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< 2 π, 0 ≤ φ ≤ π b ) ρ =1 , 0 ≤ θ< 2 π, 0 ≤ φ ≤ π, c )0 ≤ ρ< ∞ , 0 ≤ θ< 2 π,φ = π , d ) ρ =1 , 0 ≤ θ< 2 π,φ = π , e) ρ =1 ,θ = π , 0 ≤ φ ≤ π. f) 1 ≤ ρ ≤ 2 , 0 ≤ θ< 2 π, 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 44 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 = 2 cos ( t ) , y = 4 sin ( t ) , z ≡ 0 >, 0 ≤ t < 2 π, 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 , 3 π , 5 π , 7 π . Then draw the arrows. 4444 iii) Given the surface r ( θ,φ ) = < x = 2 sin ( φ ) cos ( θ ) , y = 2 sin ( φ ) sin ( θ ) , z = 2 cos ( φ ) >, 0 ≤ θ < 2 π, 0 ≤ φ ≤ π, in parametric form. Use trigonometric formulas to show that the following iden- tity holds iv) Draw the arrows .

1. 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
)

2. 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

3. ,
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<
2
π,
0
≤
φ
≤
π b
)
ρ
=1
,
0
≤
θ<
2
π,
0
≤
φ

4. ≤
π, c
)0
≤
ρ<
∞
,
0
≤
θ<
2
π,φ
=
π
, d
)
ρ
=1
,
0
≤
θ<
2
π,φ
=
π
,
e)
ρ
=1
,θ
=
π
,
0
≤

5. φ
≤
π.
f)
1
≤
ρ
≤
2
,
0
≤
θ<
2
π,
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

6. =
< x,y,z >
and the vector Field
(0.2)
F
(
x, y, z
) =
F
(
r
) =
<
1 +
z, yx, y >,
1
44
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 (

7. x, y, z
)
for the 4 points (
±
1
,
±
2
,
1)
. Afterwards draw the arrows.
ii) Show that the curve
(0.3)
r
(
t
) =
< x
= 2
cos
(
t
)
, y
= 4
sin
(
t
)
, z
≡
0
>,
0
≤
t <

8. 2
π,
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

9. )
, 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
(

10. t
)
, z
(
t
))
,
dr
(
t
)
, F
(
r
(
t
))
, cos
(
θ
(
t
))
for the points
(
x
(
t
)
, y
(
t
)
, z
(
t

11. ))
dt
corresponding to
t
= 0
,
π
,
3
π
,
5
π
,
7
π
. Then draw the arrows.
4444
iii) Given the surface
r
(
θ,φ
) =
< x
= 2
sin
(
φ
)
cos
(
θ
)
, y
= 2
sin

12. (
φ
)
sin
(
θ
)
, z
= 2
cos
(
φ
)
>,
0
≤
θ <
2
π,
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
)

13. , z
(
θ, φ
))
and representing the
x
2
(
θ, φ
) +
y
2
(
θ, φ
) +
z
2
(
θ, φ
)
≡
2
2
.
vectors
∂r
(
θ,φ
)
×
∂r
(
θ,φ
)
,

14. 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

15. )
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
(

16. θ,φ
)
and
F
(
r
(
θ,φ
))
, cos
(
α
(
θ,φ
))
forthepoints
(
x
(
θ,φ
)
,y
(
θ,φ
)
,z
(
θ,φ
))
∂θ ∂φ
corresponding to
(
θ,φ
)=(
π
,

17. π
)
,
(
3
π
,
π
)
,
(
5
π
,
π
)
,
(
7
π
,
π
)
,
(
3
π
,
5
π
)
,
(
5
π
,

18. 5
π
)
,
(
7
π
,
5
π
)
. 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
> .
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