[Paper Review] Course of analytical geometry

Руслан Шарипов|arXiv (Cornell University)|Nov 28, 2011

Soil, Finite Element MethodsEngineering20 citations

TL;DR

This paper presents a foundational treatment of vector algebra within analytical geometry, focusing on the cross product via Levi-Civita symbol notation. It derives the vector cross product formula using tensorial summation over basis vectors, establishing coordinate expressions for the resulting vector through ε-ijk components, thereby enabling systematic computation in orthonormal bases.

ABSTRACT

This book is a regular textbook of analytical geometry covering vector algebra and its applications to describing straight lines, planes, and quadrics in two and three dimensions. The stress is made on vector algebra by using skew-angular coordinates and by introducing some notations and prerequisites for understanding tensors. The book is addressed to students specializing in mathematics, physics, engineering, and technologies and to students of other specialities where educational standards require learning this subject.

Motivation & Objective

• To formalize vector algebra as a basis for analytical geometry, particularly focusing on vector operations.
• To derive the cross product of two vectors using the Levi-Civita symbol and tensor notation.
• To express the resulting vector in terms of its coordinates relative to an orthonormal basis.
• To provide a computational framework for vector cross products using index summation.

Proposed method

• The paper defines the cross product of vectors b and c as d = [b, c], using the Levi-Civita symbol ε_ijk.
• It applies the formula d^k = ∑_{i=1}^3 ∑_{j=1}^3 b^i c^j ε_ijk to compute each component of the resulting vector d.
• The vector d is expanded in the orthonormal basis {e₁, e₂, e₃}, with coefficients derived from the double summation over b^i and c^j.
• The method relies on the antisymmetric properties of the Levi-Civita symbol to ensure correct orientation and magnitude of the cross product.
• The coordinate expression d^k is systematically derived from the tensorial form, ensuring consistency with vector algebra principles.
• The derivation assumes a right-handed orthonormal basis, preserving geometric interpretation of the cross product.

Experimental results

Research questions

• RQ1 How can the vector cross product be systematically expressed using the Levi-Civita symbol and index notation?
• RQ2 What is the coordinate representation of the cross product vector in an orthonormal basis?
• RQ3 How do the components of the resulting vector d relate to the components of the input vectors b and c?
• RQ4 What is the role of the structure constants ε_ijk in computing vector products in three dimensions?

Key findings

• The cross product d = [b, c] is fully determined by the double summation d^k = ∑_{i=1}^3 ∑_{j=1}^3 b^i c^j ε_ijk over the Levi-Civita symbol.
• Each component d^k of the resulting vector corresponds to the algebraic sum of products of b^i and c^j weighted by ε_ijk.
• The expression ensures that d is orthogonal to both b and c, consistent with the geometric definition of the cross product.
• The method provides a coordinate-wise computational recipe for the cross product in three-dimensional space using standard basis vectors.
• The derivation confirms that the cross product is a bilinear, antisymmetric operation in the context of vector algebra.
• The formula establishes a direct link between tensor notation and vector component computation in orthonormal systems.