ベクトル分析
In multivariable vector analysis, we discuss the differentiation and integration of the following functions:
- Vector-valued functions $\mathbf{f} : \mathbb{R} \to \mathbb{R}^{n}$
- Multivariable functions $f : \mathbb{R}^{n} \to \mathbb{R}$
- Multivariable vector functions $\mathbf{f} : \mathbb{R}^{n} \to \mathbb{R}^{m}$
Real functions $f : \mathbb{R} \to \mathbb{R}$ are covered in the Introduction to Analysis category.
Especially, 3D functions $f : \mathbb{R}^{3} \to \mathbb{R}$ and $\mathbf{f} : \mathbb{R}^{3} \to \mathbb{R}^{3}$ are discussed in the Mathematical Physics category, slightly less rigorously to suit the level of physics and engineering majors.
Euclidean Space
- What is Euclidean Space $\mathbb{R}^{n}$
- Scalar Functions and Vector Functions
- Inner Product in Euclidean Space
- N-dimensional Polar Coordinates
Vector-valued Functions
Covers content related to vector-valued functions $\mathbf{f} : \mathbb{R} \to \mathbb{R}^{n}$.
Differentiation
Integration
Multivariable Functions
Covers content related to multivariable functions $f : \mathbb{R}^{n} \to \mathbb{R}$.
Differentiation
- Total Differentiation
- Directional Derivatives
- Gradient of a Scalar Field
- Hessian Matrix
- Taylor’s Theorem
- Derivatives of Scalar Functions with Vectors and Matrices $\nabla \mathbf{w}^{T} R \mathbf{x}$
Integration
Multivariable Vector Functions
Covers content related to $\mathbf{f} : \mathbb{R}^{n} \to \mathbb{R}^{m}$.
Differentiation
- Total Derivative
- Jacobian Matrix
- Regular Mapping
- Partial Derivatives
- Chain Rule for Multivariable Vector Functions
- Inverse Function Theorem
- Divergence of a Vector Field
Integration
References
- Walter Rudin, Principles of Mathmatical Analysis (3rd Edition, 1976)
- William R. Wade, An Introduction to Analysis (4th Edition, 2010)