FINAL PROJECT, MATH 251, FALL 2015
[The project is Due Monday after the thanks giving recess]
.NAME(PRINT).________________ SHOW ALL WORK. Explain and SKETCH (everywhere anytime and especially as you try to comprehend the prob- lems below) whenever possible and/or necessary. Please carefully recheck your answers. Leave reasonable space between lines on your solution sheets. Number them and print your name.
Please sign the following. I hereby affirm that all the work in this project was done by myself ______________________.
1) i) Explain how to derive the representation of the Cartesian coordinates x, y, z in terms of the spherical coordinates ρ, θ, φ to obtain
(0.1) r =< x = ρsin(φ)cos(θ), y = ρsin(φ)sin(θ), z = ρcos(φ) > .
What are the conventional ranges of ρ, θ, φ?
ii) Conversely, explain how to express ρ, sin(θ), cos(θ), cos(φ), sin(φ) as functions of x, y, z.
iii) Consider the spherical coordinates ρ, θ, φ. Sketch and describe in your own words the set of all points x, y, z in x, y, z space such that:
a)0≤ρ≤1,0≤θ<2π,0≤φ≤π b)ρ=1, 0≤θ<2π,0≤φ≤π, c)0≤ρ<∞, 0≤θ<2π,φ=π, d)ρ=1, 0≤θ<2π,φ=π,
e) ρ=1,θ=π,0≤φ≤π.f)1≤ρ≤2,0≤θ<2π,π ≤φ≤π. 463
iv) In a different set of Cartesian Coordinates ρ, θ, φ sketch and describe in your own words the set of points (ρ, θ, φ) given above in each item a) to f). For example the set in a) in x, y, z space is a ball with radius 1 and center (0,0,0). However, in the Cartesian coordinates ρ, θ, φ the set in a) is a rectangular box.
2) [Computation and graphing of vector fields]. Given r =< x,y,z > and the vector Field
(0.2) F (x, y, z) = F (r) =< 1 + z, yx, y >, 1
