Cauchy's Theorem for a disk

Explores Cauchy's theorem for analytic functions in an open disk, proving integrals over closed contours vanish by constructing an antiderivative via L-shaped paths. Covers the Anti-Derivative Theorem and provides examples using the Cauchy Integral Formula and Residue Theorem. Read more... 3793 words,🕔18 minutes read, Oct 25, 2025.

Sketching the Graph of a function

Sketching the Graph of a function. Solved examples. Read more... 1467 words,🕔7 minutes read, May 14, 2022.

Sketching the Graph of a function II

General strategy to plot functions. Solved examples. Basic Transformations. Plotting inverse and piecewise functions. Read more... 1909 words,🕔9 minutes read, May 14, 2022.

Bolzano–Weierstrass theorem

Bolzano–Weierstrass theorem. Every subsequence of a convergent sequence converges and to the same limit. Read more... 1197 words,🕔6 minutes read, May 14, 2022.

Boundedness theorem

Boundedness theorem. Solved homework examples. Read more... 939 words,🕔5 minutes read, May 14, 2022.

Derivatives as Rates of Change

Applications of Derivatives. Related rates. Solved homework exercises. How to solve rates of change problems. Read more... 1536 words,🕔8 minutes read, May 14, 2022.

Derivatives as Rates of Change II

Applications of Derivatives. Related rates. Solved homework exercises. How to solve rates of change problems. Read more... 1519 words,🕔8 minutes read, May 14, 2022.

Derivatives as Rates of Change III

Applications of Derivatives. Related rates. Solved homework exercises. How to solve rates of change problems. Read more... 1544 words,🕔8 minutes read, May 14, 2022.

Fermat's Theorem

Fermat's Theorem. How to find Absolute Extrema given a function f on a close interval [a, b]. Solved homework exercises. Read more... 1573 words,🕔8 minutes read, May 14, 2022.

Newton's Method

Applications of Derivatives. Newton's Method. Basic principle. Solved homework exercises. Read more... 1189 words,🕔6 minutes read, May 14, 2022.

Optimization Problems

Applications of Derivatives. Optimization Problems. Solved homework exercises. Read more... 1532 words,🕔8 minutes read, May 14, 2022.

Optimization Problems II

Applications of Derivatives. Optimization Problems. Solved homework exercises. Steps to solve an optimization problem. Read more... 2070 words,🕔10 minutes read, May 14, 2022.

Rolle's and the Mean Value Theorems

Rolle's Theorem. The Mean Value Theorem. Increasing and decreasing functions. Solved homework exercises. Read more... 2074 words,🕔10 minutes read, May 14, 2022.

Th. Extreme Value. Nested Interval Property.

Extreme Value Theorem. Proof. Solved homework exercises. How to find Absolute Extrema given a function f on a close interval [a, b]. Axiom of Completeness/the least-upper-bound property. Nested Interval Property. Read more... 1590 words,🕔8 minutes read, May 14, 2022.

Antiderivates

Antiderivates or indefinite integrals. Definition and Examples. Uniqueness of Antiderivatives. Read more... 818 words,🕔4 minutes read, May 14, 2022.

Definite integrals. Fundamental Theorem of Calculus

Definite integrals. Definition and Examples. Fundamental Theorem of Calculus. Intuitive interpretation of Fundamental Theorem. Read more... 1498 words,🕔8 minutes read, May 14, 2022.

Properties of integrals

Properties of integrals. Integration by substitution or change of variables. Linearity of Integrals. Read more... 1242 words,🕔6 minutes read, May 14, 2022.

Fundamental Theorems of Calculus. MVT for Integrals

Alternative version of the Fundamental Theorem of Calculus. Proof of the Fundamental Theorem of Calculus. The Second Fundamental Theorem of Calculus. The Mean Value Theorem for Integrals Read more... 1351 words,🕔7 minutes read, May 14, 2022.

The Logarithm Defined as an Integral. The error function

Alternative definition of the natural logarithm as a definite integral. The error function. Read more... 999 words,🕔5 minutes read, May 14, 2022.

Areas between curves

Determine the area of a region between two curves by integrating with respect to the independent variable. Read more... 1222 words,🕔6 minutes read, May 14, 2022.

Average Value Theorem

Average Function Value. Average Value Theorem. Find the Average Value with the Mean Value Theorem for Integrals. Solved exercises. Read more... 1167 words,🕔6 minutes read, May 14, 2022.

Determining volumes

Determining Volumes by Slicing. Volumes of solid of revolution. The shell method. Washer method. Read more... 1317 words,🕔7 minutes read, May 14, 2022.

Determining volumes II

Determining Volumes by Slicing. Volumes of solid of revolution. The shell method. Washer method. Read more... 1195 words,🕔6 minutes read, May 14, 2022.

Determining volumes III

Determining Volumes by Slicing. Volumes of solid of revolution. The shell method. Washer method. Read more... 1240 words,🕔6 minutes read, May 14, 2022.

Numerical integration.

Riemann Sums. Trapezoid Sums. Simpson's Rule. Read more... 956 words,🕔5 minutes read, May 14, 2022.

Weighted Average

Weighted Average. Solved exercises. Average temperature. Calculus Probability Modeling. Read more... 1436 words,🕔7 minutes read, May 14, 2022.

Weighted Average II

Weighted Average. Solved exercises. Average temperature. Calculus Probability Modeling. Read more... 1445 words,🕔7 minutes read, May 14, 2022.

Integration of Trigonometric Functions

Integration of Trigonometric Functions. Formulas and solved examples. Trigonometry substitution for integrals. Completing the square. Read more... 1209 words,🕔6 minutes read, May 14, 2022.

Integration of Trigonometric Functions II

Integration of Trigonometric Functions. Formulas and solved examples. Trigonometry substitution for integrals. Completing the square. Read more... 814 words,🕔4 minutes read, May 14, 2022.

Arc lengths

Arc lengths. Surface Area. Solved exercises. Read more... 1197 words,🕔6 minutes read, May 14, 2022.

Integration by parts

Integration by parts. Solved homework exercises. Geometrical interpretation. LIATE mnemonic. Read more... 1107 words,🕔6 minutes read, May 14, 2022.

Integration by parts II

Integration by parts. Solved homework exercises. Geometrical interpretation. LIATE mnemonic. Read more... 821 words,🕔4 minutes read, May 14, 2022.

Integration of rational functions.

Integration of rational functions. Integration of improper rational fraction. Partial fraction decomposition. Distinct linear factors. Repeated linear factors. Read more... 1130 words,🕔6 minutes read, May 14, 2022.

Parametric curves. Polar Coordinates.

Parametric curves. Polar Coordinates. The area of the sector of a curve in polar coordinates. Read more... 1311 words,🕔7 minutes read, May 14, 2022.

L'Hôpital's Rule

L'Hôpital's Rule. Motivation. General form. Solved examples. Evaluating Limits of Indeterminate Forms. Read more... 1020 words,🕔5 minutes read, May 14, 2022.

L'Hôpital's Rule II

L'Hôpital's Rule. Motivation. General form. Solved examples. Evaluating Limits of Indeterminate Forms. Rates of growth and decay. Asymptotic Complexity. Read more... 1022 words,🕔5 minutes read, May 14, 2022.

Comparison Test For Improper Integrals

Comparison Test For Improper Integrals. Solved examples. Read more... 1124 words,🕔6 minutes read, May 14, 2022.

Improper integrals type 2

Improper integrals of second type. Read more... 1209 words,🕔6 minutes read, May 14, 2022.

Improper integration.

Improper integration. Limit Comparison. Improper integrals of second type. Read more... 1064 words,🕔5 minutes read, May 14, 2022.

Alternating Series

Definition. Alternating Series Test. Solved exercises. Read more... 1398 words,🕔7 minutes read, May 14, 2022.

Convergence/Divergence of series

Necessary condition for the convergence of a series. Divergence test. Integral Test For Convergence and Divergence of Series. Read more... 1547 words,🕔8 minutes read, May 14, 2022.

Direct & Limit Comparison test

Direct Comparison test. Limit Comparison Test. Solved homework exercises. Theorem p-series. Integral Comparison. Read more... 1588 words,🕔8 minutes read, May 14, 2022.

Infinite Series

Infinite Series. Arithmetic and Geometric series. Convergent and divergent series. Solved examples. Algebraic Properties of Convergent Series. Read more... 1485 words,🕔7 minutes read, May 14, 2022.

Power Series. Convergence, derivatives, and Integrals

Power Series. Power Series Convergence. Derivatives and Integrals of Power Series. Read more... 1684 words,🕔8 minutes read, May 14, 2022.

Root and Ratio Test

Root Test. Ratio Test. Solved examples. Read more... 1308 words,🕔7 minutes read, May 14, 2022.

Taylor’s Formula.

Taylor’s Formula. Taylor's theorem. Lagrange form of the remainder. Read more... 1614 words,🕔8 minutes read, May 14, 2022.

Vectors

Vectors. Definition, components, representation, examples, solved exercises, and Properties Vector Arithmetic. Dot product. Read more... 1545 words,🕔8 minutes read, May 14, 2022.

Cross Products

Vectors II. Determinant in space. Cross products. Properties. Solved exercises. Read more... 2701 words,🕔13 minutes read, May 14, 2022.

Vectors II. The Dot Product.

Vectors. The Dot Product. Solved exercises. Read more... 2144 words,🕔11 minutes read, May 14, 2022.

Vectors III. Planes and Areas

Calculating the Area of a Triangle Using Vectors Read more... 1543 words,🕔8 minutes read, May 14, 2022.

Equations of Planes

Find the Equation of a Plane Given Three Points. Vectors and the Geometry of Space. Solved exercises. Read more... 2219 words,🕔11 minutes read, May 14, 2022.

Matrices: definitions, types, examples, and properties.

Rotation Matrices. Inverse matrix. Properties. Trace of a matrix. Diagonal matrix. Lower and upper triangular matrix. Symmetric matrix. Read more... 3281 words,🕔16 minutes read, May 14, 2022.

Systems of Linear Equations.

Equations of planes. Solving Systems of Linear Equations. Exercises of vectors and planes. Read more... 3041 words,🕔15 minutes read, May 14, 2022.

Parametric equations for lines and curves II

Parametric equations for lines and curves. Graphing a Parametrically Defined Curve. A cycloid. Read more... 1815 words,🕔9 minutes read, May 14, 2022.

Parametric equations for lines and curves.

Parametric equations for lines and curves. Graphing a Parametrically Defined Curve. Eliminating the Parameter. Read more... 1978 words,🕔10 minutes read, May 14, 2022.

Systems of Linear Equations II

Equations of planes. Solving Systems of Linear Equations. Exercises of vectors and planes. Read more... 3627 words,🕔18 minutes read, May 14, 2022.

Functions of two variables II

Functions of two variables. Sketching graphs. Read more... 1510 words,🕔8 minutes read, May 14, 2022.

Functions of two variables.

Functions of two variables. Definition, examples, domain, graphs, contour plots. Read more... 1795 words,🕔9 minutes read, May 14, 2022.

Least Squares Interpolation

Least Squares Interpolation. Fitting a linear and a quadratic model. Moore's Law. Read more... 1812 words,🕔9 minutes read, May 14, 2022.

Partial derivatives

Definition. Formal Definition. Geometric interpretation. Solved examples. Local extrema and critical points in multivariable functions. Read more... 2105 words,🕔10 minutes read, May 14, 2022.

Tangent Planes And Linear Approximations

Tangent Planes And Linear Approximations Read more... 1969 words,🕔10 minutes read, May 14, 2022.

Velocity and Acceleration.

Definition and Properties of Vectors in Motion. Kepler's Laws and Planetary Motion. Newton's explanation using Vector Calculus. Solved exercises. Read more... 3593 words,🕔17 minutes read, May 14, 2022.

Second derivative test

Quadratic Functions and Critical Points. Steps to Perform the Second Derivative Test. Non-rigorous proof. Solved examples. Read more... 3505 words,🕔17 minutes read, May 14, 2022.

Total differential

Total differential. Using Different Notation. The Chain Rule for multivariable functions. Proof of the Chain Rule. Solved exercises. Read more... 1378 words,🕔7 minutes read, May 14, 2022.

Directional derivatives

Step to Compute the Directional derivative. Formal and alternative definition. Solved exercises. Read more... 3364 words,🕔16 minutes read, May 14, 2022.

Gradient vector

The Gradient Vector is Perpendicular to Level Surfaces. Solved exercises. Read more... 3481 words,🕔17 minutes read, May 14, 2022.

Lagrange Multipliers

Lagrange Multipliers. Solved exercises. Read more... 3917 words,🕔19 minutes read, May 14, 2022.

Double integrals

Double integrals. Formal and informal definition, examples, and properties. Calculating a double integral. Fubini's Theorem. Read more... 2765 words,🕔13 minutes read, May 14, 2022.

Double integrals II

Double integrals II. Solved examples. Exchanging order of integration. Read more... 2292 words,🕔11 minutes read, May 14, 2022.

Implicit partial differentiation

Calculating ∂f/∂x, ∂f/∂y for a given implicit function Read more... 1624 words,🕔8 minutes read, May 14, 2022.

Partial Derivative with Constrained Variables

Non-independent variables. Solving a Constraint. Avoiding confusion. Calculate the area of a triangle. Read more... 3756 words,🕔18 minutes read, May 14, 2022.

Partial differential equations

Classification of differential equations. Ordinary Differential Equations. The heat and the harmonic oscillator equations. First-order differential equations. Read more... 3003 words,🕔15 minutes read, May 14, 2022.

Applications of Double Integrals

Area Region. Volume under Surface. Average Value Function. Total mass of a flat object (lamina) over a region. Read more... 4029 words,🕔19 minutes read, May 14, 2022.

Applications of Double Integrals II

Weighted Average of a Function with a Density Function. Center of Mass. Moment of Inertia. Kinetic Energy of a Rotating Object. Read more... 4441 words,🕔21 minutes read, May 14, 2022.

Double integrals in Polar Coordinates

Double integrals in Polar Coordinates Read more... 3372 words,🕔16 minutes read, May 14, 2022.

Change of variables in double integrals

Understanding the Jacobian in Change of Variables. Applying the Change of Variables Formula. Transformations with polar coordinates. Read more... 4070 words,🕔20 minutes read, May 14, 2022.

Vector fields

Vector fields Read more... 2960 words,🕔14 minutes read, May 14, 2022.

Conservative vector fields

Open, connected, and simply connected regions. The Fundamental theorem of Calculus for Line Integral. Equivalent Properties of Conservative Vector Fields. Read more... 2931 words,🕔14 minutes read, May 17, 2022.

Conservative vector fields II

Path Independence and Conservative Vector Fields. Criterion for a Conservative Vector Field. Curl and Torque Read more... 2620 words,🕔13 minutes read, May 17, 2022.

Find potential functions for conservative fields

Finding a potential function for conservative vector fields. Solved exercises. Read more... 3298 words,🕔16 minutes read, May 17, 2022.

Green's theorem

Green's theorem. Fully Explained. Step by Step examples. Read more... 3678 words,🕔18 minutes read, May 17, 2022.

Surface Integrals of Vector Fields. Flux.

Surface Integrals of Vector Fields. Flux Form of Green's Theorem. Explanation of divergence. Limitations of Green's Theorem. Read more... 3960 words,🕔19 minutes read, May 18, 2022.

Surface Integrals of Vector Fields. Flux. II

Example with a Non-Simply Connected Region. Definition and Importance of Simply Connected Regions in Green's Theorem. Why Simply Connected Regions Matter. Criteria for Conservative Fields. Solved mixed exercises. Read more... 3257 words,🕔16 minutes read, May 18, 2022.

Triple Integrals

Triple Integrals. Solved Exercises. Cylindrical Coordinates. Read more... 3410 words,🕔17 minutes read, May 17, 2022.

Triple Integrals 2. Applications.

Triple Integrals. Solved Exercises. Applications. Read more... 4916 words,🕔24 minutes read, May 17, 2022.

Triple Integrals 3. Spherical coordinates

Spherical coordinates. Solved Exercises. Applications. Calculation of Gravitational Force Exerted by an object. Read more... 4701 words,🕔23 minutes read, May 17, 2022.

Vector fields & flux 3D

Vector fields in 3D. Flux in 3D. Solved Exercises. Read more... 4767 words,🕔23 minutes read, May 17, 2022.

Vector fields & Flux in 3D II

Vector fields & flux in 3D. Solved Exercises. Explanation of Surface Area Elements Using Normal Vectors. Read more... 5023 words,🕔24 minutes read, May 17, 2022.

The Divergence Theorem

The Divergence Theorem. Examples, Physical Interpretation, and Proof. Step by Step Solved Exercises. Read more... 4926 words,🕔24 minutes read, May 17, 2022.

The Divergence Theorem II

The Divergence Theorem. Solved Exercises. The diffusion equation Read more... 4220 words,🕔20 minutes read, May 17, 2022.

Curl in 3D

Curl in 3d, definition and solved exercises. Conditions for a Vector Field to be Conservative. Read more... 2751 words,🕔13 minutes read, May 17, 2022.

Line integrals in space

Line integrals in space. Solved Exercises. Test for conservative fields. Find potential functions. Read more... 4028 words,🕔19 minutes read, May 17, 2022.

Stoke's Theorem

Formal Statement, Proof of Stoke's Theorem. Orientation and the Right-Hand Rule. Intuitive explanation. Solved Exercises. Comparing Stokes’ Theorem with Green’s Theorem. Read more... 3433 words,🕔17 minutes read, May 17, 2022.

Stoke's Theorem II

Stoke's Theorem II. Solved Exercises. Read more... 2699 words,🕔13 minutes read, May 17, 2022.

Stoke and Surface Independence. Curl.

Solved examples. Stoke Theorem and Surface Independence. Curl and Topological considerations. Physical interpretation of Curl. Conservative Fields and Rotation. Irrotational and Conservative Fields. Read more... 4637 words,🕔22 minutes read, May 17, 2022.

Faraday's law. Maxwell's equations

Faraday's law. Maxwell's equations. Read more... 1630 words,🕔8 minutes read, May 17, 2022.

Differential equations

Differential equations versus algebraic equations. Famous Examples of Differential Equations. Classification of Differential Equations. Solving First-Order Ordinary Differential Equations. Read more... 1595 words,🕔8 minutes read, May 14, 2022.

Separable differential equations

Solving differential equations. Simple Harmonic Motion, A Pendulum (EDO Intuition). Separation of Variables. Steps to Solve Separable Differential Equations. Solved exercises. Read more... 1882 words,🕔9 minutes read, May 14, 2022.

Separable differential equations 2

Separation of Variables. Steps to Solve Separable Differential Equations. Finding a Function with a Given Slope Condition. Finding Orthogonal Trajectories to a Family of Parabolas. Applying Newton’s Law of Cooling. Read more... 2014 words,🕔10 minutes read, May 14, 2022.

Bernoulli Equations

Substitutions for non-separable differential equations. Bernoulli Equations. How to solve a Bernoulli equation. Solved Exercises Read more... 1470 words,🕔7 minutes read, May 14, 2022.

Integral Factors

The Method of Integrating Factors. The Method Step-by-Step. Solved exercises. First-Order Linear Differential Equation for Newton's Law of Cooling. Read more... 1663 words,🕔8 minutes read, May 14, 2022.

Exact differential equation

Exact differential equation. Transforming Non-exact Equations into Exact ODEs. Solved examples. Read more... 1982 words,🕔10 minutes read, May 14, 2022.

Geometric Interpretation of ODEs

Geometric Interpretation of ODEs. What is a solution to an ODE. Integral Curves. Plotting the Direction Field using Isoclines. Existence and Uniqueness Theorem Read more... 3360 words,🕔16 minutes read, May 17, 2022.

Geometric Interpretation of ODEs 2

Geometric Interpretation of ODEs. What is a solution to an ODE. Integral Curves. Sketching and plotting ODEs Read more... 2092 words,🕔10 minutes read, May 17, 2022.

Numerical Solutions

Euler's Numerical Method for y'=f(x,y). Improved Euler's method or RK2. Pitfalls. Singularity in Solutions. Implications for Numerical Methods. Read more... 2820 words,🕔14 minutes read, May 17, 2022.

Applications First-order Linear ODE's

First-Order Linear Ordinary Differential Equations (ODEs). Standard Form. Applications of First-Order Linear ODEs. Newton’s Law of Cooling. Diffusion Model. Salt Concentration. Solving First-Order Linear Ordinary Differential Equations (ODEs) Read more... 2692 words,🕔13 minutes read, May 17, 2022.

First-order Substitution Methods

Types of Substitutions. Rescaling in a Temperature model. Solving a Bernoulli Differential Equation Using Direct Substitution Read more... 2703 words,🕔13 minutes read, May 17, 2022.

Homogeneous First-Order ODE's

Homogeneous First-Order ODE's. Examples of Homogeneous ODEs. Solving Homogeneous ODEs. Solved exercises. The Path of a Boat Under a Lighthouse Beam Read more... 1835 words,🕔9 minutes read, May 17, 2022.

First-order Autonomous ODE's

First-order Autonomous ODE's. Definition. Characteristics of Autonomous ODEs. Challenges in Solving Autonomous ODEs. Qualitative Analysis. Steps for Qualitative Analysis of Autonomous ODEs. Solved examples. Bank Account with Embezzlement. Read more... 2953 words,🕔14 minutes read, May 17, 2022.

First-order Autonomous ODE's II

Logistic Equation. Basic Exponential Growth model. Limitation of the Exponential Model. Logistic Growth Model. Analysis of the Logistic Differential Equation. Logistic Equation with Harvesting. Analysis of the Logistic Equation with Harvesting. Read more... 2480 words,🕔12 minutes read, May 17, 2022.

First-order Linear with Constant Coefficients

Solving the Equation Using Integrating Factors. Equations with Constant Coefficients. The Superposition Principle. Differential Equations with Periodic Inputs Read more... 3444 words,🕔17 minutes read, May 17, 2022.

Solving Differential Equations Involving Complex Numbers

Polar representation of Complex Numbers. Euler's equation. Laws of Exponents for Complex Numbers. Multiplication and Division of Complex Numbers in Polar Form. Differentiation and Integration of Complex Functions. Solving Differential Equations Involving Complex Numbers. Exponential Expressions with Complex Exponents. Integration of Real Functions Using Complex Exponentials. N-th root of unity, examples, properties. Read more... 2509 words,🕔12 minutes read, May 17, 2022.

Basic linear ODE

Basic linear ODE. Understanding Linearity in ODEs. Common Forms of First-Order Linear ODEs. Flow Rate of Salt in a Tank. Intuitive Understanding of the Model. Special Case, Constant Input Concentration. Read more... 2453 words,🕔12 minutes read, May 17, 2022.

Basic linear ODE II.

Electrical Circuit (RC Circuit). Components of the RC Circuit. Governing Differential Equation. Constant Voltage Source ε(t)=E. Radioactive Decay. Solving First-Order Linear Ordinary Differential Equations with Sinusoidal Inputs Read more... 3424 words,🕔17 minutes read, May 17, 2022.

Second-order Linear ODE's with Constant Coefficients

General Solution Method. Real and Distinct Roots (Overdamped System). Complex roots (Underdamped System). Theorem, Real and Imaginary Parts are Solutions. Two equal roots (Critically damped system). Reduction of Order Method. Read more... 3262 words,🕔16 minutes read, May 17, 2022.

Second-order Linear ODE's with Constant Coefficients II

Complex Characteristic Root. Alternative Approach Using Complex Constants. Undamped and Damped Oscillations. Solving the Damped Oscillator Equation. Different Cases Based on Damping. Read more... 2833 words,🕔14 minutes read, May 17, 2022.

Solution of a Second-Order Linear Homogeneous ODE

General Solution of a Second-Order Linear Homogeneous ODE. Test for Linear independence (the Wronskians determinant). The Superposition principle. Solving the initial value problem and Uniqueness. Completeness of the Solution Set. Existence and Uniqueness Theorem for Differential Equations. Read more... 3991 words,🕔19 minutes read, May 17, 2022.

First-order linear inhomogeneous differential equation

General Theory for Inhomogeneous ODE's. Stability Criteria for Second-Order Linear Differential Equations with Constant Coefficients. Analysis of Stability Based on Characteristic Roots. Physical Interpretation of Stability Read more... 3756 words,🕔18 minutes read, May 17, 2022.

Inhomogeneous ODE's

Inhomogeneous Second-Order Linear Differential Equations. Solving the Inhomogeneous Equation. General Solution to the Inhomogeneous Equation. Solved examples. Note on Choosing the Particular Solution. Read more... 3123 words,🕔15 minutes read, May 17, 2022.

Finding Particular Solutions Inhomogeneous ODE's

Finding Particular Solutions Inhomogeneous ODE's. The substitution Rule. Exponential input theorem. Exponential-Shift Rule. Resonance in Differential Equations Read more... 3231 words,🕔16 minutes read, May 17, 2022.

Fourier Series

Introduction to Fourier Series. Superposition of Solutions. Orthogonality of Sine and Cosine Functions. Fourier Series and Fourier Coefficients. Example. Fourier Series of the Square-Wave Function. Uniqueness of Fourier Series. Conditions for the Existence of Fourier Coefficients. Read more... 2873 words,🕔14 minutes read, May 17, 2022.

Fourier Series 2

Even and Odd Functions. Fourier Series for Even and Odd Functions. Convergence of Fourier Series. Fourier Series Extension. Find the Fourier coefficients and the Fourier series Read more... 3080 words,🕔15 minutes read, May 17, 2022.

Finding Particular Solutions via Fourier Series

Finding Particular Solutions via Fourier Series. Damped harmonic oscillator. Damping Regimes. Solution for Each Damping Regime. Finding the General Solution to y’’ + 3y = 2x. Read more... 3814 words,🕔18 minutes read, May 17, 2022.

Introduction to the Laplace Transform

Introduction to the Laplace Transform. Basic Formulas, properties, and solved examples. Laplace Transform of cos(at), sin(at), eᵃ⁺ᵇⁱ, δ(t), etc. Exponential shift formula. Read more... 2198 words,🕔11 minutes read, May 17, 2022.

Inverse Laplace Transform. Existence Laplace Transform.

Inverse Laplace Transform. Laplace Transforms of Polynomials. Conditions for the Existence of the Laplace Transform. Examples and Counterexamples of Functions of Exponential Type Read more... 2265 words,🕔11 minutes read, May 17, 2022.

Solving Linear ODE's using the Laplace Theorem

Solving Linear ODE's using the Laplace Theorem. Laplace Transform of a Derivative. Solved exercises. Read more... 2486 words,🕔12 minutes read, May 17, 2022.

Convolution

Definition, Properties, Laplace Transforms, Examples. The Convolution Theorem. Laplace Transform of the Heaviside Step Function. T-axis translation formula Read more... 3928 words,🕔19 minutes read, May 17, 2022.

Convolution II: Impulse inputs

Impulse inputs. The Sifting Property. Laplace Transform of the Dirac Delta Function. Convolution with the Delta Function. Undamped Mass-Spring System with an Impulse. Solving a Second-Order Linear Differential Equation with Periodic Impulses Using Laplace Transforms. General Second-Order Linear Differential Equation and Its Solution Using Laplace Transforms. Read more... 3693 words,🕔18 minutes read, May 17, 2022.

Linear First-order Systems of ODEs

Introduction to First-order Systems of ODE's. Solution by Elimination. Autonomous Systems of First-Order ODEs. Geometric Interpretation of a System. Read more... 2661 words,🕔13 minutes read, May 17, 2022.

Complex Eigenvalues in Systems of Differential Equations

Complex Eigenvalues in Systems of Differential Equations. Examples. Modelling a love affair. Read more... 1831 words,🕔9 minutes read, May 17, 2022.

Homogeneous Linear Systems with Constant Coefficients

Homogeneous Linear Systems with Constant Coefficients. Solving the System Using the Eigenvalue and Eigenvector Method. Repeated Real Eigenvalues. The Spectral or Principal Axis Theorem. Read more... 3375 words,🕔16 minutes read, May 17, 2022.

Sketching Solutions of Homogeneous Linear System

Sketching Solutions of 2x2 Homogeneous Linear System with Constant Coefficients Read more... 3566 words,🕔17 minutes read, May 17, 2022.

Sketching Solutions of Homogeneous Linear System II

Sketching Solutions of a Homogeneous Linear System with Complex Eigenvalues. General Homogeneous Linear System. Determining Linear Independence, The Wronskian. Fundamental Matrix Solution Read more... 2724 words,🕔13 minutes read, May 17, 2022.

Inhomogeneous Systems

The Inhomogeneous System. Matrix Representation. General Solution of the Inhomogeneous System. Method to find a particular solution to the inhomogeneous system. Solving nonhomogeneous systems of linear differential equations with constant coefficients. Read more... 2940 words,🕔14 minutes read, May 17, 2022.

Matrix Exponentials

Matrix Exponentials; Application to Solving Systems. Fundamental Matrix Solution. General solution. Exponential Law for Matrices. Methods for Computing ₑᴬᵗ. Applying the Fundamental Matrix Method Read more... 2720 words,🕔13 minutes read, May 17, 2022.

Decoupling Linear Systems with Constant Coefficients

General Method for Decoupling Linear Systems of Differential Equations. Summary of the Method. Solved examples. Read more... 3453 words,🕔17 minutes read, May 17, 2022.

Non-linear Autonomous Systems

Non-linear Autonomous Systems. Finding Critical Points. Linearizing the System Near the Critical Points. Sketching Non-linear Autonomous Systems. Lightly Damped Pendulum. Predator-Prey System Analysis Read more... 3817 words,🕔18 minutes read, May 17, 2022.

Limit Cycles

Limit Cycles in Non-Linear Autonomous System. Closed Trajectories and Limit Cycles. Existence and Non-existence Criteria. Bendixson's Criterion. Critical Point Criterion. Read more... 4692 words,🕔23 minutes read, May 17, 2022.

Relation Between Non-linear Autonomous Systems and First-order ODEs

Relation Between Autonomous Non-linear Systems and First-order ODEs. Solved examples. The Simple Harmonic Oscillator. Read more... 1737 words,🕔9 minutes read, May 17, 2022.

The Trace-Determinant Plane

Regions in the T-D Plane. Change of Coordinates and Similarity Transformations. Types of Equilibrium Points. Detailed Analysis of the Matrix with Complex Eigenvalues, Two distinct real Eigenvalues, and repeated real Eigenvalues. Read more... 3798 words,🕔18 minutes read, May 17, 2022.

Predator-Prey Model (Lotka-Volterra)

Predator-Prey Model. The Lotka-Volterra Equations and Principle. Introducing External Factors. Fishing with a Constant Ratek Analogy. Mosquito Plague and Pesticides Read more... 3056 words,🕔15 minutes read, May 17, 2022.

Open mapping theorem

Proves the Open Mapping Theorem, which states that non-constant analytic functions map open sets to open sets. The proof is based on the Local Mapping Theorem, which describes the behavior of an analytic function near a zero. Also discusses conformal maps and their connection to one-to-one analytic functions. Read more... 2204 words,🕔11 minutes read, Oct 11, 2025.

Inverse Function, Maximum Principle, and Schwarz's Lemma

Proves the Inverse Function Theorem, showing the inverse of a one-to-one analytic function is analytic. Covers the Maximum Principle via a proof by contradiction using the Open Mapping Theorem. Provides a detailed proof of Schwarz's Lemma by applying the Maximum Principle to a constructed function. Read more... 2577 words,🕔13 minutes read, Oct 11, 2025.

Möbius Transformations

Defines Möbius transformations and their key properties. Explains how they act on the extended complex plane (Riemann sphere), proving they are bijective functions. Covers the decomposition of these transformations into elementary geometric operations like translation, rotation, dilation, and inversion. Read more... 1645 words,🕔8 minutes read, Oct 11, 2025.

Fixed Points and Group Properties of Möbius Transformations

Explores the fixed points of Möbius transformations and how they classify the map's geometry. Proves the Uniqueness Theorem (3-Point Rule) and demonstrates that the set of Möbius transformations forms a group under composition, connecting it to matrix groups like GL(2, ℂ), PGL(2, ℂ), and PSL(2, ℂ) Read more... 2124 words,🕔10 minutes read, Oct 11, 2025.

The Cross Ratio

Defines the cross-ratio of four points in the extended complex plane. Explains how to construct a Möbius transformation that maps three points to a standard reference frame (1, 0, ∞) and proves that Möbius transformations preserve the cross-ratio. Includes the 3-Point Theorem for the uniqueness of such transformations. Read more... 1322 words,🕔7 minutes read, Oct 11, 2025.

Stereographic Projection and Generalized Circles

Introduces the stereographic projection, which provides a bijection between the extended complex plane and the Riemann sphere. Proves that the cross-ratio of four points is real if and only if they lie on a generalized circle (line or circle) and that Möbius transformations map generalized circles to generalized circles. Read more... 3029 words,🕔15 minutes read, Oct 11, 2025.

Symmetry with Respect to a Line and Circle

Explores the concept of symmetry for two points with respect to a line and a circle. Defines symmetry with respect to a line as the perpendicular bisector property and symmetry with respect to a circle via inversion. Discusses the implications for Möbius transformations and the invariance of the cross-ratio. Read more... 1110 words,🕔6 minutes read, Oct 11, 2025.

Maximum Modulus Principle and Extrema of |sin(z)|

Explores the Minimum Modulus Principle for non-zero analytic functions and demonstrates the Maximum Modulus Principle through a detailed problem finding the maximum of |sin(z)| on a square domain. Uses complex analysis techniques including decomposition into real and imaginary parts, hyperbolic functions, and boundary analysis. Read more... 652 words,🕔4 minutes read, Oct 11, 2025.

Extended Liouville's Theorem, polynomial Growth of Entire Functions

A detailed proof of the Extended Liouville's Theorem showing that entire functions with polynomial growth bounds are polynomials. Uses mathematical induction, Taylor series expansions, and careful bounding arguments to prove it. Read more... 712 words,🕔4 minutes read, Oct 11, 2025.

Analyticity of the Laplace Transform on a Finite Interval

Proves that the Laplace transform of a complex-valued continuous function on a finite interval [a, b] is analytic (entire) on the complex plane. Uses rigorous limit interchange arguments, uniform convergence via the Mean Value Theorem, and careful bounding to justify differentiation under the integral sign. Read more... 879 words,🕔5 minutes read, Oct 11, 2025.

Finding power series

Demonstrates finding the power series expansion of $f(z) = \frac{1}{z^2-3z+2}$ using partial fractions and geometric series, and determines the radius of convergence. Also proves that if the product of two analytic functions is zero, one function must be identically zero using the Identity Theorem. Read more... 593 words,🕔3 minutes read, Oct 11, 2025.

Removable Singularities

Defines isolated singularities in complex analysis, removable, poles, and essential singularities. Discusses the vanishing integral lemma, the Cauchy Integral Formula for mild singularities, and Riemann's Removable Singularity Theorem. Read more... 2850 words,🕔14 minutes read, Oct 11, 2025.

Classification of Isolated Singularities. Removable, Poles, and Essential Singularities.

Comprehensive classification of isolated singularities in complex analysis covering removable singularities, poles (including order determination), zeros, and essential singularities. Includes the Grand Classification Theorem and relationships between zeros and poles of analytic functions. Read more... 2812 words,🕔14 minutes read, Oct 11, 2025.

Essential Singularities and the Casorati-Weierstrass Theorem

Explores the Casorati-Weierstrass theorem on essential singularities, classification of isolated singularities using Laurent series and principal part, and examines examples including $e^{1/z}$ and $\csc(1/z)$ to illustrate chaotic behavior and non-isolated singularities. Read more... 1622 words,🕔8 minutes read, Oct 11, 2025.

Laurent Series

Introduction to Laurent Series covering the fundamental theorem, coefficient derivation via contour integrals, annulus of convergence, residues, relationship with Taylor series, and detailed proofs including path independence and uniform convergence arguments. Read more... 3673 words,🕔18 minutes read, Oct 11, 2025.

Classification of isolated singularities

Corollary for classifying isolated singularities via Laurent series (Rosetta Stone), including removable singularities, poles of order m, essential singularities, residues, and Cauchy's Residue Theorem with proof. Read more... 2384 words,🕔12 minutes read, Oct 11, 2025.

The Argument Principle

The Argument Principle for meromorphic functions, relating the integral of f'(z)/f(z) to the difference between the number of zeros and poles inside a contour, with geometric interpretation via winding number. Read more... 1295 words,🕔7 minutes read, Oct 11, 2025.

Application of Cauchy Residue Theorem to evaluation of definitive integral

Evaluation of definite and improper integrals using Cauchy's Residue Theorem, including the pizza slice contour method for rational functions and trigonometric integrals. Read more... 2092 words,🕔10 minutes read, Oct 11, 2025.

Calculating Residues of Complex Functions

Techniques for computing residues at simple poles and poles of order 2, including the derivative formula, factorization method, and worked examples with rational and transcendental functions. Read more... 1545 words,🕔8 minutes read, Oct 11, 2025.