Understanding Mathematical Formulas
Master mathematical concepts with comprehensive step-by-step guides, examples, and explanations. Learn how formulas work, when to use them, and practice with real-world applications.
Categories
Difficulty
Quadratic Formula
Solution to quadratic equations ax² + bx + c = 0
Binomial Theorem
Expansion of (x+y)ⁿ
Difference of Squares
Factorization of difference of two squares
Derivative Definition
Definition of derivative as a limit
Chain Rule
Derivative of composite functions
Fundamental Theorem of Calculus
Connection between derivatives and integrals
Normal Distribution
Gaussian distribution probability density
Bayes' Theorem
Update probability based on new evidence
Eigenvalue Equation
Matrix eigenvalue and eigenvector
Dot Product
Inner product of two vectors
Pythagorean Identity
Fundamental trigonometric identity
Euler's Formula
Complex exponential and trigonometric functions
Newton's Second Law
Force equals mass times acceleration
Mass-Energy Equivalence
Einstein's famous equation
Schrödinger Equation
Fundamental equation of quantum mechanics
Area of Circle
Area of circle with radius r
Volume of Sphere
Volume of sphere with radius r
Cauchy-Riemann Equations
Necessary conditions for complex differentiability
Laplace Transform
Integral transform for solving ODEs
Taylor Series
Taylor series expansion around point a
Fourier Series
Representation of periodic functions as sum of sines and cosines
Stokes' Theorem
Connection between line integrals and surface integrals
Central Limit Theorem
Sample means approach normal distribution as sample size increases
3D Pythagorean Theorem
Distance from origin in 3D space
L'Hôpital's Rule
Evaluate limits of indeterminate forms
Integration by Parts
Product rule for integration
Matrix Inverse (2×2)
Inverse of 2×2 matrix
Confidence Interval
Interval estimate for population mean
Maxwell's Equations
Fundamental equations of electromagnetism
Completing the Square
Method to convert quadratic to vertex form
Geometric Series
Sum of finite geometric series
Arithmetic Series
Sum of arithmetic sequence
Definite Integral
Fundamental theorem of calculus
Product Rule
Derivative of product of functions
Quotient Rule
Derivative of quotient of functions
Power Rule
Derivative of power function
Mean (Average)
Arithmetic mean of a dataset
Variance
Measure of data spread
Standard Deviation
Square root of variance
Covariance
Measure of joint variability
Correlation Coefficient
Pearson correlation coefficient
Binomial Probability
Probability of k successes in n trials
Matrix Multiplication
Definition of matrix product
2×2 Determinant
Determinant of 2×2 matrix
Cross Product
Vector product in 3D space
Matrix Trace
Sum of diagonal elements
Law of Sines
Relationship between sides and angles in triangles
Law of Cosines
Generalization of Pythagorean theorem
Double Angle (Sine)
Sine of double angle
Double Angle (Cosine)
Cosine of double angle
Sum of Angles
Sine of sum of two angles
Kinetic Energy
Energy of motion
Gravitational Potential Energy
Energy due to position in gravitational field
Momentum
Linear momentum of an object
Work-Energy Theorem
Relationship between work and kinetic energy
Ohm's Law
Relationship between voltage, current, and resistance
Coulomb's Law
Electrostatic force between charges
Universal Gravitation
Newton's law of universal gravitation
Wave Equation
Relationship between wave speed, frequency, and wavelength
Lorentz Force
Force on charged particle in electromagnetic field
Green's Theorem
Connection between line integral and double integral in the plane
Divergence Theorem
Connection between surface integral and volume integral
De Moivre's Theorem
Powers of complex numbers in polar form
Riemann Sum
Approximation of definite integral using rectangles
Poisson Distribution
Probability distribution for rare events