Multivariable Vector Calculus I: Parameterization of a Sphere.

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.

 

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.

 

The end result of this parameterization is a vector function r(θ,φ).I have come to think of its integration(for volume as an example) like this (might not be exactly accurate but it helps):  φ rotates down generating half a disc of vectors. Then θ rotates it around in a full circle, creating the final representation of the sphere in vector form. This will create infinite vectors separated by dθ and dφ. This of course, has nothing to do with the surface integration, but it helps visualize the way parameterization works.

 

In Part II the use of this parameterization will be introduced and concepts and formulas related explained.