Coordinate Systems

In vector calculus and tensor analysis, coordinate systems are foundational for describing geometric space and physical quantities. This note introduces basis vectors, coordinate axes, and covariant/contravariant bases in three-dimensional space, along with an interactive 3D visualization.

1. Basic Concepts

In 3D Euclidean space, any point’s position can be identified by coordinates . To describe vectors, we establish a set of basis vectors:

• Tangent Basis Vectors (Covariant Basis ): Vectors tangent to coordinate curves.
• Normal Basis Vectors (Contravariant Basis ): Vectors normal to coordinate surfaces.

When coordinate axes are mutually perpendicular with unit length, they form a Cartesian coordinate system. In general curvilinear or oblique coordinates, basis vectors are not necessarily orthogonal or unit length.

2. Interactive 3D Coordinate System Demonstration

The interactive figure below displays a 3D coordinate system. Here, vectors ( axis) and ( axis) define the plane, while vector ( axis) is calculated via cross product to ensure it is orthogonal to the plane.

Instruction: Drag with your mouse or touch screen on the canvas below to rotate the camera view.

Legend

● x₁ Axis

● x₂ Axis

● x³ Axis (Orthogonal to Ox₁x₂)

● Dashed: Ox₁x₂ Plane Boundary

💡 Drag to rotate view

3. Cross Product and Orthogonal Basis

Given two non-collinear basis vectors and in 3D space, we can uniquely determine a basis vector orthogonal to the plane formed by and using the cross product:

This ensures that is strictly perpendicular to the plane:

This construction is key in general curvilinear coordinates and tensor analysis for defining dual bases (contravariant bases).