312 Chapter 10Functions in 3 dimensions
Then, separating variables,
0 Exercises 40, 41
10.7 Exercises
Section 10.2
Find the cartesian coordinates (x, y, z) of the points whose spherical polar coordinates are:
1.(r, 1 θ, 1 φ) 1 = 1 (1, 1 0, 1 0) 2.(r, 1 θ, 1 φ) 1 = 1 (2, 1 π 2 2, 1 π 2 2) 3.(r, 1 θ, 1 φ) 1 = 1 (2, 12 π 2 3, 13 π 2 4)
Find the spherical polar coordinates (r, 1 θ, 1 φ) of the points:
4.(x, 1 y, 1 z) 1 = 1 (1, 1 0, 1 0) 5.(x, 1 y, 1 z) 1 = 1 (0, 1 1, 1 0) 6.(x, 1 y, 1 z) 1 = 1 (1, 1 2, 1 2)
7.(x, 1 y, 1 z) 1 = 1 (1, 1 −4, 1 −8) 8.(x, 1 y, 1 z) 1 = 1 (−2, 1 −3, 1 6) 9.(x, 1 y, 1 z) 1 = 1 (−3, 1 4, 1 −12)
Section 10.3
Express in spherical polar coordinates:
10.x
2
1 − 1 y
2
11.(x
2
1 + 1 y
2
) 2 z
2
- 2 z
2
1 − 1 x
2
1 − 1 y
2
13.Express in spherical polar coordinates:
(i)(x
2
1 + 1 y
2
1 + 1 z
2
)
− 122
, (ii)
Section 10.4
Find the total mass of a mass distribution of densityρin region V of space:
14.ρ 1 = 1 x
2
1 + 1 y
2
1 + 1 z
2
, V: the cube 01 ≤ 1 x 1 ≤ 1 1, 0 1 ≤ 1 y 1 ≤ 1 1, 0 1 ≤ 1 z 1 ≤ 11
15.ρ 1 = 1 xy
2
z
3
, V: the box 01 ≤ 1 x 1 ≤ 1 a, 0 1 ≤ 1 y 1 ≤ 1 b, 0 1 ≤ 1 z 1 ≤ 1 c
16.ρ 1 = 1 x
2
, V: the region 11 − 1 y 1 ≤ 1 x 1 ≤ 1 1, 0 1 ≤ 1 y 1 ≤ 1 1, 0 1 ≤ 1 z 1 ≤ 12
17.ρ 1 = 1 e
−(x+y+z), V: the infinite regionx 1 ≥ 1 0, y 1 ≥ 1 0, z 1 ≥ 10
18.ρ 1 = 1 x
2
1 + 1 y
2
1 + 1 z
2
, V: the sphere of radius a, centre at the origin
- V: the sphere of radius a, centre at the origin
20.ρ 1 = 1 r
3
e
−r, V: all space
21.The 2 sorbital of the hydrogen atom is. Show that the
integral of over all space is unity.
22.The 3 p
zorbital of the hydrogen atom isψ
3 pz1 = 1 C(6 1 − 1 r)re
−r 231 cos 1 θ(in atomic units)
where Cis a constant. Find the value of Cthat normalizes the 3 p
zorbital.
ψ
2
2
sψ
2
0
3
0
2
1
42
2
sraa
=−ra e
−
π
() 2
2
0ρ
θ φ
=
sin cos
,
22
r
∂
∂
++
−
x
()xyz
22212
=++
( )
−13
2RR e
R=×× ++
( )
−
−Re
R
RR
3 R328
22 2 2
π
π 22
2
3
π××
−e
R
RS
R
dded d
RAB=
−−+−−+30211120218 π
ππZZZ ZZ
φη ξξ φ
ξ∞1 121ηη ξ
ξded
RZ
∞−