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.
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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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.
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 identity 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 > .