Glossary — Math Placement Prep

Arithmetic

10

Order of Operations (PEMDAS)

$$P \to E \to MD \to AS$$

Parentheses first, then Exponents, then Multiplication/Division (left to right), then Addition/Subtraction (left to right).

Greatest Common Divisor (GCD)

$$\gcd(a,b)$$

The largest positive integer that divides both $a$ and $b$. Found via the Euclidean algorithm: $\gcd(a,b) = \gcd(b, a \bmod b)$.

Least Common Multiple (LCM)

$$\text{lcm}(a,b) = \frac{|a \cdot b|}{\gcd(a,b)}$$

The smallest positive integer divisible by both $a$ and $b$.

Fraction Addition

$$\frac{a}{b} + \frac{c}{d} = \frac{ad + bc}{bd}$$

To add fractions, find a common denominator, add the numerators, and simplify.

Fraction Multiplication

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

Multiply numerators together and denominators together, then simplify.

Percent

$$\text{Percent} = \frac{\text{Part}}{\text{Whole}} \times 100$$

A ratio expressed as a fraction of 100. To find X% of a number: multiply by X/100.

Proportion

$$\frac{a}{b} = \frac{c}{d} \implies ad = bc$$

Two equal ratios. Cross-multiplication is used to solve for an unknown.

Absolute Value

$$|a| = \begin{cases} a & a \geq 0 \\ -a & a < 0 \end{cases}$$

The distance of a number from zero on the number line. Always non-negative.

Exponent Rules

$$a^m \cdot a^n = a^{m+n}, \quad \frac{a^m}{a^n} = a^{m-n}, \quad (a^m)^n = a^{mn}$$

Fundamental rules for multiplying, dividing, and raising powers of the same base.

Scientific Notation

$$a \times 10^n, \quad 1 \leq |a| < 10$$

A compact way to write very large or small numbers. The coefficient $a$ is between 1 and 10.

Algebra

15

Slope

$$m = \frac{y_2 - y_1}{x_2 - x_1}$$

The rate of change of a line — rise over run between two points.

Slope-Intercept Form

$$y = mx + b$$

A linear equation where $m$ is the slope and $b$ is the y-intercept.

Point-Slope Form

$$y - y_1 = m(x - x_1)$$

A linear equation using a known point $(x_1, y_1)$ and slope $m$.

Standard Form (Line)

$$Ax + By = C$$

A linear equation with integer coefficients; useful for finding intercepts.

Quadratic Formula

$$x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$$

Solves $ax^2 + bx + c = 0$. The discriminant $b^2 - 4ac$ determines the number and type of roots.

Discriminant

$$\Delta = b^2 - 4ac$$

$\Delta > 0$: two real roots. $\Delta = 0$: one repeated root. $\Delta < 0$: two complex roots.

Vertex of a Parabola

$$\left(-\frac{b}{2a},\; f\!\left(-\frac{b}{2a}\right)\right)$$

The maximum or minimum point of $f(x) = ax^2 + bx + c$.

Factoring Difference of Squares

$$a^2 - b^2 = (a-b)(a+b)$$

A binomial that is the difference of two perfect squares factors into conjugates.

Factoring Perfect Square Trinomial

$$a^2 \pm 2ab + b^2 = (a \pm b)^2$$

A trinomial that is the square of a binomial.

Completing the Square

$$ax^2+bx+c = a\left(x+\frac{b}{2a}\right)^2 - \frac{b^2-4ac}{4a}$$

Rewriting a quadratic in vertex form to find the vertex or solve the equation.

Distance Formula

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

The straight-line distance between two points in a coordinate plane (from the Pythagorean theorem).

Midpoint Formula

$$M = \left(\frac{x_1+x_2}{2}, \frac{y_1+y_2}{2}\right)$$

The point exactly halfway between two given points.

Direct Variation

$$y = kx$$

A linear relationship through the origin where $k$ is the constant of proportionality.

Inverse Variation

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

As $x$ increases, $y$ decreases proportionally, with constant product $xy = k$.

Binomial Theorem

$$(a+b)^n = \sum_{k=0}^{n} \binom{n}{k} a^{n-k} b^k$$

Expands any power of a binomial into a sum involving binomial coefficients.

Exponentials & Logarithms

9

Exponential Growth / Decay

$$y = y_0 e^{kt}$$

$k > 0$: growth. $k < 0$: decay. $y_0$ is the initial value, $t$ is time.

Half-Life

$$N(t) = N_0 \left(\frac{1}{2}\right)^{t/t_{1/2}}$$

The time for a quantity to reduce to half. $t_{1/2} = \ln 2 / \lambda$.

Logarithm Definition

$$\log_b a = c \iff b^c = a$$

The logarithm answers: "To what power must $b$ be raised to get $a$?"

Log Product Rule

$$\log_b(xy) = \log_b x + \log_b y$$

The log of a product is the sum of the logs.

Log Quotient Rule

$$\log_b\!\left(\frac{x}{y}\right) = \log_b x - \log_b y$$

The log of a quotient is the difference of the logs.

Log Power Rule

$$\log_b(x^n) = n \log_b x$$

An exponent inside a log can be brought out front as a multiplier.

Change of Base

$$\log_b a = \frac{\ln a}{\ln b} = \frac{\log a}{\log b}$$

Converts between logarithm bases using natural or common logs.

Compound Interest

$$A = P\left(1 + \frac{r}{n}\right)^{nt}$$

$P$ = principal, $r$ = annual rate, $n$ = compounds/year, $t$ = years. Continuous: $A = Pe^{rt}$.

Euler's Number

$$e = \lim_{n\to\infty}\left(1+\frac{1}{n}\right)^n \approx 2.71828$$

The base of natural logarithms; fundamental constant in calculus, growth/decay, and complex analysis.

Trigonometry

10

SOH-CAH-TOA

$$\sin\theta = \frac{\text{opp}}{\text{hyp}}, \quad \cos\theta = \frac{\text{adj}}{\text{hyp}}, \quad \tan\theta = \frac{\text{opp}}{\text{adj}}$$

The primary trig ratios for a right triangle.

Pythagorean Identity

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

The fundamental trigonometric identity, valid for all angles $\theta$.

Related Identities

$$1 + \tan^2\theta = \sec^2\theta, \quad 1 + \cot^2\theta = \csc^2\theta$$

Derived from dividing $\sin^2 + \cos^2 = 1$ by $\cos^2$ or $\sin^2$.

Law of Sines

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

Relates sides to opposite angles in any triangle. Used for AAS, ASA, and SSA problems.

Law of Cosines

$$c^2 = a^2 + b^2 - 2ab\cos C$$

Generalizes the Pythagorean theorem to any triangle. Used for SAS and SSS problems.

Double Angle Formulas

$$\sin 2\theta = 2\sin\theta\cos\theta, \quad \cos 2\theta = \cos^2\theta - \sin^2\theta$$

Express trig functions of $2\theta$ in terms of $\theta$.

Unit Circle

$$x^2 + y^2 = 1, \quad (\cos\theta, \sin\theta)$$

A circle of radius 1 centered at the origin. Any point on it has coordinates $(\cos\theta, \sin\theta)$.

Radians ↔ Degrees

$$\theta_{\text{rad}} = \theta_{\text{deg}} \times \frac{\pi}{180}$$

There are $2\pi$ radians in a full rotation (360°). $\pi$ rad = 180°.

Arc Length

$$s = r\theta$$

Arc length equals radius times central angle (in radians).

Area of a Triangle (Trig)

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

Area using two sides and their included angle — no need for height.

Geometry

10

Pythagorean Theorem

$$a^2 + b^2 = c^2$$

In a right triangle, the square of the hypotenuse equals the sum of the squares of the other two sides.

Area of a Circle

$$A = \pi r^2$$

The area enclosed by a circle of radius $r$.

Circumference

$$C = 2\pi r = \pi d$$

The perimeter (distance around) a circle.

Volume of a Sphere

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

The space enclosed by a sphere of radius $r$.

Surface Area of a Sphere

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

The total outer area of a sphere of radius $r$.

Volume of a Cylinder

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

Base area times height for a circular cylinder.

Volume of a Cone

$$V = \frac{1}{3}\pi r^2 h$$

One-third of the cylinder with the same base and height.

Area of a Trapezoid

$$A = \frac{1}{2}(b_1 + b_2)h$$

Average of the two parallel sides (bases) times the height.

Heron's Formula

$$A = \sqrt{s(s-a)(s-b)(s-c)}, \quad s = \frac{a+b+c}{2}$$

Area of a triangle from its three side lengths, where $s$ is the semi-perimeter.

Equation of a Circle

$$(x-h)^2 + (y-k)^2 = r^2$$

Standard form of a circle with center $(h,k)$ and radius $r$.

Sequences & Series

8

Arithmetic Sequence

$$a_n = a_1 + (n-1)d$$

Each term differs from the previous by a constant common difference $d$.

Arithmetic Series Sum

$$S_n = \frac{n}{2}(a_1 + a_n) = \frac{n}{2}(2a_1+(n-1)d)$$

Sum of the first $n$ terms of an arithmetic sequence.

Geometric Sequence

$$a_n = a_1 \cdot r^{n-1}$$

Each term is multiplied by a constant common ratio $r$.

Geometric Series Sum (finite)

$$S_n = a_1 \cdot \frac{1-r^n}{1-r}, \quad r \neq 1$$

Sum of the first $n$ terms of a geometric sequence.

Infinite Geometric Series

$$S = \frac{a_1}{1-r}, \quad |r| < 1$$

Converges only when $|r| < 1$. Otherwise the series diverges.

Sigma Notation

$$\sum_{k=1}^{n} a_k = a_1 + a_2 + \cdots + a_n$$

Compact notation for writing sums. The index $k$ runs from the lower to upper bound.

Sum of First n Integers

$$\sum_{k=1}^{n} k = \frac{n(n+1)}{2}$$

Gauss's formula. Also called the triangular number formula.

Sum of First n Squares

$$\sum_{k=1}^{n} k^2 = \frac{n(n+1)(2n+1)}{6}$$

The sum of $1^2 + 2^2 + \cdots + n^2$.

Calculus

12

Definition of a Limit

$$\lim_{x \to a} f(x) = L$$

As $x$ approaches $a$, $f(x)$ approaches $L$. The foundation of all calculus.

Definition of the Derivative

$$f'(x) = \lim_{h \to 0} \frac{f(x+h) - f(x)}{h}$$

The instantaneous rate of change — the slope of the tangent line at a point.

Power Rule

$$\frac{d}{dx} x^n = nx^{n-1}$$

The most fundamental derivative rule. Works for all real $n$.

Product Rule

$$\frac{d}{dx}[f \cdot g] = f'g + fg'$$

Derivative of a product of two functions.

Quotient Rule

$$\frac{d}{dx}\left[\frac{f}{g}\right] = \frac{f'g - fg'}{g^2}$$

Derivative of a quotient. Remember: "low d-high minus high d-low, over low squared."

Chain Rule

$$\frac{d}{dx}f(g(x)) = f'(g(x)) \cdot g'(x)$$

Derivative of a composite function: differentiate the outer, multiply by the derivative of the inner.

Common Derivatives

$$\frac{d}{dx}\sin x = \cos x, \quad \frac{d}{dx}\cos x = -\sin x, \quad \frac{d}{dx}e^x = e^x, \quad \frac{d}{dx}\ln x = \frac{1}{x}$$

Essential derivatives of transcendental functions used throughout calculus.

Fundamental Theorem of Calculus

$$\int_a^b f(x)\,dx = F(b) - F(a), \quad F'(x) = f(x)$$

Links differentiation and integration: the definite integral equals the antiderivative evaluated at the bounds.

Power Rule (Integration)

$$\int x^n\,dx = \frac{x^{n+1}}{n+1} + C, \quad n \neq -1$$

The reverse of the power rule for derivatives. $C$ is the constant of integration.

Integration by Parts

$$\int u\,dv = uv - \int v\,du$$

Used when the integrand is a product. Choose $u$ and $dv$ using LIATE.

Area Under a Curve

$$A = \int_a^b f(x)\,dx$$

The definite integral gives the signed area between a curve and the x-axis over $[a,b]$.

L'Hôpital's Rule

$$\lim_{x\to a}\frac{f(x)}{g(x)} = \lim_{x\to a}\frac{f'(x)}{g'(x)}$$

Used when a limit gives $0/0$ or $\infty/\infty$. Differentiate numerator and denominator separately.

Linear Algebra

8

Matrix Multiplication

$$(AB)_{ij} = \sum_k A_{ik}B_{kj}$$

The entry in row $i$, column $j$ of the product is the dot product of row $i$ of $A$ and column $j$ of $B$.

Determinant (2×2)

$$\det\begin{pmatrix}a&b\\c&d\end{pmatrix} = ad - bc$$

Measures how a 2D transformation scales area. Zero determinant means the matrix is singular.

Matrix Inverse (2×2)

$$A^{-1} = \frac{1}{ad-bc}\begin{pmatrix}d&-b\\-c&a\end{pmatrix}$$

Exists only when $\det(A) \neq 0$. Satisfies $AA^{-1} = I$.

Cramer's Rule

$$x_i = \frac{\det(A_i)}{\det(A)}$$

Solves a system of linear equations using determinants. $A_i$ replaces the $i$-th column with the constant vector.

Eigenvalue Equation

$$A\mathbf{v} = \lambda\mathbf{v}$$

An eigenvector $\mathbf{v}$ is scaled by factor $\lambda$ (eigenvalue) when multiplied by matrix $A$.

Characteristic Equation

$$\det(A - \lambda I) = 0$$

Solving this polynomial gives the eigenvalues of matrix $A$.

Dot Product

$$\mathbf{a} \cdot \mathbf{b} = |\mathbf{a}||\mathbf{b}|\cos\theta = \sum a_i b_i$$

A scalar measuring how aligned two vectors are. Zero means perpendicular.

Cross Product

$$\mathbf{a} \times \mathbf{b} = |\mathbf{a}||\mathbf{b}|\sin\theta\,\hat{n}$$

A vector perpendicular to both inputs. Its magnitude is the area of the parallelogram they span.

Statistics & Probability

10

Mean (Average)

$$\bar{x} = \frac{1}{n}\sum_{i=1}^{n} x_i$$

The arithmetic mean — sum of all values divided by the count.

Variance & Standard Deviation

$$\sigma^2 = \frac{1}{n}\sum(x_i - \bar{x})^2, \quad \sigma = \sqrt{\sigma^2}$$

Variance measures spread. Standard deviation is its square root, in the same units as the data.

Permutations

$$P(n,r) = \frac{n!}{(n-r)!}$$

The number of ordered arrangements of $r$ items chosen from $n$ items.

Combinations

$$\binom{n}{r} = \frac{n!}{r!(n-r)!}$$

The number of unordered selections of $r$ items from $n$ items.

Probability

$$P(A) = \frac{\text{favorable outcomes}}{\text{total outcomes}}$$

Classical probability for equally likely outcomes. $0 \leq P(A) \leq 1$.

Conditional Probability

$$P(A|B) = \frac{P(A \cap B)}{P(B)}$$

The probability of $A$ given that $B$ has occurred.

Bayes' Theorem

$$P(A|B) = \frac{P(B|A)\,P(A)}{P(B)}$$

Updates the probability of $A$ after observing evidence $B$. Fundamental to statistical inference.

Normal Distribution

$$f(x) = \frac{1}{\sigma\sqrt{2\pi}}\,e^{-\frac{(x-\mu)^2}{2\sigma^2}}$$

The bell curve. Defined by mean $\mu$ and standard deviation $\sigma$. ~68% within 1σ, ~95% within 2σ.

z-Score

$$z = \frac{x - \mu}{\sigma}$$

How many standard deviations a value $x$ is from the mean. Used to compare across different distributions.

Expected Value

$$E[X] = \sum x_i \cdot P(x_i)$$

The long-run average outcome of a random variable. Weighted mean using probabilities.

Differential Equations

6

Separable ODE

$$\frac{dy}{dx} = f(x)g(y) \implies \int \frac{dy}{g(y)} = \int f(x)\,dx$$

Separate variables to opposite sides and integrate both sides independently.

First-Order Linear ODE

$$\frac{dy}{dx} + P(x)y = Q(x)$$

Solved using an integrating factor $\mu = e^{\int P\,dx}$. Multiply through and integrate.

Second-Order Linear (const. coeff.)

$$ay'' + by' + cy = 0 \implies ar^2+br+c=0$$

Solve the characteristic equation for $r$. Roots give $e^{r_1 x}$, $e^{r_2 x}$ (or sinusoidal solutions for complex roots).

Hooke's Law / SHM

$$m\ddot{x} + kx = 0 \implies x(t) = A\cos(\omega t + \phi)$$

Simple harmonic motion. $\omega = \sqrt{k/m}$. Solutions are sinusoidal.

Damped Oscillation

$$m\ddot{x} + c\dot{x} + kx = 0$$

Adds friction/damping. Solutions decay exponentially: $x = Ae^{-\gamma t}\cos(\omega_d t + \phi)$.

Euler's Method

$$y_{n+1} = y_n + h\,f(x_n, y_n)$$

Numerical approximation of ODE solutions. Step size $h$ controls accuracy.

Complex Numbers

6

Imaginary Unit

$$i = \sqrt{-1}, \quad i^2 = -1$$

The building block of complex numbers. Allows solutions to equations like $x^2 + 1 = 0$.

Complex Number

$$z = a + bi$$

Has a real part $a$ and imaginary part $b$. Plotted on the complex plane.

Modulus

$$|z| = \sqrt{a^2 + b^2}$$

The distance from the origin to $z$ in the complex plane.

Polar Form

$$z = r(\cos\theta + i\sin\theta) = re^{i\theta}$$

Expresses a complex number using magnitude $r$ and angle $\theta$.

Euler's Formula

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

Connects the exponential function to trigonometry. One of the most beautiful equations in math.

De Moivre's Theorem

$$(\cos\theta + i\sin\theta)^n = \cos(n\theta) + i\sin(n\theta)$$

Raises a complex number in polar form to a power. Also used for finding $n$-th roots.

Physics

18

Newton's Second Law

$$\vec{F} = m\vec{a}$$

Net force equals mass times acceleration. The foundation of classical mechanics.

Kinematic Equations

$$v = v_0 + at, \quad x = v_0 t + \tfrac{1}{2}at^2, \quad v^2 = v_0^2 + 2a\Delta x$$

Constant-acceleration motion in one dimension. Three equations, five variables.

Weight

$$W = mg$$

The gravitational force on a mass. $g \approx 9.8$ m/s² on Earth's surface.

Friction

$$f = \mu N$$

Friction force equals the coefficient of friction times the normal force.

Kinetic Energy

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

Energy of motion. Proportional to the square of speed.

Potential Energy (Gravity)

$$PE = mgh$$

Energy stored due to height $h$ above a reference level.

Conservation of Energy

$$KE_1 + PE_1 = KE_2 + PE_2$$

Total mechanical energy is conserved when only conservative forces act.

Work-Energy Theorem

$$W = \Delta KE = F \cdot d \cos\theta$$

Net work done on an object equals its change in kinetic energy.

Power

$$P = \frac{W}{t} = Fv$$

The rate of doing work. Measured in watts (J/s).

Momentum

$$\vec{p} = m\vec{v}$$

The product of mass and velocity. Conserved in all collisions.

Impulse

$$\vec{J} = \vec{F}\Delta t = \Delta\vec{p}$$

Force applied over time produces a change in momentum.

Universal Gravitation

$$F = G\frac{m_1 m_2}{r^2}$$

Every mass attracts every other mass. $G = 6.674 \times 10^{-11}$ N·m²/kg².

Coulomb's Law

$$F = k\frac{|q_1 q_2|}{r^2}$$

Electrostatic force between two charges. $k = 8.99 \times 10^9$ N·m²/C².

Ohm's Law

$$V = IR$$

Voltage equals current times resistance in a conductor.

Electric Power

$$P = IV = I^2R = \frac{V^2}{R}$$

Power dissipated in a resistor. Three equivalent forms.

Projectile Range

$$R = \frac{v_0^2 \sin 2\theta}{g}$$

Horizontal range of a projectile launched at angle $\theta$ with speed $v_0$. Maximum at 45°.

Centripetal Acceleration

$$a_c = \frac{v^2}{r}$$

Acceleration toward the center of a circular path of radius $r$ at speed $v$.

Wave Equation

$$v = f\lambda$$

Wave speed equals frequency times wavelength. Universal for all wave types.

Chemistry

12

Moles

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

$n$ = moles, $m$ = mass (g), $M$ = molar mass (g/mol). 1 mole = $6.022 \times 10^{23}$ particles.

Avogadro's Number

$$N_A = 6.022 \times 10^{23}\,\text{mol}^{-1}$$

The number of atoms, molecules, or particles in one mole of substance.

pH

$$\text{pH} = -\log_{10}[\text{H}^+]$$

Measures acidity/basicity on a logarithmic scale. pH 7 = neutral. Each unit = 10× change in [H⁺].

pH + pOH

$$\text{pH} + \text{pOH} = 14$$

At 25°C, the sum of pH and pOH always equals 14.

Ideal Gas Law

$$PV = nRT$$

$P$ = pressure, $V$ = volume, $n$ = moles, $R = 8.314$ J/(mol·K), $T$ = temperature (K).

Boyle's Law

$$P_1V_1 = P_2V_2$$

At constant temperature and amount, pressure and volume are inversely proportional.

Charles's Law

$$\frac{V_1}{T_1} = \frac{V_2}{T_2}$$

At constant pressure and amount, volume is directly proportional to temperature (K).

Equilibrium Constant

$$K = \frac{[C]^c[D]^d}{[A]^a[B]^b}$$

For $aA + bB \rightleftharpoons cC + dD$. $K > 1$: products favored. $K < 1$: reactants favored.

Arrhenius Equation

$$k = Ae^{-E_a/(RT)}$$

Reaction rate constant depends exponentially on temperature. $E_a$ = activation energy.

Dilution Formula

$$M_1V_1 = M_2V_2$$

Relates concentration and volume before and after dilution. Moles of solute stay constant.

Radioactive Decay

$$N(t) = N_0 e^{-\lambda t}, \quad \lambda = \frac{\ln 2}{t_{1/2}}$$

Exponential decay of unstable nuclei. $\lambda$ is the decay constant, $t_{1/2}$ is the half-life.

Gibbs Free Energy

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

Predicts spontaneity. $\Delta G < 0$: spontaneous. Combines enthalpy and entropy.

Thermodynamics

8

Temperature Conversion

$$T_F = \frac{9}{5}T_C + 32, \qquad T_K = T_C + 273.15$$

Converting between Fahrenheit, Celsius, and Kelvin.

Heat Transfer

$$Q = mc\Delta T$$

Heat energy transferred. $m$ = mass, $c$ = specific heat capacity, $\Delta T$ = temperature change.

First Law of Thermodynamics

$$\Delta U = Q - W$$

Change in internal energy = heat added minus work done by the system. Energy is conserved.

Carnot Efficiency

$$\eta = 1 - \frac{T_C}{T_H}$$

Maximum possible efficiency of a heat engine. $T$ in Kelvin. No engine can exceed this.

Entropy Change

$$\Delta S = \frac{Q_{\text{rev}}}{T}$$

For a reversible process. Total entropy of the universe always increases (2nd Law).

Thermal Efficiency

$$\eta = \frac{W}{Q_{\text{in}}} = 1 - \frac{Q_{\text{out}}}{Q_{\text{in}}}$$

Ratio of useful work output to heat input.

Stefan-Boltzmann Law

$$P = \sigma A T^4$$

Radiative power from a blackbody. $\sigma = 5.67 \times 10^{-8}$ W/(m²·K⁴).

Fourier's Law of Heat Conduction

$$Q = -kA\frac{dT}{dx}$$

Rate of heat flow through a material. $k$ = thermal conductivity, $A$ = cross-sectional area.

Engineering

13

Ohm's Law

$$V = IR$$

Voltage = Current × Resistance. The fundamental circuit equation.

Kirchhoff's Current Law (KCL)

$$\sum I_{\text{in}} = \sum I_{\text{out}}$$

At any junction, total current in equals total current out. Conservation of charge.

Kirchhoff's Voltage Law (KVL)

$$\sum V = 0 \text{ (around any loop)}$$

The sum of all voltage drops and rises around a closed loop is zero.

Series Resistors

$$R_T = R_1 + R_2 + \cdots + R_n$$

Total resistance is the sum of individual resistances.

Parallel Resistors

$$\frac{1}{R_T} = \frac{1}{R_1} + \frac{1}{R_2} + \cdots + \frac{1}{R_n}$$

Total resistance is the reciprocal of the sum of reciprocals.

Gear Ratio

$$GR = \frac{N_{\text{driven}}}{N_{\text{driver}}} = \frac{\omega_{\text{in}}}{\omega_{\text{out}}}$$

Relates teeth count to speed ratio. Higher ratio = lower speed but higher torque.

Mechanical Advantage

$$MA = \frac{F_{\text{out}}}{F_{\text{in}}} = \frac{d_{\text{in}}}{d_{\text{out}}}$$

Force multiplication in levers, pulleys, and inclined planes.

Moment / Torque

$$M = F \times d$$

Rotational force. Force times perpendicular distance from the pivot point.

Stress

$$\sigma = \frac{F}{A}$$

Force per unit area in a material. Units: Pascals (N/m²).

Strain

$$\epsilon = \frac{\Delta L}{L_0}$$

Fractional change in length. Dimensionless.

Hooke's Law (Materials)

$$\sigma = E\epsilon$$

Stress is proportional to strain in the elastic region. $E$ = Young's modulus.

Decibel (Power)

$$L = 10\log_{10}\frac{P}{P_0}\text{ dB}$$

Logarithmic scale for power ratios. +3 dB ≈ double power, +10 dB = 10× power.

Decibel (Voltage)

$$L = 20\log_{10}\frac{V}{V_0}\text{ dB}$$

For amplitude quantities. +6 dB ≈ double voltage, +20 dB = 10× voltage.

Vibrations & Waves

9

Simple Harmonic Motion

$$x(t) = A\sin(\omega t + \phi)$$

Sinusoidal oscillation. $A$ = amplitude, $\omega$ = angular frequency, $\phi$ = phase.

Period & Frequency

$$T = \frac{1}{f} = \frac{2\pi}{\omega}$$

Period $T$ (seconds per cycle) is the reciprocal of frequency $f$ (cycles per second, Hz).

Mass-Spring Frequency

$$\omega = \sqrt{\frac{k}{m}}, \quad T = 2\pi\sqrt{\frac{m}{k}}$$

Natural frequency of a mass $m$ on a spring with constant $k$.

Pendulum Period

$$T = 2\pi\sqrt{\frac{L}{g}}$$

Period of a simple pendulum depends only on length $L$ and gravity $g$, not mass.

Damped Oscillation

$$x(t) = Ae^{-\gamma t}\sin(\omega_d t + \phi)$$

Oscillation with exponentially decaying amplitude. $\gamma$ = damping coefficient.

Wave Speed

$$v = f\lambda$$

Speed = frequency × wavelength. Universal for all wave types.

Beat Frequency

$$f_{\text{beat}} = |f_1 - f_2|$$

When two close frequencies interfere, the result oscillates at the difference frequency.

Standing Wave Harmonics

$$f_n = n \cdot \frac{v}{2L}$$

Frequencies of standing waves on a string of length $L$. $n = 1$ is the fundamental.

Doppler Effect

$$f' = f\frac{v \pm v_o}{v \mp v_s}$$

Frequency shift when source and/or observer move relative to the medium.

Sensors & Measurement

6

Sensor Transfer Function

$$V_{\text{out}} = mX + b$$

Linear relationship between measured quantity $X$ and output voltage. $m$ = sensitivity.

ADC Resolution

$$\text{Resolution} = \frac{V_{\text{range}}}{2^n}$$

An $n$-bit ADC divides the input range into $2^n$ levels.

Nyquist Theorem

$$f_s \geq 2f_{\max}$$

Sampling rate must be at least twice the highest signal frequency to avoid aliasing.

Relative Error

$$\text{RE} = \frac{\Delta x}{x} \times 100\%$$

Error as a percentage of the measured value. Measures proportional accuracy.

Error Propagation (Product)

$$\frac{\Delta(ab)}{ab} = \frac{\Delta a}{a} + \frac{\Delta b}{b}$$

For multiplication/division, add relative errors.

Signal-to-Noise Ratio

$$\text{SNR} = 10\log_{10}\frac{P_{\text{signal}}}{P_{\text{noise}}}\text{ dB}$$

Measures signal quality. Higher SNR = cleaner signal.

Financial Math

7

Simple Interest

$$I = Prt, \quad A = P(1 + rt)$$

$P$ = principal, $r$ = annual rate, $t$ = years. Interest does not compound.

Compound Interest

$$A = P\left(1 + \frac{r}{n}\right)^{nt}$$

Interest earns interest. $n$ = times compounded per year.

Continuous Compounding

$$A = Pe^{rt}$$

The limit as compounding frequency approaches infinity. Uses $e \approx 2.71828$.

Present Value

$$PV = \frac{FV}{(1+r)^n}$$

Today's value of a future sum. Discounts by the rate of return.

Future Value of Annuity

$$FV = PMT \cdot \frac{(1+r)^n - 1}{r}$$

Total value of a series of equal payments made at regular intervals.

Loan Payment (Amortization)

$$PMT = P \cdot \frac{r(1+r)^n}{(1+r)^n - 1}$$

Fixed periodic payment to fully repay a loan over $n$ periods at rate $r$ per period.

Rule of 72

$$t_{\text{double}} \approx \frac{72}{r\%}$$

Quick estimate: divide 72 by the annual interest rate to find years to double.

Equation Glossary