Order now

Draw the arrows emanating from (x, y, z) and representing the vectors F (r) = F (x, y, z) .


[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



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


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 > .

Place a similar order with us or any form of academic custom essays related subject and it will be delivered within its deadline. All assignments are written from scratch based on the instructions which you will provide to ensure it is original and not plagiarized. Kindly use the calculator below to get your order cost; Do not hesitate to contact our support staff if you need any clarifications.

Type of paper Academic level Subject area
Number of pages Paper urgency Cost per page:

Whatever level of paper you need – college, university, research paper, term paper or just a high school paper, you can safely place an order.