I’ve recently taken a multivariable vector calculus class and had a very hard time visualizing and understanding the problems until I drew it out and wrote about it. This page is an explanation of the parameterization and calculation of the surface integral of the sphere. I also try to explain where equations come from as best I can.
Part 1: Expressing the Sphere
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.
Since we want to calculate the surface area of a sphere we are also going to use some additional concepts.
A sphere is defined in cartesian coordinates by:
X2 + Y2 + Z2= r2
Now this surface integral IS solvable without parameterization; However it will be nasty.
In the past I have used the unit circle to express the equation X2 + Y2 = r2 as X = rcos(θ), Y = rsin(θ) in polar coordinates. This allowed for rotation about one axis θ with small slices of the circle with angle dθ thickness.
Because a sphere exists in 3 dimensions, we will have to rotate about an additional axis to get the surface integral. In general φ is used as this additional movement angle.
Sphere by Matthew Leingang https://www.slideshare.net/leingang/math-21a-midterm-i-review
To simplify this; Sphere X2 + Y2 + Z2= r2 Can be expressed in terms of constant r, φ, and θ. Where φ and θ control the orientation of a vector and r the magnitude. 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)
Therefore the sphere can be expressed in terms of θ,φ:
r(θ,φ) = (rSin(φ)Cos(θ), rSin(φ)Sin(θ),rCos(φ))
0≤θ≤2π 0≤φ≤π
The X, Y, Z representations come from trigonometry and geometric relationships which I will try to show later.
In Part II the use of this parameterization will be introduced and concepts and formulas related explained.