Surface Integrals II: Applying the parameterized function

In Part I I defined the surface integral as:

A surface integral is defined as:

Where r(u,v) is the surface equation, f(x,y,z) is the function for the numerical value ‘assigned’ each point on the surface such as a temperature distribution model. The cross product finds the area between the bounded parallelogram of vector Ru and Rv. For now, most of this doesn’t matter but I will try to continually elaborate on its meaning and why it is what it is.

To solve for the surface area of a sphere, the following parameterizations were defined in part I.

To simplify this; Sphere X² + Y² + Z²= r²  Can be expressed in terms of r, φ, and θ. Where φ and θ control the orientation of a vector and r the magnitude. r(r,θ,φ) -> (X,Y,Z) In this visual representation, φ  would be bounded from [0,π] and θ from [0,2π].

X = rSin(φ)Cos(θ)

Y= rSin(φ)Sin(θ)

Z = rCos(φ)