Mathematical Tools for Physics

8—Multivariable Calculus 244

8.22 Taylor’s power series expansion of a function of several variables was discussed in section2.5. The Taylor
series in one variable was expressed in terms of an exponential in problem2.30. Show that the series in three
variables can be written as
e

~h.∇

f(x,y,z)

8.23 The wave equation is (1). Change variables toz=x−vtandw=x+vtand show that in these coordinates
this equation is (2).

(1)

∂^2 u

∂x^2

−

1

v^2

∂^2 u

∂t^2

= 0 (2)

∂^2 u

∂z∂w

= 0

8.24 The equation ( 16 ) comes from taking the gradient of the Earth’s gravitational potential in an expansion to
terms in 1 /r^3.

V=−

GM

r

−

GQ

r^3

P 2 (cosθ)

whereP 2 (cosθ) =^32 cos^2 θ−^12 is the second order Legendre polynomial. Compute~g=−∇V.

8.25 In problem2.25you computed the electric potential at large distances from a pair of charges,−qat the
origin and+qatz=a(ra). The result was

V =

kqa

r^2

P 1 (cosθ)

whereP 1 (cosθ) = cosθ is the first order Legendre polynomial. Compute the electric field from this potential,
E~=−∇V. And sketch it of course.

8.26 In problem2.26you computed the electric potential at large distances from a set of three charges,− 2 qat
the origin and+qatz=±a(ra). The result was

V =

kqa^2

r^3

P 2 (cosθ)

whereP 2 (cosθ)is the second order Legendre polynomial. Compute the electric field from this potential,E~ =
−∇V. And sketch it of course.