Engineering Fundamentals

Engineering Fundamentals represent the core body of scientific principles, mathematical techniques, and problem-solving methodologies that form the bedrock of all engineering disciplines. This foundational knowledge typically includes the study of physics (such as mechanics, thermodynamics, and electricity), chemistry, advanced mathematics (including calculus and differential equations), statics, dynamics, and materials science. By mastering these essentials, students develop a universal toolkit for analyzing, designing, and creating solutions, providing the critical groundwork necessary before specializing in a specific field like mechanical, civil, electrical, or chemical engineering.

1. Foundational Mathematics for Engineering
   1. Arithmetic and Algebra Review
      1. Order of Operations
      2. Exponents and Radicals
      3. Factoring and Expanding Expressions
      4. Solving Linear Equations
      5. Solving Quadratic Equations
      6. Systems of Equations
   2. Functions and Graphs
      1. Definition of a Function
      2. Domain and Range
      3. Types of Functions
         1. Linear Functions
         2. Quadratic Functions
         3. Polynomial Functions
         4. Exponential Functions
         5. Logarithmic Functions
         6. Trigonometric Functions
      4. Graphing Techniques
      5. Transformations of Functions
   3. Calculus
      1. Limits and Continuity
         1. Concept of a Limit
         2. One-Sided Limits
         3. Infinite Limits
         4. Continuity and Discontinuities
      2. Derivatives and Differentiation Rules
         1. Definition of the Derivative
         2. Power Rule
         3. Product Rule
         4. Quotient Rule
         5. Chain Rule
         6. Implicit Differentiation
         7. Higher-Order Derivatives
         8. Related Rates
      3. Applications of Differentiation
         1. Maxima and Minima
         2. Optimization Problems
         3. Curve Sketching
         4. Concavity and Inflection Points
         5. Motion Along a Line
      4. Integrals and Integration Techniques
         1. Indefinite Integrals
         2. Definite Integrals
         3. Fundamental Theorem of Calculus
         4. Substitution Method
         5. Integration by Parts
         6. Trigonometric Substitution
         7. Partial Fractions
         8. Improper Integrals
      5. Applications of Integration
         1. Area Under a Curve
         2. Area Between Curves
         3. Volumes of Revolution
         4. Arc Length
         5. Surface Area of Revolution
         6. Work and Fluid Pressure
         7. Center of Mass
      6. Sequences and Series
         1. Definitions and Notation
         2. Arithmetic Sequences
         3. Geometric Sequences
         4. Convergence and Divergence
         5. Convergence Tests
            1. nth-Term Test
            2. Integral Test
            3. Comparison Test
            4. Ratio Test
            5. Root Test
            6. Alternating Series Test
         6. Power Series
         7. Taylor Series
         8. Maclaurin Series
         9. Applications of Series
   4. Multivariable and Vector Calculus
      1. Vectors and the Geometry of Space
         1. Vector Operations
         2. Dot Product
         3. Cross Product
         4. Lines in Space
         5. Planes in Space
         6. Distance Between Points
         7. Distance Between Lines and Planes
      2. Functions of Several Variables
         1. Graphs and Level Curves
         2. Limits in Several Variables
         3. Continuity in Several Variables
      3. Partial Derivatives
         1. Definition and Notation
         2. Higher-Order Partial Derivatives
         3. Chain Rule for Partial Derivatives
         4. Directional Derivatives
         5. Gradient
         6. Tangent Planes
         7. Linear Approximation
         8. Optimization with Constraints
         9. Lagrange Multipliers
      4. Multiple Integrals
         1. Double Integrals
            1. Over Rectangular Regions
            2. Over General Regions
            3. Change of Order of Integration
         2. Triple Integrals
            1. Cartesian Coordinates
            2. Cylindrical Coordinates
            3. Spherical Coordinates
         3. Applications in Mass and Center of Mass
         4. Surface Area Calculations
         5. Volume Calculations
      5. Vector Fields
         1. Definition and Representation
         2. Gradient
         3. Divergence
         4. Curl
      6. Line Integrals and Surface Integrals
         1. Line Integrals of Scalar Fields
         2. Line Integrals of Vector Fields
         3. Work Done by a Force Field
         4. Surface Integrals of Scalar Fields
         5. Surface Integrals of Vector Fields
      7. Fundamental Theorems of Vector Calculus
         1. Green's Theorem
         2. Stokes' Theorem
         3. Divergence Theorem
   5. Differential Equations
      1. Introduction to Differential Equations
         1. Definitions and Terminology
         2. Initial Value Problems
         3. Boundary Value Problems
      2. First-Order Differential Equations
         1. Separable Equations
         2. Linear Equations
         3. Exact Equations
         4. Integrating Factors
         5. Applications
            1. Population Growth
            2. Mixing Problems
      3. Second-Order Linear Differential Equations
         1. Homogeneous Equations with Constant Coefficients
         2. Characteristic Equation
         3. Nonhomogeneous Equations
            1. Method of Undetermined Coefficients
            2. Variation of Parameters
         4. Applications
            1. Mechanical Vibrations
            2. Electrical Circuits
      4. Higher-Order Differential Equations
      5. Systems of Differential Equations
      6. Laplace Transforms
         1. Definition and Properties
         2. Inverse Laplace Transform
         3. Solving Differential Equations with Laplace Transforms
   6. Linear Algebra
      1. Systems of Linear Equations
         1. Substitution Method
         2. Elimination Method
         3. Gaussian Elimination
         4. Gauss-Jordan Elimination
      2. Matrices and Matrix Operations
         1. Matrix Addition
         2. Matrix Multiplication
         3. Transpose
         4. Inverse
         5. Row Echelon Form
         6. Determinants
            1. Properties of Determinants
            2. Cramer's Rule
      3. Vector Spaces
         1. Definition and Examples
         2. Subspaces
         3. Linear Independence
         4. Basis and Dimension
      4. Eigenvalues and Eigenvectors
         1. Characteristic Equation
         2. Diagonalization
         3. Applications in Engineering
   7. Statistics and Probability
      1. Descriptive Statistics
         1. Measures of Central Tendency
         2. Measures of Variability
         3. Data Visualization
      2. Probability Theory
         1. Basic Probability Concepts
         2. Conditional Probability
         3. Bayes' Theorem
         4. Probability Distributions
            1. Discrete Distributions
            2. Continuous Distributions
      3. Statistical Inference
         1. Sampling Distributions
         2. Confidence Intervals
         3. Hypothesis Testing
      4. Regression Analysis
         1. Linear Regression
         2. Correlation Analysis
         3. Model Validation