Vector Analysis
In multivariable vector analysis, we discuss the differentiation and integration of the following functions:
- Vector-valued functions
- Multivariable functions
- Multivariable vector functions
Real functions are covered in the Introduction to Analysis category.
Especially, 3D functions and 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
- Scalar Functions and Vector Functions
- Inner Product in Euclidean Space
- N-dimensional Polar Coordinates
Vector-valued Functions
Covers content related to vector-valued functions .
Differentiation
Integration
Multivariable Functions
Covers content related to multivariable functions .
Differentiation
- Total Differentiation
- Directional Derivatives
- Gradient
- Laplacian
- Hessian Matrix
- Taylor’s Theorem
- Derivatives of Scalar Functions with Vectors and Matrices
Integration
Multivariable Vector Functions
Covers content related to .
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)
- James Stewart, Daniel Clegg, and Saleem Watson, Calculus (early transcendentals, 9E)
All posts
- Inner product in Euclidean space
- Proof of the Pappus-Guldin Theorem
- Scalar Functions and Vector-valued Functions
- Jacobian Matrix or Jacobi Matrix
- What is a Hessian Matrix?
- Gradient of Scalar Field
- Volume in Vector Fields
- Divergence in Vector Fields
- Integration of Vector-Valued Functions
- Derivatives of Vectors and Matrices
- Partial Derivatives
- Integration of Multivariable Functions
- Laplacian of a Scalar Field
- Partial Derivatives: Derivatives of Multivariable Vector Functions
- Conformal Mapping
- Definition of Directional Derivative
- Chain Rule for Multivariable Vector Functions
- Jacobian of Composite Functions
- Inverse Function Theorem in Analysis
- Taylor's Theorem for Multivariable Functions
- n-Dimensional Polar Coordinates
- Sum of Squared Residuals' Gradient
- Why Notation of Partial Differential is Different?
- Derivative of a Vector-Valued Function
- Limits and Continuity of Vector-Valued Functions
- Angle Between Two Vectors in an n-Dimensional Euclidean Space