The Chemistry Maths Book, Second Edition

10.7 Exercises 313

23.Calculate the average distance from the nucleus of an electron in the 2 sorbital

(see Exercise 21).

24.Calculate the average value ofr

2

for the 3 p

z

orbital (see Exercise 22).

Section 10.5

Find∇

2

f:

25.f 1 = 1 x

2

y

3

z

4

26.r

n

e

−ar

27. 28.e

−r 23

1 sin 1 θ 1 sin 1 φ 29.xze

−r 22

30.Iff 1 = 1 (2 1 − 1 r)e

−r 22

show that

31.Iff 1 = 1 ze

− 3 r 22

show that

32.Iff 1 = 1 xye

−r

show that

33.Show that where ais an arbitrary number, satisfies∇

2

f 1 = 1 a

2

f.

Show that the following functions satisfy the Laplace equation:

34. 2 x

3

1 − 13 x(y

2

1 + 1 z

2

) 35. 36.r

n

1 sin

n

1 θ 1 cos 1 nφ,n 1 = 1 1, 1 2, 1 3,1=

37.The d’Alembertian operator is

where x, y, zare coordinates, tis the time, and cis the speed of light. Show that if the

functionf(x, y, z)satisfies Laplace’s equation then the functiong(x, 1 y, 1 z, 1 t) 1 = 1 f(x, 1 y, 1 z)e

ikct

satisfies the equation

Section 10.6

38.What are the parametric equations of a left-handed helix in (i) cartesian coordinates,

(ii) cylindrical polar coordinates?

39.Integrate the functionf 1 = 1 y

2

z

3

over the cylindrical region of radius a, betweenz 1 = 10 and

z 1 = 11 , and symmetric about the z-axis.

Use confocal elliptic coordinates, φ, to integrate the following

functions over all space:

40. 
    
    
    
    41. 
        
    
    
    
    

e

r

−+()rr

cos

AB

B

2

φ

e

r

−+()rr

AB

A

ξη=

• 
  

,=

−

,

rr

R

rr

R

AB AB



22

gkg=.



2

2

2

2

2

2

22

2

2

1

=

∂

∂

• 
  

∂

∂

• 
  

∂

∂

−

∂

xyzct∂

cos

sin

2

22

φ

r θ

fr

e

r

ar

() , =

∇+ =

2

6

f

f

r

f.

∇+ =

2

69

4

f

f

r

f

.

∇+ =

2

2

4

f

f

r

f

.

e

r

−r