Vector Analysis

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
  • Angle Between Two Vectors 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

• Limits and Continuity
• Derivatives
  • A Constant-Magnitude Vector-Valued Function is Orthogonal to Its Derivative

Integration

• Integration

Multivariable Functions

Covers content related to multivariable functions $f : \mathbb{R}^{n} \to \mathbb{R}$.

Differentiation

• Total Differentiation
• Directional Derivatives
• Gradient $\nabla u = \operatorname{grad}u$
• Laplacian $\Delta u = \nabla^{2} u$
• Hessian Matrix
• Taylor’s Theorem

Integration

• Integration of Multivariable Functions

Multivariable Vector Functions

Covers content related to $\mathbf{f} : \mathbb{R}^{n} \to \mathbb{R}^{m}$.

Differentiation

• Total Derivative
• Jacobian Matrix
  • Jacobian of Composite Functions
• Regular Mapping
• Partial Derivatives
  • Why Use Different Notation for Partial Derivatives $d \overset{?}{\to} \partial$
• Chain Rule for Multivariable Vector Functions
• Inverse Function Theorem
• Divergence of a Vector Field

Integration

• Volume of a Vector Field

Vector and Matrix Calculus

Vector Calculus

• Derivatives of Scalar Functions with Vectors and Matrices $\nabla \mathbf{w}^{T} R \mathbf{x}$
  • Gradient of Residual Sum of Squares $\nabla \left| \mathbf{y} - X \mathbf{w} \right|_{2}$

Matrix Calculus

• Generalizing Differentiation: Gradient Matrices and Matrix Calculus $\nabla_{\mathbf{X}}f$
• Matrix Differentiation Table for Scalar Functions
  • Matrix Calculus of Quadratic and Bilinear Forms
  • Matrix Calculus of Trace
• Total Differential of Function of a Matrix $\mathrm{d}f = \Tr \left( \left( \nabla_{\mathbf{X}}f \right)^{\mathsf{T}} \mathrm{d}\mathbf{X} \right)$
• Matrix infinitesimal $\mathrm{d}\mathbf{X}$
• Jacobi’s Formula $\mathrm{d}(\det A) = \Tr \left( (\operatorname{adj}A) \mathrm{d}A \right) = \det A \cdot \Tr(A^{-1} \mathrm{d}A)$
• Trace Trick $\mathrm{d}f = \Tr (\mathrm{d}f) = \mathrm{d}\Tr (f)$

References

• Walter Rudin, Principles of Mathmatical Analysis (3rd Edition, 1976)
• William R. Wade, An Introduction to Analysis (4th Edition, 2010)
• James Stewart, Daniel Clegg, and Saleem Watson, Calculus (early transcendentals, 9E)