# importing-data.ipynb

In [1]:
import pandas as pd
from IPython.display import HTML

In [2]:
df = pd.read_csv('equations.csv')

In [3]:
df.head()

Out[3]:
equations field_of_study topic title latex_formula
0 Equations Trigonometry Trigonometric Properties Pythagorean Property - Sine and Cosine \sin ^2 \theta + \cos ^2 \theta = 1
1 Equations Trigonometry Trigonometric Definitions Sine Definition for a Right Triangle \sin \theta = \frac{{{\rm{Opposite Side}}}}{{...
2 Equations Trigonometry Trigonometric Definitions Cosine Definition for a Right Triangle \cos \theta = \frac{{{\rm{Adjacent Side}}}}{{...
3 Equations Trigonometry Trigonometric Definitions Tangent Definition for a Right Triangle \tan \theta = \frac{{{\rm{Opposite Side}}}}{{...
4 Equations Trigonometry Trigonometric Identities Double Angle Identity - Sine \sin 2\theta = 2\sin \theta \cos \theta
In [4]:
s = df['latex_formula'][0]

In [5]:
HTML(f'$${s}$$')

Out[5]:
$$\sin ^2 \theta + \cos ^2 \theta = 1$$
In [6]:
for s in df:

display(HTML(f'$${s}$$'))

$$equations$$
$$field_of_study$$
$$topic$$
$$title$$
$$latex_formula$$
In [7]:
next(df.iterrows())

Out[7]:
(0,
equations                                      Equations
field_of_study                              Trigonometry
topic                           Trigonometric Properties
title             Pythagorean Property - Sine and Cosine
latex_formula      \sin ^2 \theta  + \cos ^2 \theta  = 1
Name: 0, dtype: object)
In [ ]:


In [8]:
for index, row in df.iterrows():
title = row['title']
latex_formula = row['latex_formula']

print(title)
display(HTML(f'$${latex_formula}$$'))
print('\n')


Pythagorean Property - Sine and Cosine

$$\sin ^2 \theta + \cos ^2 \theta = 1$$

Sine Definition for a Right Triangle

$$\sin \theta = \frac{{{\rm{Opposite Side}}}}{{{\rm{Hypotenuse}}}}$$

Cosine Definition for a Right Triangle

$$\cos \theta = \frac{{{\rm{Adjacent Side}}}}{{{\rm{Hypotenuse}}}}$$

Tangent Definition for a Right Triangle

$$\tan \theta = \frac{{{\rm{Opposite Side}}}}{{{\rm{AdjacentSide}}}}$$

Double Angle Identity - Sine

$$\sin 2\theta = 2\sin \theta \cos \theta$$

Double Angle Identity - Cosine

$$\cos 2\theta = \cos ^2 \theta - \sin ^2 \theta = 2\cos ^2 \theta - 1$$

Half Angle Identity - Sine

$$\sin \frac{\theta }{2} = \sqrt {\frac{{1 - \cos \theta }}{2}}$$

Half Angle Identity - Cosine

$$\cos \frac{\theta }{2} = \sqrt {\frac{{1 + \cos \theta }}{2}}$$

Sum and Difference of Angles Identity - Sine

$$\sin \left( {\theta _1 \pm \theta _2 } \right) = \sin \theta _1 \cos \theta _2 \pm \cos \theta _1 \sin \theta _2$$

Sum and Difference of Angles Identity - Cosine

$$\cos \left( {\theta _1 \pm \theta _2 } \right) = \cos \theta _1 \cos \theta _2 \mp \sin \theta _1 \sin \theta _2$$


$${\rm{a + 0 = a}}$$


$$a + \left( { - a} \right) = 0$$


$$\left( {a + b} \right) + c = a + \left( {b + c} \right)$$


$$a + b = b + a$$

Definition of Subtraction

$$a - b = a + \left( { - b} \right)$$

Multiplicative Identity

$$a \times 1 = a$$

Multiplicative Inverse

$$a \times \frac{1}{a} = 1$$

Zero Multiplication Property

$$a \times 0 = 0$$

Associative Property of Multiplication

$$\left( {a \times b} \right) \times c = a \times \left( {b \times c} \right) = a \times b \times c$$

Definition of Division

$$\frac{a}{b} = a \times \left( {\frac{1}{b}} \right)$$

Square of a First Order Polynomial

$$\left( {a + b} \right)^2 = a^2 + 2ab + b^2$$

Polynomial FOIL operation

$$\left( {a + b} \right)\left( {c + d} \right) = ac + ad + bc + bd$$

Difference of Squares Factorization

$$a^2 - b^2 = \left( {a + b} \right)\left( {a - b} \right)$$

Sum of Cubes Factorization

$$a^3 + b^3 = \left( {a + b} \right)\left( {a^2 - ab + b^2 } \right)$$

Difference of Cubes Factorization

$$a^3 - b^3 = \left( {a - b} \right)\left( {a^2 + ab + b^2 } \right)$$

Second Order Polynomial Factorization

$$x^2 + x\left( {a + b} \right) + ab = \left( {x + a} \right)\left( {x + b} \right)$$


$$\begin{array}{*{20}c} {x = \frac{{ - b \pm \sqrt {b^2 - 4ac} }}{{2a}}} & {{\rm{when}}} & {ax^2 + bx + c = 0} \\ \end{array}$$

Exponent Equal to Zero Rule

$$x^0 = 1$$

Exponent Equal to One Rule

$$x^1 = x$$


$$x^a x^b = x^{\left( {a + b} \right)}$$

Distributive Property of Exponents

$$x^a y^a = \left( {xy} \right)^a$$

Power Rule of Exponents

$$\left( {x^a } \right)^b = x^{\left( {ab} \right)}$$

Fractional Exponent to Fractional Root Relationship

$$x^{\left( {\frac{a}{b}} \right)} = \sqrt[b]{{x^a }}$$

Definition of Square Root

$$x^{\left( {\frac{1}{2}} \right)} = \sqrt x$$

Negative Exponent Definition

$$x^{ - a} = \frac{1}{{x^a }}$$

Subtraction of Exponents Rule

$$x^{\left( {a - b} \right)} = \frac{{x^a }}{{x^b }}$$

Definition of a Logarithm

$$y = \log _b \left( x \right){\rm{ iff }}x = b^y$$

Logarithm of One

$$\log _b \left( 1 \right) = 0$$

Logarithmic Identity Property

$$\log _b \left( b \right) = 1$$

Sum of Logarithms Property

$$\log _b \left( {xy} \right) = \log _b \left( x \right) + \log _b \left( y \right)$$

Difference of Logarithms Property

$$\log _b \left( {\frac{x}{y}} \right) = \log _b \left( x \right) - \log _b \left( y \right)$$

Logarithm of an Exponential

$$\log _b \left( {x^n } \right) = n\log _b \left( x \right)$$

Logarithm Base Conversion

$$\log _b \left( x \right) = \log _b \left( c \right)\log _c \left( x \right) = \frac{{\log _c \left( x \right)}}{{\log _c \left( b \right)}}$$

SURD Multiplication

$$a\sqrt b \times c\sqrt d = ac\sqrt {bd}$$

SURD Division

$$\frac{{a\sqrt b }}{{c\sqrt d }} = \frac{a}{c}\sqrt {\frac{b}{d}}$$


$$a\sqrt b \pm c\sqrt b = \left( {a \pm c} \right)\sqrt b$$

Integral of powers not equal to -1

$$\int {x^n } dx = \frac{{x^{n + 1} }}{{n + 1}},(n \ne - 1)$$

Integration by parts

$$\int {u\frac{{dv}}{{dx}}} dx = uv - \int {\frac{{du}}{{dx}}} vdx$$

Integral of reciprocal

$$\int {\frac{1}{x}} dx = \ln \left| x \right| + c$$

Integral of cosine

$$\int {\cos (ax)} dx = \frac{1}{a}\sin (ax) + c$$

Integral of sine

$$\int {\sin (ax)} dx = - \frac{1}{a}\cos (ax) + c$$

Integral of tangent

$$\int {\tan (ax)} dx = - \frac{1}{a}\ln \left| {\cos (ax)} \right| + c$$

Integral of cosecant

$$\int {\csc (ax)} dx = \frac{1}{a}\ln \left| {\tan \left( {\frac{{ax}}{2}} \right)} \right| + c$$

Integral of secant

$$\int {\sec (ax)} dx = \frac{1}{a}\ln \left| {\tan \left( {\frac{{ax}}{2} + \frac{\pi }{4}} \right)} \right| + c$$

Limit of Sine X over X as X Approaches Zero

$$\mathop {\lim }\limits_{x \to 0} \frac{{\sin x}}{x} = 1$$

Limit of Tangent X over X as X Approaches Zero

$$\mathop {\lim }\limits_{x \to 0} \frac{{\tan x}}{x} = 1$$

Area of a circle

$$A = \pi r^2$$

Area of an ellipse

$$A = \pi r_1 r_2$$

Area of an equilateral triangle

$$A = \frac{{h^2 \sqrt 3 }}{3}$$

Area of a parallelogram

$$A = bh$$

Area of a rectangle

$$A = lw$$

Area of a regular polygon

$$A = \frac{{nsr}}{2} = \frac{{pr}}{2}$$

Area of a rhombus

$$A = \frac{{x_1 x_2 }}{2}$$

Area of a sector

$$A = \frac{{\theta r^2 }}{2}$$

Area of a square

$$A = x^2$$

Area of a trapezoid

$$A = \frac{1}{2}(x_1 + x_2 )h$$

Area of a triangle

$$A = \frac{1}{2}bh$$

Volume of a cone

$$V = \frac{{Bh}}{3} = \frac{{\pi r^2 h}}{3}$$

Volume of a sphere

$$V = \frac{{4\pi r^3 }}{3}$$

Volume of a pyramid

$$V = \frac{{Bh}}{3}$$

Volume of a cube

$$V = x^3$$

Volume of a cuboid

$$V = lhw$$

Volume of a cylinder

$$V = Bh = \pi r^2 h$$

Volume of a prism

$$V = Bh$$

Ideal gas equation

$$PV = nRT$$

kinetic energy

$$E_k = \frac{1}{2}mv^2$$

equation of linear motion

$$x\left( t \right) = x_o + vt + \frac{1}{2}at^2$$

Surface area of a sphere

$$S = 4\pi r^2$$

Surface area of a cylinder

$$S = 2\pi r^2 + 2\pi rh$$

Surface area of a cuboid

$$S = 2lw + 2lh + 2wh$$

Surface area of a cube

$$S = 6x^2$$

Refractive index

$$n = \frac{{\sin {\rm{ }}i}}{{\sin {\rm{ }}r}} = \frac{{{\rm{depth}}}}{{{\rm{apparent depth}}}}$$

Einstein's relativistic mass-energy relation

$$E = mc^2$$

spring constant

$$k = \frac{F}{x}$$

simple harmonic motion acceleration

$$a = - \omega ^2 x = - \omega ^2 r\sin (\omega t)$$


$$PV = k$$

Ampere's law

$$\oint_C {Bd\ell = \mu _0 I_C }$$

Angular Momentum

$$M = I\omega$$

Beat frequency

$$f = f_1 - f_2$$

Charles' Law

$$\frac{V}{t} = k$$

de Broglie wavelength

$$\lambda = \frac{h}{{mv}}$$

Voltage equation

$$V = IR$$

Zurich sunspot number

$$R = k(f + 10g)$$

Yukawa Potential

$$V = \frac{{V_\theta e^{ - kr} }}{r}$$

Young's modulus

$$\Upsilon = \frac{{{F \mathord{\left/ {\vphantom {F A}} \right. \kern-\nulldelimiterspace} A}}}{{{{\Delta L} \mathord{\left/ {\vphantom {{\Delta L} L}} \right. \kern-\nulldelimiterspace} L}}} = \frac{{{\rm{Stress}}}}{{{\rm{Strain}}}}$$

Newton's Second Law (Force)

$$F = ma$$

Z-transform time domain convolution (z domain multiplication) property

$$h(n) * x(n) \Leftrightarrow H(z)X(z)$$

Z-transform linearity property

$$a_1 x_1 (n) + a_2 x_2 (n) \Leftrightarrow a_1 X_1 (z) + a_2 X_2 (z)$$

Z-transform translation (time shift) property

$$x(n - m) \Leftrightarrow z^{ - m} X(z)$$

Z-transform multiplication by an exponential (z domain scaling) property

$$a^n x\left( n \right) \Leftrightarrow X\left( {\frac{z}{a}} \right)$$

Z-transform multiplication by a ramp (z domain differentiation) property

$$nx\left( n \right) \Leftrightarrow - z\frac{{dX(z)}}{{dz}}$$

Z-transform time domain multiplication (z domain convolution) property

$$h(n)x(n) \Leftrightarrow \frac{1}{{2\pi j}}\oint_C {H\left( v \right)X\left( {z/v} \right)\mathop v\nolimits^{ - 1} dv}$$

Z-transform initial value theorem

$$x\left( {0^ - } \right) = \mathop {\lim }\limits_{z \to \infty } X\left( z \right)$$

Z-transform final value theorem.  Valid only if polues of (z-1)X(z) are inside the unit circle.

$$x\left( \infty \right) = \mathop {\lim }\limits_{z \to 1} \left( {z - 1} \right)X\left( z \right)$$

Z-transform of delta

$$\delta (n) \Leftrightarrow 1$$

Z-transform of shifted delta

$$\delta (n - m) \Leftrightarrow z^{ - m} ,\left| z \right| > 1$$

Z-transform of unit step function

$$u(n) \Leftrightarrow \frac{z}{{z - 1}},\left| z \right| > 1$$

Z-transform involving the unit step function

$$nu(n) \Leftrightarrow \frac{z}{{\left( {z - 1} \right)^2 }},\left| z \right| > 1$$

Z-transform involving the unit step function

$$n^2 u(n) \Leftrightarrow \frac{{z\left( {z + 1} \right)}}{{\left( {z - 1} \right)^3 }},\left| z \right| > 1$$

Z-transform involving the unit step function

$$n^3 u(n) \Leftrightarrow \frac{{z\left( {z^2 + 4z + 1} \right)}}{{\left( {z - 1} \right)^4 }},\left| z \right| > 1$$

Z-transform involving the unit step function and an exponential

$$a^n u(n) \Leftrightarrow \frac{z}{{\left( {z - a} \right)}},\left| z \right| > \left| a \right|$$

Z-transform involving the unit step function and an exponential

$$na^n u(n) \Leftrightarrow \frac{{az}}{{\left( {z - a} \right)^2 }},\left| z \right| > \left| a \right|$$

Z-transform involving the unit step function and an exponential

$$n^2 a^n u(n) \Leftrightarrow \frac{{az(z + a)}}{{\left( {z - a} \right)^3 }},\left| z \right| > \left| a \right|$$

Z-transform involving the unit step function and an exponential

$$n^3 a^n u(n) \Leftrightarrow \frac{{az(z^2 + 4az + a^2 )}}{{\left( {z - a} \right)^4 }},\left| z \right| > \left| a \right|$$

Z-transform involving the unit step function and an exponential

$$na^{n - 1} u(n) \Leftrightarrow \frac{z}{{\left( {z - a} \right)^2 }},\left| z \right| > \left| a \right|$$

Z-transform involving the unit step function and an exponential

$$\frac{1}{2}n(n - 1)a^{n - 2} u(n) \Leftrightarrow \frac{z}{{\left( {z - a} \right)^3 }},\left| z \right| > \left| a \right|$$

Z-transform involving the unit step function and an exponential

$$\frac{1}{6}n(n - 1)(n - 2)a^{n - 3} u(n) \Leftrightarrow \frac{z}{{\left( {z - a} \right)^4 }},\left| z \right| > \left| a \right|$$

Z-transform involving the unit step function and an exponential

$$e^n u(n) \Leftrightarrow \frac{z}{{\left( {z - e} \right)}},\left| z \right| > \left| e \right|$$

Z-transform involving the unit step function and sine

$$\sin (\omega _o n)u(n) \Leftrightarrow \frac{{z\sin \omega _o }}{{\left( {z^2 - 2z\cos \omega _o + 1} \right)}},\left| z \right| > 1$$

Z-transform involving the unit step function and cosine

$$\cos (\omega _o n)u(n) \Leftrightarrow \frac{{z(z - \cos \omega _o )}}{{\left( {z^2 - 2z\cos \omega _o + 1} \right)}},\left| z \right| > 1$$

Z-transform involving the unit step function and cosine

$$\cos (\omega _o n + \theta )u(n) \Leftrightarrow \frac{{z[z\cos \theta - \cos (\omega _o - \theta )]}}{{\left( {z^2 - 2z\cos \omega _o + 1} \right)}},\left| z \right| > 1$$

Z-transform involving the unit step function and sine

$$\sin (\omega _o n + \theta )u(n) \Leftrightarrow \frac{{z[z\sin \theta + \sin (\omega _o - \theta )]}}{{\left( {z^2 - 2z\cos \omega _o + 1} \right)}},\left| z \right| > 1$$

Z-transform involving the unit step function and hyperbolic sine

$$\sinh (\omega _o n)u(n) \Leftrightarrow \frac{{z\sinh \omega _o }}{{\left( {z^2 - 2z\cosh \omega _o + 1} \right)}},\left| z \right| > e^{\omega _o }$$

Z-transform involving the unit step function and hyperbolic cosine

$$\cosh (\omega _o n)u(n) \Leftrightarrow \frac{{z(z - \cosh \omega _o )}}{{\left( {z^2 - 2z\cosh \omega _o + 1} \right)}},\left| z \right| > e^{\omega _o }$$

Laplace transform of Kroeneker delta function

$$\delta (t) \Leftrightarrow 1$$

Laplace transform of unit step function times a constant (K)

$$Ku(t) \Leftrightarrow \frac{K}{s}$$

Laplace transform involving the unit step function

$$tu(t) \Leftrightarrow \frac{1}{{s^2 }}$$

Laplace transform involving the unit step function

$$t^n u(t) \Leftrightarrow \frac{{n!}}{{s^{n + 1} }}$$

Laplace transform involving the unit step function and an exponential

$$Ke^{ - at} u(t) \Leftrightarrow \frac{K}{{s + a}}$$

Laplace transform involving the unit step function and an exponential

$$t^n e^{ - at} u(t) \Leftrightarrow \frac{{n!}}{{(s + a)^{n + 1} }}$$

Laplace transform involving the unit step function and sine

$$\sin (\Omega t)u(t) \Leftrightarrow \frac{\Omega }{{(s^2 + \Omega ^2 )}}$$

Laplace transform involving the unit step function and cosine

$$\cos (\Omega t)u(t) \Leftrightarrow \frac{s}{{(s^2 + \Omega ^2 )}}$$

Laplace transform involving the unit step function, cosine, and an exponential

$$e^{ - at} \cos (\Omega t)u(t) \Leftrightarrow \frac{{s + a}}{{(s + a)^2 + \Omega ^2 }}$$

Laplace transform involving the unit step function, sine, and an exponential

$$e^{ - at} \sin (\Omega t)u(t) \Leftrightarrow \frac{\Omega }{{(s + a)^2 + \Omega ^2 }}$$

Laplace transform linearity property

$$a_1 x_1 (t) + a_2 x_2 (t) \Leftrightarrow a_1 X_1 (s) + a_2 X_2 (s)$$

Laplace transform Nth time domain derivative property

$$\frac{{d^n x(t)}}{{dt^n }} \Leftrightarrow s^n X(s)$$

Laplace transform integral property

$$\int\limits_0^t {x(\tau )d} \tau \Leftrightarrow \frac{1}{s}X(s)$$

Laplace transform time domain shifting property

$$x(t - a)u(t - a) \Leftrightarrow e^{ - as} X(s + a)$$

Laplace transform time domain scaling property

$$x(at)u(t) \Leftrightarrow \frac{1}{a}X\left( {\frac{s}{a}} \right)$$

Laplace transform time varying coefficient (s domain differentiation) property

$$tx(t)u(t) \Leftrightarrow \frac{{ - dX(s)}}{{ds}}$$

Laplace transform time domain linear convolution (s domain multiplication) property

$$\int\limits_0^\infty {x_1 (\tau )x_2 (t - \tau )d\tau } \Leftrightarrow X_1 (s)X_2 (s)$$

Laplace transform final value theorem (valid if poles of sX(s) are in left half of s plane).

$$x(\infty ) = \mathop {\lim }\limits_{s \to 0} sX(s)$$

Laplace transform initial value theorem

$$x(0^ + ) = \mathop {\lim }\limits_{s \to \infty } sX(s)$$

Laplace transform definition

$$X(s) = \int\limits_0^\infty {x(t)e^{ - st} dt}$$

Z transform definition

$$X(z) \buildrel \Delta \over = \sum\limits_{n = - \infty }^\infty {x(n)z^{ - n} }$$

Discrete-Time Fourier linearity theorem

$$ax(n) + by(n) \Leftrightarrow aX(e^{j\omega } ) + bY(e^{j\omega } )$$

Discrete-Time Fourier time shift property

$$x(n - n_o ) \Leftrightarrow e^{ - j\omega n_o } X(e^{j\omega } )$$

Discrete-Time Fourier frequency shift property

$$e^{ + j\omega _o n} x(n) \Leftrightarrow X(e^{j(\omega - \omega _o )} )$$

Discrete-Time Fourier time reversal property

$$x( - n) \Leftrightarrow X(e^{ - j\omega } )$$

Discrete-Time Fourier time reversal property; x(n) real

$$x( - n) \Leftrightarrow X^* (e^{j\omega } )$$

Discrete-Time Fourier frequency differentiation property

$$nx(n) \Leftrightarrow j\frac{{dX(e^{j\omega } )}}{{d\omega }}$$

Discrete-Time Fourier time/space convolution property

$$x(n) * y(n) \Leftrightarrow X(e^{j\omega } )Y(e^{j\omega } )$$

Discrete-Time Fourier windowing, modulation, frequency convolution property

$$x(n)y(n) \Leftrightarrow \frac{1}{{2\pi }}\int\limits_{ - \pi }^\pi {X(e^{j\theta } )Y(e^{j(\omega - \theta )} )d\theta }$$

Discrete-Time Fourier transform - Parseval's Theorem

$$\sum\limits_{n = - \infty }^\infty {\left| {x(n)} \right|^2 } = \frac{1}{{2\pi }}\int\limits_{ - \pi }^\pi {\left| {X(e^{j\omega } )} \right|^2 d\omega }$$

Discrete-Time Fourier transform - Parseval's Theorem

$$\sum\limits_{n = - \infty }^\infty {x(n)y^* (n)} = \frac{1}{{2\pi }}\int\limits_{ - \pi }^\pi {X(e^{j\omega } )Y^* (e^{j\omega } )d\omega }$$

Discrete-Time Fourier transform of delta

$$a\delta (n) \Leftrightarrow a$$

Discrete-Time Fourier transform of shifted delta

$$\delta (n - n_o ) \Leftrightarrow e^{ - j\omega n_o }$$

Discrete-Time Fourier transform of a constant

$$a \Leftrightarrow 2\pi a\sum\limits_{k = - \infty }^\infty {\delta (\omega + 2\pi k)} ,( - \infty < n < \infty )$$

Discrete-Time Fourier transform of unit step function

$$u(n) \Leftrightarrow \frac{1}{{(1 - e^{ - j\omega } )}} + \sum\limits_{k = - \infty }^\infty {\pi \delta (\omega + 2\pi k)}$$

Discrete-Time Fourier transform of unit step function and exponential

$$a^n u(n) \Leftrightarrow \frac{1}{{(1 - ae^{ - j\omega } )}},\left| a \right| < 1$$

Discrete-Time Fourier transform of unit step function and exponential

$$(n + 1)a^n u(n) \Leftrightarrow \frac{1}{{(1 - ae^{ - j\omega } )^2 }},\left| a \right| < 1$$

Discrete-Time Fourier transform of complex exponential

$$e^{j\omega _o n} \Leftrightarrow 2\pi \sum\limits_{k = - \infty }^\infty {\delta (\omega - \omega _o + 2\pi k)}$$

Discrete-Time Fourier transform of a sinc sequence

$$\frac{{\sin (\omega _c n)}}{n} \Leftrightarrow X(e^{j\omega } ) = \left\{ {\begin{array}{*{20}c} {1,\left| \omega \right| \le \omega _c } \\ {0,\omega _c < \left| \omega \right| \le \omega _c } \\ \end{array}} \right.$$

Discrete-Time Fourier transform of a boxcar sequence

$$x(n) = \left\{ {\begin{array}{*{20}c} {1,0 \le n \le N - 1} \\ {0,{\rm{otherwise}}} \\ \end{array} \Leftrightarrow } \right.\frac{{\sin (\omega N/2)}}{{\sin (\omega /2)}}e^{ - j\omega (N - 1)/2}$$

Discrete-Time Fourier transform definition

$$X(e^{j\omega } ) = x(n)e^{ - j\omega n} ,\omega {\rm{in radians}}$$

Black Hole Entropy as derived by Stephen Hawking

$$S = \frac{{Akc^3 }}{{4\hbar G}}$$

Time-independent, one-dimensional Schr

$$- \frac{{\hbar ^2 }}{{2m}}\frac{{d^2 \psi (x)}}{{dx^2 }} + U(x)\psi (x) = E\psi (x)$$

Time-dependent, one-dimensional Schr

$$- \frac{{\hbar ^2 }}{{2m}}\frac{{\partial ^2 \psi (x,t)}}{{\partial x^2 }} + U(x)\psi (x,t) = i\hbar \frac{{\partial \psi (x,t)}}{{\partial t}}$$

Ohm's Law

$$V = IR = I\left( {\frac{L}{{\sigma A}}} \right) = I\left( {\frac{{\rho L}}{A}} \right)$$


$$a_0 = \frac{{\hbar ^2 }}{{m_e ke^2 }}$$

Radii of stable orbits in the Bohr model

$$r = n^2 \frac{{\hbar ^2 }}{{m_e kZe^2 }} = n^2 \frac{{a_0 }}{Z}$$

Phase difference between the first and last waves for a single-slit diffraction pattern

$$\phi = \frac{{2\pi }}{\lambda }a\sin \theta$$

single-slit diffraction pattern points of zero intensity

$$a\sin \theta = m\lambda$$

Definition of intensity

$$I = \frac{{P_{av} }}{A}$$

Superposition of standing waves on a string with both ends fixed

$$y(x,t) = \sum\limits_n {A_n \cos (\omega _n t + \delta _n )\sin (k_n x)}$$

Standing-wave function

$$y(x,t) = A_n \cos (\omega _n t + \delta _n )\sin (k_n x)$$

Energy transmitted by a harmonic wave

$$\Delta E = \frac{1}{2}\mu \omega ^2 A^2 \Delta x = \frac{1}{2}\mu \omega ^2 A^2 \upsilon \Delta t$$

Power transmitted by a harmonic wave

$$P = \frac{{dE}}{{dt}} = \frac{1}{2}\mu \omega ^2 A^2 \upsilon$$

Harmonic wave function

$$y(x,t) = A\sin \left[ {2\pi \left( {\frac{x}{\lambda } - \frac{t}{T}} \right)} \right]$$

Harmonic wave function

$$y(x,t) = A\sin \left[ {k(x - \upsilon t)} \right]$$

Velocity at resonance frequency of a driven oscillator

$$\upsilon = + A\omega \cos \left( {\omega t} \right) = - A\omega \sin \left( {\omega t - \frac{\pi }{2}} \right)$$

Amplitude of a driven oscillation

$$A = \frac{{F_0 }}{{\sqrt {m^2 \left( {\omega _0^2 - \omega ^2 } \right)^2 + b^2 \omega ^2 } }}$$

Displacement of a driven oscillator

$$x = A\cos \left( {\omega t + \delta } \right)$$

Displacement of a slightly damped oscillator

$$x = A_0 \exp \left( { - \frac{b}{{2m}}t} \right)\cos \left( {\omega 't + \delta } \right)$$

Kinetic energy of simple harmonic motion

$$K = \frac{1}{2}kA^2 \sin ^2 \left( {\omega t + \delta } \right)$$

Potential energy of simple harmonic motion

$$U = \frac{1}{2}kA^2 \cos ^2 \left( {\omega t + \delta } \right)$$

Total energy of simple harmonic motion

$$E_{Total} = \frac{1}{2}kA^2$$

Viscous flow

$$F = \eta \frac{{\upsilon A}}{z}$$

Continuity equation

$$I_V = \upsilon A = {\rm{constant}}$$

Hydraulic lift

$$F_2 = \frac{{F_1 }}{{A_1 }}A_2 = \frac{{A_2 }}{{A_1 }}F_1$$

Shear modulus defined

$$M_S = \frac{{{{F_S } \mathord{\left/ {\vphantom {{F_S } A}} \right. \kern-\nulldelimiterspace} A}}}{{{{\Delta x} \mathord{\left/ {\vphantom {{\Delta x} L}} \right. \kern-\nulldelimiterspace} L}}} = \frac{{{{F_S } \mathord{\left/ {\vphantom {{F_S } A}} \right. \kern-\nulldelimiterspace} A}}}{{{\rm{tan}}\theta }} = \frac{{{\rm{Shear Stress}}}}{{{\rm{Shear Strain}}}}$$

Shear stress

$${\rm{Shear Stress = }}\frac{{{\rm{F}}_{\rm{S}} }}{{\rm{A}}}$$

Stress

$${\rm{Stress}} = \frac{F}{A}$$

Centripetal acceleration

$$a = \frac{{\upsilon ^2 }}{r}$$

Instantaneous Acceleration

$$a = \frac{{d\upsilon }}{{dt}} = \frac{{d^2 x}}{{dt^2 }}$$

Velocity

$$\upsilon ^2 = \upsilon _0 ^2 + 2a\Delta x$$

Displacement

$$\Delta x = x - x_0 = \upsilon _0 t + \frac{1}{2}at^2$$

Velocity

$$\upsilon = \upsilon _0 + at$$

Average acceleration

$$a_{av} = \frac{{\Delta \upsilon }}{{\Delta t}}$$


$$\oint_C {E \cdot d\ell = - \frac{d}{{dt}}} \int_S {B_n dA}$$

Maxwell's equation - Gauss's law

$$\oint_S {E_n dA = \frac{1}{{\varepsilon _0 }}} Q_{inside}$$

Self inductance of a solenoid

$$L = \frac{{\phi _m }}{I} = \mu _0 n^2 A\ell$$

Magnetic flux defined

$$\phi _m = \int_S {N{\bf{B}} \cdot {\bf{\hat n}}dA = } \int_S {NB_n dA}$$

EMF (Electromotive Force) defined

$$\xi = \oint_C {E \cdot d\ell }$$

Resistance

$$R = \frac{L}{{\sigma A}} = \frac{{\rho L}}{A}$$

Electric current

$$I = \frac{{\Delta Q}}{{\Delta t}} = nqA\upsilon _d$$

Number density of charge carriers

$$n = \frac{I}{{Aq\upsilon _d }} = \frac{I}{{wte\upsilon _d }}$$

Current in a conducting strip

$$I = nAq\upsilon _d$$

Magnetic dipole moment of a current loop

$$m = NIA\hat n$$

Capacitance of a cylindrical capacitor

$$C = \frac{Q}{V} = \frac{{2\pi \varepsilon _0 L}}{{\ln \left( {{b \mathord{\left/ {\vphantom {b a}} \right. \kern-\nulldelimiterspace} a}} \right)}}$$

Capacitance of a parallel-plate capacitor

$$C = \frac{Q}{V} = \frac{{\varepsilon _0 A}}{s}$$

Potential due to a line charge

$$V = - 2k\lambda \ln \frac{r}{a}$$

Change in potential

$$\Delta V = V_b - V_a = \frac{{\Delta U}}{{q_0 }} = - \int_a^b {E \cdot d\ell }$$

Change in electrostatic potential energy for a point charge

$$\Delta U = U_b - U_a = \int_a^b {dU} = - \int_a^b {q_0 E \cdot d\ell }$$

Gauss's law

$$\phi _{net} = \int_S {E_n dA = } \frac{1}{{\varepsilon _0 }}Q_{inside} = 4\pi kQ_{inside}$$

Electric flux defined

$$\phi = \int_S {{\bf{E}} \cdot {\bf{\hat n}}dA}$$

Electric field on the axis of a ring charge

$$E_x = \frac{{kQx}}{{\left( {x^2 + a^2 } \right)^{{3 \mathord{\left/ {\vphantom {3 2}} \right. \kern-\nulldelimiterspace} 2}} }}$$

Acceleration of a particle in an electric field

$$a = \frac{q}{m}E$$

Net power radiated by an object

$$P_{net} = e\sigma A\left( {T^4 - T_0^4 } \right)$$


$$P_a = e\sigma AT_0^4$$

Stefan-Boltzmann law

$$P = e\sigma AT^4$$

Thermal resistance

$$R = \frac{{\Delta x}}{{kA}}$$

Thermal conduction

$$I = \frac{{\Delta Q}}{{\Delta t}} = kA\frac{{\Delta T}}{{\Delta x}}$$

Van der Waals equation

$$\left( {P + \frac{{an^2 }}{{V^2 }}} \right)\left( {V - bn} \right) = nRT$$

Speed of sound waves in a fluid

$$\upsilon = \sqrt {\frac{B}{\rho }}$$

Phase constant of a driven oscillation

$$\tan \delta = \frac{{b\omega }}{{m\left( {\omega _0^2 - \omega ^2 } \right)}}$$

Angular frequency for a damped oscillation

$$\omega ' = \omega _0 \sqrt {1 - \left( {\frac{b}{{2m\omega _0 }}} \right)^2 } = \omega _0 \sqrt {1 - \frac{1}{{4Q^2 }}}$$

Energy change in a damped oscillation

$$\frac{{\Delta E}}{E} = - \frac{b}{m}T$$

Energy change in a damped oscillation

$$E = E_0 \exp \left( { - \frac{b}{m}t} \right) = E_0 \exp \left( { - \frac{t}{\tau }} \right)$$

Compressibility

$$k = \frac{1}{B} = - \frac{{{{\Delta V} \mathord{\left/ {\vphantom {{\Delta V} V}} \right. \kern-\nulldelimiterspace} V}}}{P}$$

Bulk modulus defined

$$B = - \frac{P}{{{{\Delta V} \mathord{\left/ {\vphantom {{\Delta V} V}} \right. \kern-\nulldelimiterspace} V}}}$$

Poynting vector

$$S = \frac{{E \times B}}{{\mu _0 }}$$

Electric field and magnetic field relationship for an electromagnetic wave

$$E = cB$$

Wave equation for magnetic field

$$\frac{{\partial ^2 B}}{{\partial x^2 }} = \frac{1}{{c^2 }}\frac{{\partial ^2 B}}{{\partial t^2 }}$$

Magnetic field inside a solenoid

$$B = \mu _0 nI$$

Biot-Savart law

$$d{\bf{B}} = \frac{{\mu _0 }}{{4\pi }}\frac{{Id\ell \times {\bf{\hat r}}}}{{r^2 }}$$

Torque on a current loop

$$\tau = m \times B$$

Magnetic force on a moving charge

$$F = q{\bf{v}} \times {\bf{B}}$$

Rydberg constant

$$R = \frac{{m_e k^2 e^4 }}{{4\pi c\hbar ^3 }}$$


$$R = \frac{{2GM}}{{c^2 }}$$

Black Hole Temperature

$$T = \frac{{\hbar c^3 }}{{8\pi kGM}}$$

Binomial Coefficient

$$\left( {\begin{array}{*{20}c} n \\ k \\ \end{array}} \right) = \frac{{n!}}{{k!\left( {n - k} \right)!}}$$

Binomial Equation

$$y = \frac{{n!}}{{k!\left( {n - k} \right)!}}p^k q^{n - k} = \left( {\begin{array}{*{20}c} n \\ k \\ \end{array}} \right)p^k q^{n - k}$$

Mean of Binomial Distribution

$$M_b = np$$

Variance of Binomial Distribution

$$\sigma ^2 _b = npq$$

Standard Normal Distribution

$$y = \frac{1}{{\sqrt {2\pi } }}e^{ - \frac{{z^2 }}{2}} = .3989e^{ - 5z^2 }$$

Euler's Constant

$$e = \mathop {\lim }\limits_{n \to \infty } \left( {1 + \frac{1}{n}} \right)^n$$

Gaussian Normal Distribution

$$P(x) = \frac{1}{{\sigma \sqrt {2\pi } }}e^{{{ - \left( {x - \mu } \right)^2 } \mathord{\left/ {\vphantom {{ - \left( {x - \mu } \right)^2 } {2\sigma ^2 }}} \right. \kern-\nulldelimiterspace} {2\sigma ^2 }}}$$


$$\mu _n \left( a \right) = \sum {\left( {x - a} \right)^n P\left( x \right)}$$


$$\mu = \mu _1 = \sum {xP\left( x \right)}$$

Variance or second moment about the Mean

$$\sigma ^2 = \mu _2 = \sum {\left( {x - \mu _1 } \right)^2 P\left( x \right)}$$

Fisher Skewness

$$\gamma _1 = \frac{{\mu _3 }}{{\mu _2 ^{{3 \mathord{\left/ {\vphantom {3 2}} \right. \kern-\nulldelimiterspace} 2}} }} = \frac{{\mu _3 }}{{\sigma ^3 }}$$

Standard Deviation

$$\sigma = \sqrt {\mu _2 }$$

Sample Variance (Biased)

$$s_N = \sqrt {\frac{1}{N}\sum\limits_{i = 1}^N {\left( {x_i - \bar x} \right)^2 } }$$

Unbiased Estimator of Populatoin Sample Variance

$$s_{N - 1} = \sqrt {\frac{1}{{N - 1}}\sum\limits_{i = 1}^N {\left( {x_i - \bar x} \right)^2 } }$$

Standard Error

$${\mathop{\rm var}} \left( {\bar x} \right) = \frac{{\sigma ^2 }}{n}$$

Poisson Distribution

$$P\left( x \right) = \frac{{e^{ - \lambda } \lambda ^x }}{{x!}}$$

Gamma Distribution

$$\Gamma \left( a \right) = \int\limits_0^\infty {s^{a - 1} } e^{ - s} ds$$

Pythagorean Property - Tangent and Secant

$$1 + \tan ^2 \theta = \sec ^2 \theta$$

Pythagorean Property - Cotangent and Cosecant

$$\cot ^2 \theta + 1 = \csc ^2 \theta$$

Reciprocal Property - Tangent and Cotangent

$$\cot \theta = \frac{1}{{\tan \theta }}$$

Reciprocal Property - Sine and Cosecant

$$\csc \theta = \frac{1}{{\sin \theta }}$$

Reciprocal Property - Cosine and Secant

$$\sec \theta = \frac{1}{{\cos \theta }}$$

Quotient Property - Tangent, Sine, and Cosine

$$\tan \theta = \frac{{\sin \theta }}{{\cos \theta }}$$

Quotient Property - Tangent, Secant, and Cosecant

$$\tan \theta = \frac{{\sec \theta }}{{\csc \theta }}$$

Quotient Property - Cotangent, Cosecant, and Secant

$$\cot \theta = \frac{{\csc \theta }}{{\sec \theta }}$$

Quotient Property - Cotangent, Cosine, and Sine

$$\cot \theta = \frac{{\cos \theta }}{{\sin \theta }}$$

Odd Symmetry Property - Sine

$$\sin \left( { - \theta } \right) = - \sin \left( \theta \right)$$

Odd Symmetry Property - Tangent

$$\tan \left( { - \theta } \right) = - \tan \left( \theta \right)$$

Odd Symmetry Property - Cosecant

$$\csc \left( { - \theta } \right) = - \csc \left( \theta \right)$$

Even Symmetry Property - Cosine

$$\cos \left( { - \theta } \right) = \cos \left( \theta \right)$$

Even Symmetry Property - Cotangent

$$\cot \left( { - \theta } \right) = \cot \left( \theta \right)$$

Even Symmetry Property - Secant

$$\sec \left( { - \theta } \right) = \sec \left( \theta \right)$$

Area of Arbitrary Triangle

$$A = \frac{1}{2}ab\sin C$$

Law of Sines

$$\frac{{\sin A}}{a} = \frac{{\sin B}}{b} = \frac{{\sin C}}{c}$$

Law of Cosines

$$a^2 = b^2 + c^2 - 2bc\cos A$$

Sum and Difference of Angles Identity - Tangent

$$\tan \left( {\theta _1 \pm \theta _2 } \right) = \frac{{\tan \theta _1 \pm \tan \theta _2 }}{{1 \mp \tan \theta _1 \tan \theta _2 }}$$

Cotangent Definition for a Right Triangle

$$\cot \theta = \frac{{{\rm{Adjacent Side}}}}{{{\rm{Opposite Side}}}}$$

Cosecant Definition for a Right Triangle

$$\csc \theta = \frac{{{\rm{Hypotenuse}}}}{{{\rm{Opposite Side}}}}$$

Secant Definition for a Right Triangle

$$\sec \theta = \frac{{{\rm{Hypotenuse}}}}{{{\rm{Adjacent Side}}}}$$

Distributive Property

$$a\left( {b + c} \right) = ab + ac$$

Derivative of a Constant

$$\frac{d}{{dx}}C = 0$$

Derivative of a Variable to the First Power

$$\frac{d}{{dx}}x = 1$$

Derivative of a Variable to the nth Power

$$\frac{d}{{dx}}x^n = nx^{\left( {n - 1} \right)}$$

Derivative of an Exponential

$$\frac{d}{{dx}}e^{ax} = ae^{ax}$$

Derivative of an Arbitrary Base Exponential

$$\frac{d}{{dx}}b^x = b^x \ln \left( b \right)$$

Derivative of a Natural Logarithm

$$\frac{d}{{dx}}\ln \left( x \right) = \frac{1}{x}$$

Derivative of Sine

$$\frac{d}{{dx}}\sin x = \cos x$$

Derivative of Cosine

$$\frac{d}{{dx}}\cos x = - \sin x$$

Derivative of Tangent

$$\frac{d}{{dx}}\tan x = \sec ^2 x$$

Derivative of Cotangent

$$\frac{d}{{dx}}\cot x = - \csc ^2 x$$

Derivative of Secant

$$\frac{d}{{dx}}\sec x = \sec x\tan x$$

Derivative of Cosecant

$$\frac{d}{{dx}}\csc x = - \csc x\cot x$$

Derivative of Inverse Sine (Arcsine)

$$\frac{d}{{dx}}\arcsin x = \frac{d}{{dx}}sin^{ - 1} x = \frac{1}{{\sqrt {1 - x^2 } }}$$

Derivative of Inverse Cosine (Arccosine)

$$\frac{d}{{dx}}\arccos x = \frac{d}{{dx}}\cos ^{ - 1} x = \frac{{ - 1}}{{\sqrt {1 - x^2 } }}$$

Derivative of Inverse Tangent (Arctangent)

$$\frac{d}{{dx}}\arctan x = \frac{d}{{dx}}\tan ^{ - 1} x = \frac{1}{{1 + x^2 }}$$

Derivative of Inverse Cosecant (Arccosecant)

$$\frac{d}{{dx}}arc\csc x = \frac{d}{{dx}}\csc ^{ - 1} x = \frac{{ - 1}}{{\left| x \right|\sqrt {x^2 - 1} }}$$

Derivative of Inverse Secant (Arcsecant)

$$\frac{d}{{dx}}arc\sec x = \frac{d}{{dx}}\sec ^{ - 1} x = \frac{1}{{\left| x \right|\sqrt {x^2 - 1} }}$$

Derivative of Inverse Cotangent (Arccotangent)

$$\frac{d}{{dx}}arc\cot x = \frac{d}{{dx}}\cot ^{ - 1} x = \frac{{ - 1}}{{1 + x^2 }}$$

Derivative of Hyperbolic Sine

$$\frac{d}{{dx}}\sinh x = \cosh x$$

Derivative of Hyperbolic Cosine

$$\frac{d}{{dx}}\cosh x = \sinh x$$

Derivative of Hyperbolic Tangent

$$\frac{d}{{dx}}\tanh x = 1 - \tanh ^2 x$$

Derivative of Hyperbolic Cotangent

$$\frac{d}{{dx}}\coth x = 1 - \coth ^2 x$$

Derivative of Hyperbolic Secant

$$\frac{d}{{dx}}{\mathop{\rm sech}\nolimits} x = - \tanh x{\mathop{\rm sech}\nolimits} x$$

Derivative of Hyperbolic Cosecant

$$\frac{d}{{dx}}{\mathop{\rm csch}\nolimits} x = - \coth x{\mathop{\rm csch}\nolimits} x$$

Product Rule of Differentiation

$$\frac{d}{{dx}}\left( {f\left( x \right)g\left( x \right)} \right) = f\left( x \right)\frac{d}{{dx}}g\left( x \right) + \frac{d}{{dx}}f\left( x \right)g\left( x \right)$$

Quotient Rule of Differentiation

$$\frac{d}{{dx}}\left( {\frac{{f\left( x \right)}}{{g\left( x \right)}}} \right) = \frac{{\frac{d}{{dx}}f\left( x \right)g\left( x \right) - f\left( x \right)\frac{d}{{dx}}g\left( x \right)}}{{g^2 \left( x \right)}}$$

Chain Rule of Differentiation

$$\frac{d}{{dx}}\left[ {f\left( u \right)} \right] = \frac{d}{{du}}\left[ {f\left( u \right)} \right]\frac{{du}}{{dx}}$$

Fundamental Theorem for Derivatives

$$\frac{d}{{dx}}\int\limits_a^x {f\left( s \right)} ds = f\left( x \right)$$

Definition of a Derivative

$$\frac{d}{{dx}}f\left( x \right) = \mathop {\lim }\limits_{\Delta \to 0} \frac{{f\left( {x + \Delta } \right) - f\left( x \right)}}{\Delta }$$

Fundamental Theorem of Integrals of Derivatives

$$\int\limits_a^b {\frac{d}{{dx}}F\left( x \right)dx} = F\left( b \right) - F\left( a \right)$$

Gamma Function

$$\Gamma \left( x \right) = \int\limits_0^\infty {s^{x - 1} e^{ - s} ds}$$

Equation of a Line

$$y = mx + b$$

Equation of a Circle

$$\left( {x - x_0 } \right)^2 + \left( {y - y_0 } \right)^2 = R^2$$

Equation of a Sphere

$$\left( {x - x_0 } \right)^2 + \left( {y - y_0 } \right)^2 + \left( {z - z_0 } \right)^2 = R^2$$

Equation of an Ellipsoid

$$\frac{{\left( {x - x_0 } \right)^2 }}{{a^2 }} + \frac{{\left( {y - y_0 } \right)^2 }}{{b^2 }} + \frac{{\left( {z - z_0 } \right)^2 }}{{c^2 }} = 1$$

Equation of an Ellipse

$$\frac{{\left( {x - x_0 } \right)^2 }}{{a^2 }} + \frac{{\left( {y - y_0 } \right)^2 }}{{b^2 }} = 1$$

Superposition (Addition and Subtraction) of Sine of Angles

$$\sin \theta _1 \pm \sin \theta _2 = 2\sin \left( {\frac{{\theta _1 \pm \theta _2 }}{2}} \right)\cos \left( {\frac{{\theta _1 \mp \theta _2 }}{2}} \right)$$

Superposition (Addition) of Cosine of Angles

$$\cos \theta _1 + \cos \theta _2 = 2\cos \left( {\frac{{\theta _1 + \theta _2 }}{2}} \right)\cos \left( {\frac{{\theta _1 - \theta _2 }}{2}} \right)$$

Superposition (Subtraction) of Cosine of Angles

$$\cos \theta _1 - \cos \theta _2 = - 2\sin \left( {\frac{{\theta _1 + \theta _2 }}{2}} \right)\sin \left( {\frac{{\theta _1 - \theta _2 }}{2}} \right)$$

Euler's Formula

$$e^{ \pm i\theta } = \cos \theta \pm i\sin \theta$$

Cosine Definition as an Infinite Series

$$\cos x = \sum\limits_{n = 0}^\infty {\frac{{\left( { - 1} \right)^n x^{2n} }}{{\left( {2n} \right)!}}}$$

Sine Definition as an Infinite Series

$$\sin x = \sum\limits_{n = 1}^\infty {\frac{{\left( { - 1} \right)^{n - 1} x^{2n - 1} }}{{\left( {2n - 1} \right)!}}}$$

Distance Between Two Points (2-D)

$$d = \sqrt {\left( {x_1 - x_2 } \right)^2 + \left( {y_1 - y_2 } \right)^2 }$$

Distance Between Two Points (3-D)

$$d = \sqrt {\left( {x_1 - x_2 } \right)^2 + \left( {y_1 - y_2 } \right)^2 + \left( {z_1 - z_2 } \right)^2 }$$

Cartesian to Polar Coordinates (2-D)

$$\begin{array}{*{20}c} {x = r\cos \theta } & {r = \sqrt {x^2 + y^2 } } \\ {y = r\sin \theta } & {\theta = \tan ^{ - 1} \left( {\frac{y}{x}} \right)} \\ \end{array}$$

Eccentricity of an Ellipse

$$\varepsilon = \frac{{\sqrt {a^2 - b^2 } }}{a}$$

Eccentricity of a Hyperbola

$$\varepsilon = \frac{{\sqrt {a^2 + b^2 } }}{a}$$

Equation of a Hyperbola

$$\frac{{\left( {x - x_0 } \right)^2 }}{{a^2 }} - \frac{{\left( {y - y_0 } \right)^2 }}{{b^2 }} = 1$$

Equation of a Plane

$$Ax + By + Cz + D = 0$$

Equation of a Hyperboloid of One Sheet

$$\frac{{\left( {x - x_0 } \right)^2 }}{{a^2 }} + \frac{{\left( {y - y_0 } \right)^2 }}{{b^2 }} - \frac{{\left( {z - z_0 } \right)^2 }}{{c^2 }} = 1$$

Equation of an Elliptic Cone

$$\frac{{\left( {x - x_0 } \right)^2 }}{{a^2 }} + \frac{{\left( {y - y_0 } \right)^2 }}{{b^2 }} = \frac{{\left( {z - z_0 } \right)^2 }}{{c^2 }}$$

Equation of an Elliptic Cylinder

$$\frac{{\left( {x - x_0 } \right)^2 }}{{a^2 }} + \frac{{\left( {y - y_0 } \right)^2 }}{{b^2 }} = 1$$

Equation of a Hyperboloid of Two Sheets

$$\frac{{\left( {x - x_0 } \right)^2 }}{{a^2 }} - \frac{{\left( {y - y_0 } \right)^2 }}{{b^2 }} - \frac{{\left( {z - z_0 } \right)^2 }}{{c^2 }} = 1$$

Equation of an Elliptic Paraboloid

$$\frac{{\left( {x - x_0 } \right)^2 }}{{a^2 }} + \frac{{\left( {y - y_0 } \right)^2 }}{{b^2 }} = \frac{{\left( {z - z_0 } \right)}}{c}$$

Equation of a Hyperbolic Paraboloid

$$\frac{{\left( {x - x_0 } \right)^2 }}{{a^2 }} - \frac{{\left( {y - y_0 } \right)^2 }}{{b^2 }} = \frac{{\left( {z - z_0 } \right)}}{c}$$

Equation of a Parabola

$$\left( {y - y_0 } \right)^2 = 4a\left( {x - x_0 } \right)$$

Cartesian to Spherical Coordinates (3-D)

$$\begin{array}{*{20}c} {x = R\sin \theta \cos \phi } & {R = \sqrt {x^2 + y^2 + z^2 } } & {} \\ {y = R\sin \theta \sin \phi } & {\phi = \tan ^{ - 1} \left( {\frac{y}{x}} \right)} & {} \\ {z = R\cos \theta } & {\theta = \cos ^{ - 1} \left( {\frac{z}{{\sqrt {x^2 + y^2 + z^2 } }}} \right)} & {} \\ \end{array}$$

Cartesian to Cylindrical Coordinates (3-D)

$$\begin{array}{*{20}c} {x = r\cos \theta } & {r = \sqrt {x^2 + y^2 } } & {} \\ {y = r\sin \theta } & {\theta = \tan ^{ - 1} \left( {\frac{y}{x}} \right)} & {} \\ {z = z} & {z = z} & {} \\ \end{array}$$

Cylindrical to Spherical Coordinates (3-D)

$$\begin{array}{*{20}c} {r = R\sin \theta } & {R = \sqrt {r^2 + z^2 } } & {} \\ {z = R\sin \theta } & {\phi = \theta } & {} \\ {\theta = \phi } & {\theta = \tan ^{ - 1} \left( {\frac{r}{z}} \right)} & {} \\ \end{array}$$

Arithmetic Series - Sequential Integers

$$\sum\limits_{k = 1}^n {k = \frac{{n\left( {n + 1} \right)}}{2}}$$

Arithmetic Series - Sequential Odd Integers

$$\sum\limits_{k = 1}^n {2k - 1 = n^2 }$$

Arithmetic Series - Sequential Squared Integers

$$\sum\limits_{k = 1}^n {k^2 = \frac{{n\left( {n + 1} \right)\left( {2n + 1} \right)}}{6}}$$

Finite Geometric Series

$$\sum\limits_{k = 1}^n {ar^{k - 1} = \frac{{a\left( {1 - r^n } \right)}}{{1 - r}}}$$

Infinite Geometric Series

$$\sum\limits_{k = 1}^\infty {ar^{k - 1} = \frac{a}{{1 - r}}}$$

Perimeter of a Circle

$$P = 2\pi r$$

Perimeter of a Rectangle

$$P = 2l + 2w$$

Perimeter of a Square

$$P = 4s$$

Perimeter of a Triangle

$$P = a + b + c$$

Perimeter of a Regular Polygon

$$P = ns$$

Spiral of Archimedes (Archimedean Spiral) in Polar Coordinates

$$r = a\theta$$

Arclength

$$s = r\theta$$

L'Hopital's Rule

$$\mathop {\lim }\limits_{x \to c} \frac{{f\left( x \right)}}{{g\left( x \right)}} = \mathop {\lim }\limits_{x \to c} \frac{{f'\left( x \right)}}{{g'\left( x \right)}}$$

Limit of One over X to the nth Power

$$\mathop {\lim }\limits_{x \to \infty } \frac{1}{{x^n }} = 0$$

Limit of Arctangent X as X Approaches Infinity

$$\mathop {\lim }\limits_{x \to \infty } \tan ^{ - 1} \left( x \right) = \frac{\pi }{2}$$

Limit of Arctangent X as X Approaches Negative Infinity

$$\mathop {\lim }\limits_{x \to - \infty } \tan ^{ - 1} \left( x \right) = - \frac{\pi }{2}$$

Limit of e to the X power as X Approaches Negative Infinity

$$\mathop {\lim }\limits_{x \to - \infty } e^x = 0$$

Entropy Change

$$\Delta S^\circ = \sum {S^\circ {\rm{products}}} - \sum {S^\circ {\rm{reactants}}}$$

Enthalpy Change

$$\Delta H^\circ = \sum {H^\circ _f {\rm{products}}} - \sum {H^\circ _f {\rm{reactants}}}$$

Gibb's Free Energy Change Defined

$$\Delta G^\circ = \sum {G^\circ _f {\rm{products}}} - \sum {G^\circ _f {\rm{reactants}}}$$

Gibb's Free Energy Change in Terms of Enthalpy, Absolute Temperature, and Entropy

$$\Delta G^\circ = \Delta H^\circ - T\Delta S^\circ$$

Gibb's Free Energy Change in Terms of Gas Constant, Absolute Temperature, and Equilibrium Constant

$$\Delta G^\circ = - RT\ln K = - 2.303RT\log K$$

Gibb's Free Energy Change in Terms of Number of Moles, Faraday, and Standard Reduction Potential

$$\Delta G^\circ = - n\Im E^\circ$$

Reaction Quotient

$$\begin{array}{*{20}c} {Q = \frac{{\left[ C \right]^c \left[ D \right]^d }}{{\left[ A \right]^a \left[ B \right]^b }}} \\ {\begin{array}{*{20}c} {where} & {aA + bB \to cC + dD} \\ \end{array}} \\ \end{array}$$

Electric Current

$$I = \frac{q}{t}$$

Cell Voltage

$$E_{cell} = E^\circ _{cell} - \frac{{RT}}{{n\Im }}\ln Q = E^\circ _{cell} - \frac{{0.0592}}{n}\log Q$$

Relationship between Equilibrium Constant and Cell Voltage

$$\log K = \frac{{nE^\circ }}{{0.0592}}$$

Molar Heat Capacity at Constant Pressure

$$C_p = \frac{{\Delta H}}{{\Delta T}}$$

Partial Pressure of a Gas

$$\begin{array}{*{20}c} {P_A = P_{total} X_A } \\ {\begin{array}{*{20}c} {where} & {X_A = \frac{{\begin{array}{*{20}c} {moles} & A \\ \end{array}}}{{\begin{array}{*{20}c} {total} & {moles} \\ \end{array}}}} \\ \end{array}} \\ \end{array}$$

Total Gas Pressure as Sum of Partial Pressures

$$P_{total} = P_A + P_B + P_C + \ldots$$

Number of Moles

$$n = \frac{m}{M}$$

Temperature in Kelvin from Degrees Celsius Conversion

$$K = ^\circ C + 273$$

Combined Gas Law

$$\frac{{P_1 V_1 }}{{n_1 T_1 }} = \frac{{P_2 V_2 }}{{n_2 T_2 }}$$

Density of a Material

$$D = \frac{m}{V}$$

Root Mean Square Velocity of Gas Molecules

$$u_{rms} = \sqrt {\frac{{3kT}}{m}} = \sqrt {\frac{{3RT}}{M}}$$

Kinetic Energy per molecule

$$\frac{{KE}}{{molecule}} = \frac{1}{2}m\upsilon ^2$$

Kinetic Energy per Mole

$$\frac{{KE}}{{mole}} = \frac{3}{2}RTn$$

Kinetic Energy per Mole

$$\frac{{KE}}{{mole}} = \frac{3}{2}RTn$$

Graham's Law of Effusion

$$\frac{{r_1 }}{{r_2 }} = \sqrt {\frac{{M_2 }}{{M_1 }}}$$

Molarity Defined

$$\begin{array}{*{20}c} {molarity,} & {M = \frac{{\begin{array}{*{20}c} {moles} & {solute} \\ \end{array}}}{{\begin{array}{*{20}c} {liter} & {solution} \\ \end{array}}}} \\ \end{array}$$

Molality Defined

$$\begin{array}{*{20}c} {molality,} & { = \frac{{\begin{array}{*{20}c} {moles} & {solute} \\ \end{array}}}{{\begin{array}{*{20}c} {kilogram} & {solvent} \\ \end{array}}}} \\ \end{array}$$

Freezing Point Depression

$$\Delta T_f = iK_f \times molality$$

Boiling Point Elevation

$$\Delta T_b = iK_b \times molality$$

Osmotic Pressure

$$\pi = \frac{{nRT}}{V}i$$

Specific Heat Capacity to Heat Equation

$$q = mc\Delta T$$

Acid Ionization Constant

$$K_a = \frac{{\left[ {H^ + } \right]\left[ {A^ - } \right]}}{{\left[ {HA} \right]}}$$

Base Ionization Constant

$$K_b = \frac{{\left[ {OH^ - } \right]\left[ {HB^ + } \right]}}{{\left[ B \right]}}$$

Ion Product Constant for Water

$$\begin{array}{*{20}c} {K_w = \left[ {OH^ - } \right]\left[ {H^ + } \right] = K_a \times K_b } \\ {\begin{array}{*{20}c} { = 1.0 \times 10^{ - 14} } & {at} & {25^\circ C} \\ \end{array}} \\ \end{array}$$

pH Defined

$$pH = - \log \left[ {H^ + } \right]$$

pOH Defined

$$pOH = - \log \left[ {OH^ - } \right]$$

pH and pOH Relationship

$$14 = pH + pOH$$

Buffer Design Equation

$$pH \approx pK_a - \log \frac{{\left[ {HA} \right]_0 }}{{\left[ {A^ - } \right]_0 }}$$

pOH and Base Ionization Equilibrium Constant Relationship

$$pOH = pK_b + \log \frac{{\left[ {HB^ + } \right]}}{{\left[ B \right]}}$$

pKa Definition

$$pK_a = - \log K_a$$

pKb Definition

$$pK_b = - \log K_b$$

Gas Pressure and Concentration Relationship

$$K_p = K_c \left( {RT} \right)^{\Delta n}$$

Planck's Quantized (Quantum) Energy Equation

$$\Delta E = h\nu$$

Speed of Light to Wavelength and Frequency Relationship

$$c = \lambda \nu$$

De Broglie Wavelength

$$\lambda = \frac{h}{{m\upsilon }}$$

Linear Momentum

$$p = m\upsilon$$

Relationship between Energy and Principal Quantum Number

$$E_n = - R_H \left( {\frac{1}{{n^2 }}} \right) = \frac{{ - 2.178 \times 10^{ - 18} }}{{n^2 }}joule$$

Rydberg Equation

$$\Delta E = R_H \left( {\frac{1}{{n_i ^2 }} - \frac{1}{{n_f ^2 }}} \right)$$

van't Hoff equation

$$\ln \left( {\frac{{K_2 }}{{K_1 }}} \right) = - \frac{{\Delta H^\circ }}{R}\left[ {\frac{1}{{T_2 }} - \frac{1}{{T_1 }}} \right]$$


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