Representing Geometric Objects/Fields in Coordinate Systems

1. Geometric Objects like Curves and Surfaces are represented mathematically through the following
   1. Scalar Coordinate Equations: These are some Scalar Function Equations that a particular Geometric Object satisfies. The Variables in these Scalar Function Equations are Coordinates Axes of the Coordinate System in which the Geometric Object is represented.
   2. Parametric Equations / Position Vector Functions: In this representation the Cartesian Coordinates \(x, y, z\) etc. are themselves given in terms of Scalar Functions of 1 or More Parameters/Variables. The value of the \(x,y,z\) thus obtained form the Components of Position Vector of Points on the Geometric Object.
   3. Vector Equations: These are some Vector Equations that a particular Geometric Object satisfies.
   Each of the above representation can have Various Types Depending on the Actual Geometric Object.
2. Data Distributions / Fields are represented mathematically through the following
   1. Scalar Coordinate Expressions: These are Scalar Function Expressions that represent Fields. The Variables in these Scalar Function Expressions are Coordinates Axes of the Coordinate System in which the Field is represented.
   2. Vector Functions: These are Vector Function Expressions that represent Fields. Each Component of the Vector Function is a Scalar Function Expression. The Variables in these Scalar Function Expressions are Coordinates Axes of the Coordinate System in which the Field is represented.