Principle Library

Deep dives into specific concepts. Each principle guide covers derivations, applications, and common pitfalls.

259 principles across 2 domains

Mathematics

Algebra (30)

Absolute Value - Definition: The Piecewise Distance Rule

Absolute Value Cases: Split Into Two Branches

Additive Equality: Adding the Same Quantity to Both Sides

Additive Inequality: Same Quantity, Both Sides

Additive Inverse: Canceling an Offset with Its Opposite

Clear Denominators: Multiply to Remove Fractions from Equations

Combine Like Terms: Merge Coefficients of Identical Powers

Completing the Square (Rewrite Identity): Quadratic to Vertex Form by Pattern Substitution

Cross Multiplication: Solving Proportions by Cross Products

Difference of Squares: Factor a Squared Difference into Two Binomials

Distributive Property: Expand Multiplication Over Addition

Exponent Power Rule: Multiply Exponents in a Nested Power

Exponent Product Rule: Add Exponents When Multiplying Same-Base Powers

Exponential Model: Equal Multiplier per Unit Input

Factor Common Term: Reverse the Distributive Property

Linear Model: Understanding Slope and Intercept

Multiplicative Equality: Multiply or Divide Both Sides

Multiplicative Inequality: Flipping When the Multiplier Is Negative

Multiplicative Inverse: Canceling a Factor with a Reciprocal

Negative Exponent Rule: Rewrite to Reciprocal Form

Perfect Square Trinomial: Rewrite as a Squared Binomial

Proportional Model: Mastering Direct Variation with y = kx

Quadratic Factoring: Rewrite a Quadratic as a Product of Linear Factors

Quadratic Formula: Extract Both Roots Without Factoring

Quadratic Model: Parabolic Curves and Second-Degree Structure

Radical Definition: The Principal Nonnegative Root Rule

Simplify Radicals (Square Factor): Extract Perfect-Square Roots

Square Root Property: Splitting a Squared Equation into Two Branches

Substitution Property: Replace Any Expression with an Equal One

Zero Product Property: From Factored Form to Solutions

Calculus (47)

Antiderivative definition: Recognizing the Reverse Derivative

Average value of a function: Integrating to Find the Mean

Chain rule: Differentiating Composite Functions

Continuity at a Point: Definition and the Three-Part Test

Definite integral (Riemann sum form): Area as a Limit of Sums

Derivative at a Point (Definition): The Difference Quotient

Derivative Constant Multiple Rule: Factor Constants Out of Derivatives

Derivative of a constant: Any constant differentiates to zero

Derivative of a^x: Why the Log Factor Appears

Derivative of e^x: The Function Equal to Its Own Derivative

Derivative of ln(x): Differentiate the Natural Log on Its Domain

Derivative sum rule: Distribute differentiation over addition

Differentiate both sides: Preserve equality under differentiation

First derivative test (local maximum): Infer a local peak from a positive-to-negative sign change

First derivative test (local minimum): Infer a local minimum from a sign chart

Fundamental Theorem of Calculus (Part 1): Differentiate accumulation integrals directly

Fundamental Theorem of Calculus (Part 2): Evaluate a definite integral from an antiderivative

Indefinite Integral as Antiderivative: Finding Every F with F' = f

Integral constant multiple rule: Pull constants outside an integral

Integral of 1/x: Recognize the Logarithmic Antiderivative Away From Zero

Integral of a^x: Why the Log Divisor Appears

Integral of e^x: Antidifferentiate the natural exponential in one step

Integral sum rule: Split an integral before integrating each part

Integration by parts: Reassign derivative and antiderivative roles in a product

L'Hopital's rule: Replace an indeterminate quotient with a ratio of derivatives

Left-hand limit statement: f(x) Approaching L from the Left

Limit constant multiple rule: Pull a constant factor out of a limit

Limit of a Constant: Evaluating Constant Limits Directly

Limit of a continuous composition: Push the limit through an outer function

Limit of the Identity: Direct Substitution Base Case

Limit product rule: Split a product to evaluate limits separately

Limit quotient rule: Split a quotient to evaluate limits separately

Limit statement: What It Means for f(x) to Approach L

Limit sum rule: Split a sum to evaluate limits separately

Linear Approximation: Estimating Function Values Near a Point

Power rule for integrals: Integrate x^n when n is not -1

Power rule: Differentiate real powers of x

Product Rule: Differentiating Products of Functions

Quotient Rule: Differentiating Ratios of Functions

Right-hand limit statement: f(x) Approaching L from the Right

Second derivative test (local maximum): Classify a critical point from concavity

Second derivative test (local minimum): Infer a local minimum from curvature at a critical point

Squeeze theorem: Evaluating limits by bounding

Substitution rule (definite integral): Change variables and bounds together

Substitution rule (u-substitution): Rewrite integrals by changing variables

Tangent Line Equation: Slope from the Derivative

Taylor Polynomial Definition: Approximating Functions with Polynomials

Differential Equations (47)

Autonomous Differential Equation Form: Slopes Depend on y

Boundary Value Problem Form: Conditions at Both Ends

Characteristic Equation Relation: Build the Auxiliary Equation

Complex Roots Solution Family: Oscillating Modes

Differential Equation Solution Condition: Testing a Candidate Function

Direction Field Slope Relation: Read Slopes from a Point

Distinct Real Roots Solution Family: Two Exponential Modes

Eigenvalue Boundary-Condition Problem: Find Allowed Modes

Eigenvalue-Eigenvector Solution Mode: Exponential Modes for Linear Systems

Equilibrium Jacobian Linearization: Local Linear Models

Euler Method Step: Advance Using the Current Slope

Exact Equation Potential Relation: Build the Potential Function

First-Order Explicit Differential Equation Form: Reading an ODE Slope Rule

First-Order Linear Standard Form: Put Linear ODEs in Shape

First-Order Linear System Form: Matrix Form for ODE Systems

Fourier Series Mode Expansion: Build Functions from Modes

Fundamental Solution Set General Form: Span Homogeneous Solutions

General Solution Family Parameter: One Formula, Many Curves

Homogeneous First-Order Equation Form: Slope Depends on y over x

Homogeneous First-Order Reduction: Substitute y equals v times x

Initial Condition Particular Solution: Choosing One Family Member

Integrating Factor Definition: Build the Multiplier

Integrating Factor Product Derivative: Collapse the Left Side

Inverse Laplace Transform Relation: Return to the Time Domain

Laplace Convolution Theorem: Multiply Transforms

Laplace Derivative Transform: Turn Derivatives into Algebra

Laplace Frequency-Shift Transform: Shift the Transform Argument

Laplace Second-Derivative Transform: Carry Initial Data

Laplace Time-Shift Transform: Delay with Unit Steps

Laplace Transform Definition: Move Functions into the s-Domain

Linear Homogeneous Superposition: Combine Solutions Safely

Linear Nonhomogeneous Solution Structure: Build Full Solutions

Logistic Differential Model: Growth Slows Near Capacity

Matrix Exponential Solution Form: Full System Solution Family

Nonhomogeneous Linear System Solution Structure: Build the Full Solution Family

Repeated Root Solution Family: Extra x Mode

Resonance Modified Trial Factor: Clear Homogeneous Overlap

Scalar Equilibrium Solution Condition: Constant ODE Solutions

Second-Order Linear Constant-Coefficient Form: Fix the Coefficients

Second-Order Linear Standard Form: Recognize the Template

Separable Equation Product Form: Recognize Separable Differential Equations

Separation of Variables: The Safe Rewrite Step

System Equilibrium Condition: Constant Solutions of Systems

System Variation Of Constants Formula: Solve Forced Linear Systems

Undetermined Coefficients Trial Family: Match the Forcing

Unit Step Forcing Representation: Model a Single Switch

Variation Of Parameters Particular Solution: Build Forced Solutions

Functions (12)

Affine Transform Form: Parameters of a Transformed Function

Apply Inverse to Both Sides: Isolate a Variable Inside a Function

Composition Definition: Feeding One Function's Output into Another

Composition Expansion: Evaluate a Composed Function in Two Steps

Evaluate by Substitution: Plug an Input into a Function Rule

Function Rule Definition: What It Means to Define a Function by a Rule

Inverse Cancellation: Recover the Original Input or Output

Inverse Definition: Reversing a Function's Input and Output

Log-Exponential Rewrite: Convert Between Log and Exponential Forms

Logarithm Model: The Inverse Relationship to Exponentiation

Piecewise Branch Selection: Evaluating the Right Rule

Piecewise Definition: Define Functions with Multiple Rules Across Regions

Physics

Classical Mechanics (60)

Acceleration - Derivative Definition: Instantaneous Rate of Change

Angular Acceleration - Derivative Definition: Instantaneous Rotational Change

Angular Displacement Integral Relation

Angular Impulse Formula: When Torque Over Time Changes Angular Momentum

Angular Impulse-Angular Momentum Theorem (Integral): Time-Varying Torque

Angular Momentum (Particle): Rotational Motion About a Point

Angular Momentum of a Rigid Body: Fixed-Axis L = Iω

Angular Velocity - Derivative Definition: How Rotation Rate Changes

Angular Velocity Change: Integral Form for Rotational Kinematics

Arc Length-Angle Relation: Connecting Linear and Rotational Motion

Center of Mass (Position): Locating the Balance Point

Center of Mass (Velocity): Finding the Average Motion

Centripetal Acceleration: Understanding Acceleration in Circular Motion

Conservation of Angular Momentum

Conservation of Linear Momentum

Displacement from Average Velocity — Kinematics 4

Displacement-Integral Relation: Finding Position from Velocity

Gravitational Potential Energy (Near Surface): Mastering the mgh Formula

Hooke's Law: Understanding Spring Force and Elastic Behavior

Impulse-Momentum Theorem (Algebraic): Impulse Equals Momentum Change

Impulse-Momentum Theorem (Integral): General Time-Varying Forces

Kinematics 1 - Algebraic: Predicting Position Under Constant Acceleration

Kinematics 2 - Algebraic: Velocity as a Function of Time

Kinetic Friction: Understanding Sliding Resistance in Mechanics

Linear Momentum (Definition): Quantifying Motion's Persistence

Mechanical Energy Conservation: Master Energy Transformations

Mechanical Energy with Non-Conservative Work: Tracking Energy Transfers

Moment of Inertia - Integral Definition: Continuous Mass Distributions

Moment of Inertia (Discrete): Quantifying Rotational Inertia

Newton's First Law (Rotation): Rotational Equilibrium

Newton's First Law (Translation): When Forces Balance

Newton's Law of Gravitation: Understanding Universal Attraction

Newton's Second Law - Momentum Form: Force as Rate of Change

Newton's Second Law (Rotation): Connecting Torque to Angular Acceleration

Newton's Second Law (Translation): The Foundation of Force and Motion

Newton's Third Law: Identify Action–Reaction Pairs

Parallel Axis Theorem: Shifting Rotational Inertia

Power - Derivative Form: Instantaneous Rate of Energy Transfer

Power (Algebraic): Calculating Energy Transfer Rate

Power (Rotation): Instantaneous Rate of Rotational Energy Transfer

Rotational Kinematics 1: Angular Position Under Constant Angular Acceleration

Rotational Kinematics 2: Angular Velocity with Constant Angular Acceleration

Rotational Kinematics 3: Angular Velocity Without Time

Rotational Kinematics 4: Angular Displacement via Average Angular Velocity

Rotational Kinetic Energy: Master Energy in Rotating Systems

Rotational Work: Energy from Constant Net Torque

Spring Potential Energy: Elastic Energy Storage

Static Friction: Understanding the Force that Prevents Sliding

Tangential Acceleration Formula: When and How to Use It

Tangential Speed: Formula, Meaning, and Tangential Velocity

Torque - Angular Momentum Form: Rotational Dynamics in Calculus Form

Torque (Definition): Measuring Rotational Force Effectiveness

Translational Kinetic Energy Formula: When 1/2mv^2 Applies

Translational Work: Energy from Constant Force

Velocity - Derivative Definition: Instantaneous Rate of Position Change

Velocity Change - Integral Relation: Find Velocity from Acceleration

Velocity-Position Relation: Kinematics 3

Weight (Near Surface): Understanding Gravitational Force in Local Contexts

Work - Integral Definition: Computing Work Along Arbitrary Paths

Work-Energy Theorem: Connect Force to Speed

Electromagnetism (63)

Ampere Law: Current Sets Magnetic Circulation

Ampere-Maxwell Law: Magnetic Circulation From Two Sources

Biot-Savart Law: Source Geometry to Magnetic Field

Capacitance Definition: Charge Stored per Volt

Capacitor Energy: Stored Energy from Voltage

Capacitor Impedance: Frequency-Dependent AC Opposition

Capacitor Time Constant: Time Scale for RC Circuits

Charge Density Differential Relation: Turn Density Into dq

Coulomb Force: Start With Magnitude, Keep Direction Explicit

Current Density Definition: Add Flow Through Oriented Area

Electric Current Definition: Charge Flow per Time

Electric Field Energy Density: Field Energy Per Volume

Electric Field From Continuous Charge Distribution: Add dq Fields

Electric Field From Point Charge: Start With Magnitude, Keep Radial Direction Explicit

Electric Field From Potential Gradient: Field Points Downhill

Electric Field Superposition: Add Source Fields As Vectors

Electric Field-Force Relation: Use One Relation in Both Directions

Electric Flux In A Uniform Field: Use the Area Vector

Electric Flux Integral: Add Local Field Through Surface

Electric Potential Energy From Potential: Track Charge Sign and Delta V

Electric Potential Energy Of Two Point Charges: Track Sign And Reference

Electric Potential From Continuous Charge Distribution: Add dq Potentials

Electric Potential Line Integral: Field Along Path Sets Voltage

Electric Potential Of A Point Charge: Keep Scalar Value And Reference Explicit

Electric Power: Current, Voltage, and Energy Rate

Electromagnetic Wave Field Relation: Linked by the Wave Speed

Electromagnetic Wave Speed: Vacuum Constants Set c

Equivalent Capacitance In Parallel: Shared Voltage Rule

Equivalent Capacitance In Series: Shared Charge Rule

Equivalent Resistance In Parallel: Reciprocal Sum Rule

Equivalent Resistance In Series: Direct Sum Rule

Faraday Law Finite Change: Average Induced EMF

Faraday Law Integral: Circulation From Changing Flux

Gauss Law For Magnetism: No Net Flux Through a Closed Surface

Gauss Law: Net Flux Reveals Enclosed Charge

Inductance-Flux Relation: The Flux Linkage Model of a Coil

Inductor Energy: Magnetic Energy Stored by Current

Inductor Impedance: AC Opposition That Rises With Frequency

Inductor Voltage Relation: Voltage From Changing Current

Kirchhoff Junction Rule: Balance Current at a Node

Kirchhoff Loop Rule: Balance Voltage Around a Closed Path

Lorentz Force: Add Electric and Magnetic Force

Magnetic Dipole Energy: Alignment and Potential Energy

Magnetic Dipole Moment Of A Current Loop: Area Vector

Magnetic Field Energy Density: Field Energy Per Volume

Magnetic Field in a Long Solenoid: Interior Field Formula

Magnetic Field Near A Long Straight Wire: Formula and Direction

Magnetic Flux In A Uniform Field: Area-Vector Orientation

Magnetic Flux Integral: Field Through an Oriented Surface

Magnetic Force On A Moving Charge: Use the Cross Product

Magnetic Force On A Wire: Formula and Direction

Microscopic Ohm's Law: Local Current Density From Electric Field

Motional EMF: Moving Conductors in Magnetic Fields

Ohm's Law: Voltage, Current, and Resistance

Parallel-Plate Capacitance: Area, Gap, and Medium

Poynting Vector Definition: Direction of Energy Flow

RC Charging Voltage: Exponential Rise to Source Voltage

RC Discharging Voltage: Exponential Decay from Initial Voltage

Resistance From Geometry: Length, Area, and Material

Resistor Impedance: AC Resistance in Phasor Circuits

RL Time Constant: Time Scale for Inductor Circuits

Torque On A Magnetic Dipole: Direction, Size, and Alignment

Uniform-Field Potential Difference: Track the Sign of Voltage

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