Representation of Surfaces

1. Surfaces are represented in Cartesian Coordinate Systems using Explicit Scalar Equations, Implicit Scalar Equations and / or Parametric Position Vector Functions.
2. Explicit Scalar Equations: These equations can be used to represent Surfaces in 3D Cartesian Coordinate System. The Surfaces are represented as
   
   \(z=f(x,y)\)
   
   where \(x\) and \(y\) are Independent Variables and \(z\) which is a Function of Variables \(x\) and \(y\) is the Dependent Variable
   
   OR
   
   \(y=f(z,x)\)
   
   where \(z\) and \(x\) are Independent Variables and \(y\) which is a Function of Variables \(z\) and \(x\) is the Dependent Variable
   
   OR
   
   \(x=f(y,z)\)
   
   where \(y\) and \(z\) are Independent Variables and \(x\) which is a Function of Variables \(y\) and \(z\) is the Dependent Variable.
   
   Please note the although Explicit Scalar Equations allow us to determine the Orientation of the Surfaces, not all kinds of 3D Surfaces can be represented by using these equations, especially in which the Surfaces Intersect/Meet themselves.
3. Implicit Scalar Equations: These equations can also be used to represent Surfaces in 3D Cartesian Coordinate System. The Surfaces are represented as
   
   \(f(x,y,z)=0\)
   
   Please note that Although any kind of 3D Surfaces can be represented by using the Implicit Scalar Equations, the Orientation of the Surfaces cannot be determined by by these equations.
4. Parametric Position Vector Functions: These functions can be used to represent any and all kinds of Surfaces in 3D Cartesian Coordinate System. Also representing Surfaces using these functions allow us to determine the Orientation of the Surfaces.
   
   The Surfaces are represented using Position Vectors whose Components are Functions of 2 Variable Parameters. For example, any 3D Surface is represented as
   
   \(\vec{R}=x\hat{\mathbf{i}} + y\hat{\mathbf{j}} + z\hat{\mathbf{k}}\)
   
   where \(x\), \(y\) and \(z\) are Functions of 2 Variable Parameters, \(u\) and \(v\) given as follows
   
   \(x=f_x(u,v)\),   \(y=f_y(u,v)\),   \(z=f_z(u,v)\)
   
   and Vector \(\vec{R}\) is the Position Vector of any point on the Surface.