Video Lecture
The whole series is available as a single video with chapter markers. Each chapter page starts this video at that chapter.
AI Terakoya Top›Fundamentals of Mathematics›Calculus Vector Analysis
🌐 EN | 🇯🇵 JP | Last sync: 2025-11-16
← Fundamentals of Mathematics Top
🎯 Series Overview
Calculus and vector analysis are the essential mathematical foundations for all areas of materials science, process engineering, and machine learning. This series covers single-variable and multivariable differential and integral calculus, vector fields, gradients, divergence, curl, line integrals, and surface integrals, with paired theory and implementation (Python/NumPy/SymPy).
Learning Path
Differentiation &
Numerical Differentiation"] B["Chapter 2
Integration &
Numerical Integration"] C["Chapter 3
Multivariable Calculus"] D["Chapter 4
Vector Fields &
Differential Operators"] E["Chapter 5
Line/Surface Integrals &
Integral Theorems"] A --> B --> C --> D --> E style A fill:#667eea,stroke:#764ba2,stroke-width:2px,color:#fff style B fill:#667eea,stroke:#764ba2,stroke-width:2px,color:#fff style C fill:#667eea,stroke:#764ba2,stroke-width:2px,color:#fff style D fill:#667eea,stroke:#764ba2,stroke-width:2px,color:#fff style E fill:#667eea,stroke:#764ba2,stroke-width:2px,color:#fff
📋 Learning Objectives
- Understand and implement differentiation and integration of single and multivariable functions
- Understand the concepts and physical meaning of vector fields
- Calculate and interpret gradients, divergence, and curl
- Calculate and apply line integrals and surface integrals
- Implement numerical and symbolic calculus using NumPy/SymPy
📖 Prerequisites
High school level single-variable differentiation and integration, plus basic vectors, are sufficient. Understanding basic Python usage (variables, functions, lists) is recommended.
Chapter 1
Fundamentals of Differentiation and Numerical Differentiation
Learn from the definition of differentiation to calculation rules for derivatives and higher-order derivatives, and implement numerical differentiation using NumPy (forward difference, central difference, Richardson extrapolation). Applications to temperature dependence of material properties and reaction rate analysis are also introduced.
Definition of Differentiation Derivatives Numerical Differentiation Higher-Order Derivatives NumPy Implementation
💻 7 Code Examples ⏱️ 18-22 minutes
Chapter 2
Fundamentals of Integration and Numerical Integration
Learn the definition of definite integrals, calculation of indefinite integrals, and the relationship between integration and differentiation (fundamental theorem of calculus), and implement numerical integration methods such as the trapezoidal rule and Simpson's rule, together with SciPy's adaptive quadrature. Applications to heat calculation and improper/singular integrals are also covered.
Definite & Indefinite Integrals Fundamental Theorem Trapezoidal Rule Simpson's Rule SciPy Implementation
💻 7 Code Examples ⏱️ 18-22 minutes
Chapter 3
Multivariable Calculus
Learn partial derivatives, the total differential and the chain rule, and the gradient, and handle extremum problems of multivariable functions (gradient descent, Lagrange multipliers). Double integrals and the change of variables to polar coordinates are also implemented.
Partial Derivatives Total Differential Chain Rule Gradient Double Integrals Extremum Problems
💻 7 Code Examples ⏱️ 18-22 minutes
Chapter 4
Vector Fields and Differential Operators
Learn the concept of vector fields, definitions and physical meanings of gradient (grad), divergence (div), and curl (rot). Implementation of Laplacian, vector field visualization, and determination of conservative fields and potential functions.
Vector Fields Gradient (grad) Divergence (div) Curl (rot) Laplacian
💻 7 Code Examples ⏱️ 18-22 minutes
Chapter 5
Line Integrals, Surface Integrals, and Integral Theorems
Learn calculation methods for line integrals (scalar and vector fields) and surface integrals (scalar and vector fields). Understand Green's theorem, Gauss's divergence theorem, and Stokes' theorem, and implement an application to atomic diffusion via the continuity equation.
Line Integrals Surface Integrals Green's Theorem Divergence Theorem Stokes' Theorem
💻 7 Code Examples ⏱️ 18-22 minutes
📚 Recommended Learning Paths
Pattern 1: Beginner - Theory and Practice Balanced (5-7 days)
- Day 1: Chapter 1 (Fundamentals)
- Day 2: Chapter 2 (Core Concepts)
- Day 3: Chapter 3 (Advanced Theory)
- Day 4: Chapter 4 (Applications)
- Day 5: Chapter 5 (Python Practice) + Review
Pattern 2: Intermediate - Fast Track (3 days)
- Day 1: Chapters 1-2 (Fundamentals and Core Concepts)
- Day 2: Chapters 3-4 (Advanced Theory and Applications)
- Day 3: Chapter 5 (Practice) + All Exercises
Pattern 3: Topic-Focused - Computational Skills (1 day)
- Focus: Code examples from all chapters
- Execute all Python implementations
- Modify parameters and analyze results
- Light theory review as needed
🎯 Overall Learning Outcomes
Upon completing this series, you will achieve:
Knowledge Level
- ✅ Understand fundamental theoretical concepts and mathematical formulations
- ✅ Explain relationships between key equations and physical phenomena
- ✅ Interpret results in context of real-world applications
- ✅ Connect concepts across chapters systematically
Practical Skills
- ✅ Implement algorithms from scratch using Python
- ✅ Utilize NumPy, SciPy, and Matplotlib effectively
- ✅ Visualize complex data and results
- ✅ Debug and optimize numerical code
Application Ability
- ✅ Apply theoretical concepts to practical problems
- ✅ Design computational experiments
- ✅ Analyze and interpret simulation results
- ✅ Extend learned methods to new domains
🛠️ Technologies and Tools Used
Main Libraries
- numpy
- scipy
- matplotlib
- sympy
Development Environment
- Python : 3.8 or higher
- Jupyter Notebook : Interactive development and visualization
- IDE : VSCode, PyCharm, or similar
Recommended Tools
- Google Colab (cloud-based, no setup required)
- Anaconda Distribution (complete environment)
- Git (version control for exercises)
🚀 Next Steps
Deep Dive Learning
For more advanced study in this field:
- Real Analysis
- Differential Geometry
- Tensor Calculus
Related Series
Expand your knowledge with related topics:
- Linear Algebra and Tensor Analysis
- Complex Functions and Special Functions
Practical Projects
Apply your skills to hands-on projects:
- 3D vector field visualization
- Numerical PDE solver
- Gradient descent optimizer
⚠️ Disclaimer
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