Vector Calculus: Geometry of Euclidean Space ℝⁿ and Vector Algebra · Introduction

Introduction

What this guide covers

An integral and axiomatic treatise on the linear, metric, topological and differential structure of the n-dimensional Euclidean space. From the foundations of pre-Hilbert spaces to bilinear forms, Grassmann's exterior algebra, geometric theory of curves and surfaces, and tensor differential operators.

Overview

Topics covered

Introduction
Vector Calculus Geometry of ℝⁿ and Vector Algebra
The Real Vector Space ℝⁿ: Axiomatic Structure
Linear Combinations, Independence and Bases
The Euclidean and Hilbert Metric Scaling Product
Angle between vectors in ℝⁿ

Related concepts

Main connections

Compatible notation

Mathematical sample in LaTeX

The expression is rendered accessibly, with its LaTeX source preserved below.

\R^n = \underbrace{\R \times \R \times \dots \times \R}_{n\text{ veces}} = \left\{ \mathbf{x} = (x_1, x_2, \dots, x_n) \mid x_i \in \R, \, i=1, 2, \dots, n \right\}.

View LaTeX source

\R^n = \underbrace{\R \times \R \times \dots \times \R}_{n\text{ veces}} = \left\{ \mathbf{x} = (x_1, x_2, \dots, x_n) \mid x_i \in \R, \, i=1, 2, \dots, n \right\}.