2/t1/2 Caltech stat 5 x- 0 and half life (time) y- 100 and 50 (mass remaining)
Application DE (formula and caltech): For Mixing with same or constant flowrate q
(C-Ci/Co-Ci)=e^-(q/v)(t)
Application DE (formula and caltech): For Mixing with qo not equal to qi
V=(qi-qo)t+Vo
(C-Ci/Co-Ci)=(V/Vo)^-qi/qi-qo
Application DE (formula and caltech): For Population Growth
P=Poe^kt
k=ln2/t2
dp/dt= mortality(Kp) - morbidity (death) + migration - immigration
Caltech stat 5
x= 0 and t
Y= 100 and population @ t
Application DE (formula and caltech): For Newton’s law of cooling/heating
(T-Ta/To-Ta) = e^kt
Ta= ambient T Caltech Mode Stat 5 (x time, y temp) For cooling (T-Ta) For heating (Ta-T)
Econ: simple interest formula
F=P(1+in)
Econ: Compound interest
F=P(1+r/m)^mt
Econ: i is equal to
r/m
Econ: r stands for
Nominal interest rate
Econ: ieff for compounded
ieff= [(1+r/m)^m] - 1
Econ: ieff for continuous
ieff=e^r - 1
Econ: continuous interest
F=Pe^rn
Econ: Interest formula
I=Pin
Depreciation: Caltech mode
Straight line method
Stat 2
Depreciation: Caltech mode
Declining Balance Method
Stat 6
Depreciation: Caltech mode
Double Declining Balance
Stat 6
Depreciation: Caltech mode
Sum of Years digit
Stat 3
Depreciation: Caltech mode
Sinking Fund Method
Stat 6
Break even in econ means
No loss or gain (balik capital)
Econ: book value formula for SLM, DBM, DDBM and SOYD
BVn=nÿ
Econ: Depreciation @ nth year formula for SLM, DBM, DDBM and SOYD
dn=(n-1)ÿ-nÿ
Econ: Total Depreciation formula for SLM, DBM, DDBM and SOYD
Dtn=FC-nÿ
FC- first cost
Econ: book value formula for SFM
BVn=(FC-SV)x(1-nÿ)/(1-Lÿ)
L- economic life
Econ: depreciation @ nth year formula for SFM
dn=(FC-SV)x(0ÿ-1ÿ)/(1-Lÿ)
Econ: total depreciation formula for SFM
Dtn=FC-BVn
Perpetuity Formula
P’=A/i
Ordinary Annuity Present Formula
P= A( (1+i)^-1/i(1+i)^n )
Ordinary Annuity Future Formula
F= A ( (1+i)^n-1/i )
Deferred Annuity Formula
P= A ( (1+i)^n -1/i (1+i)^n) (1+i)^-(k-1)
