6—Vector Spaces 1416.29 Do the same calculation as in problem6.7, but use the scalar product
〈f,g
〉
=
∫ 1
0x^2 dxf*(x)g(x)
6.30 Show that the following is a scalar product.
〈f,g
〉
=
∫badx
[
f(x)g(x) +λf′(x)g′(x)
]
whereλis a constant. What restrictions if any must you place onλ? The name Sobolev is associated
with this scalar product.
6.31 (a) With the scalar product of problem6.29, find the angle between the vectors 1 andx. Here
the word angle appears in the sense ofA~.B~=ABcosθ. (b) What is the angle if you use the scalar
product of problem6.7? (c) With the first of these scalar products, what combination of 1 andxis
orthogonal to 1? Ans: 14. 48 ◦
6.32 In the online text linked on the second page of this chapter, you will find that section two of
chapter three has enough additional problems to keep you happy.
6.33 Show that the sequence of rational numbersan=
∑nk=1^1 /kis not a Cauchy sequence. What
about
∑nk=1^1 /k
(^2)?
6.34 In the vector space of polynomials of the formαx+βx^3 , use the scalar product
〈
f,g
〉
=
∫ 1
0 dxf(x)
∗g(x)and construct an orthogonal basis for this space. Ans: One pair isx, x (^3) − 3
5 x.
6.35 You can construct the Chebyshev polynomials by starting from the successive powers,xn,n=
0 , 1 , 2 ,...and applying the Gram-Schmidt process. The scalar product in this case is
〈
f,g
〉
=
∫ 1
− 1dx
f(x)∗g(x)
√
1 −x^2
The conventional normalization for these polynomials isTn(1) = 1, so you should not try to make the
norm of the resulting vectors one. Construct the first four of these polynomials, and show that these
satisfyTn(cosθ) = cos(nθ). These polynomials are used in numerical analysis because they have the
property that they oscillate uniformly between− 1 and+1on the domain− 1 < x < 1. Verify that your
results for the first four polynomials satisfy the recurrence relation:Tn+1(x) = 2xTn(x)−Tn− 1 (x).
Also show thatcos
(
(n+ 1)θ
)
= 2 cosθcos
(
nθ
)
−cos(
(n−1)θ
)
.
6.36 In spherical coordinates(θ,φ), the angle θ is measured from the z-axis, and the function
f 1 (θ,φ) = cosθcan be written in terms of rectangular coordinates as (section8.8)
f 1 (θ,φ) = cosθ=
z
r
=
z
√
x^2 +y^2 +z^2
Pick up the functionf 1 and rotate it by 90 ◦counterclockwise about the positivey-axis. Do this rotation
in terms of rectangular coordinates, but express the result in terms of spherical coordinates: sines and
cosines ofθandφ. Call itf 2. Draw a picture and figure out where the original and the rotated function
are positive and negative and zero.