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

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FINAL PROJECT, MATH 251, FALL 2015 2

i) Draw the arrows emanating from (x, y, z) and representing the vectors F (r) = F (x, y, z) . First draw a 2 raw table recording F (r) versus (x, y, z) for the 4 points (±1, ±2, 1) . Afterwards draw the arrows.

ii) Show that the curve

(0.3) r(t) =< x = 2cos(t), y = 4sin(t), z ≡ 0 >, 0 ≤ t < 2π,

is an ellipse. Draw the arrows emanating from (x(t), y(t), z(t)) and representing the vector values of dr(t) , F (r(t)) = F (x(t), y(t), z(t)) . Let θ(t) be the angle

dt

between the arrows representing dr(t) and F(r(t)) . First draw a 5 raw table dt

recording t, (x(t), y(t), z(t)), dr(t) , F (r(t)), cos(θ(t)) for the points (x(t), y(t), z(t)) dt

corresponding to t = 0,π , 3π , 5π , 7π . Then draw the arrows. 4444

iii) Given the surface

r(θ,φ) =< x = 2sin(φ)cos(θ), y = 2sin(φ)sin(θ), z = 2cos(φ) >,0 ≤ θ < 2π, 0 ≤ φ ≤ π, in parametric form. Use trigonometric formulas to show that the following iden-

tity holds

iv) Draw the arrows emanating from (x(θ, φ), y(θ, φ), z(θ, φ)) and representing the

x2(θ, φ) + y2(θ, φ) + z2(θ, φ) ≡ 22.

vectors ∂r(θ,φ) × ∂r(θ,φ) , F (r(θ, φ)) = F (x(θ, φ), y(θ, φ), z(θ, φ)) . Let α(θ, φ) be

∂θ ∂φ

the angle between the arrows representing ∂r(θ,φ) × ∂r(θ,φ) and F(r(θ,φ)) . First ∂θ ∂φ

draw a table with raws and columns recording (θ, φ),(x(θ, φ), y(θ, φ), z(θ, φ)), ∂r(θ,φ)×∂r(θ,φ) andF(r(θ,φ)), cos(α(θ,φ))forthepoints(x(θ,φ),y(θ,φ),z(θ,φ))

∂θ ∂φ

corresponding to(θ,φ)=(π,π),(3π,π),(5π,π),(7π,π),(3π,5π),(5π,5π),(7π,5π). Thendraw

the arrows in (x, y, z) space. Repeat iv) with

3) Given the integral (0.4)

F(x,y,z) = F(r) =< 1,x,0 > .