Module MA1132 (Frolov), Advanced Calculus
Homework Sheet 4
Each set of homework questions is worth 100 marks
Due: at the beginning of the tutorial session Thursday/Friday, 18/19 February 2016
Name:
1. Consider the function
π
π
z = 3ey− 4 cos x − 2e 2 −x sin y
(a) Find
iii)
∂ 2z π π
( , ),
∂x∂y 2 4
iv)
∂ 2z π π
( , ).
∂y∂x 2 4
π
π
(b) Find the slope of the surface z = 3ey− 4 cos x − 2e 2 −x sin y in the y-direction at the
point ( π3 , π6 ).
π
π
(c) Show that the function z = 3ey− 4 cos x − 2e 2 −x sin y satisfies Laplace’s equation
∂ 2z ∂ 2z
+
= 0.
∂x2 ∂y 2
2. The equations of motion of a system of n particles are given by
mi ẍi = −
∂U (x1 , . . . , xn )
,
∂xi
ẍi =
d 2 xi
,
dt2
i = 1, 2, . . . , n ,
where mi is the mass and xi is the coordinate of the i-th particle, and U (x1 , . . . , xn ) is
the potential energy of the system.
(a) Consider a system of n particles moving in a central field
U (x1 , . . . , xn ) = V (r) ,
n
X
r=
xi e i ,
i=1
where V is a smooth function of a single variable.
i.
ii.
iii.
iv.
v.
Find the equations of motion of the first particle (x1 ).
Find the equations of motion of the second particle (x2 ).
Find the equations of motion of the last particle (xn ).
Find the equations of motion of the i-th particle (xi ) for 1 < i < n.
Write the equations of motion of the i-th particle (xi ) for 1 ≤ i ≤ n by using
the Kronecker delta δij .
(b) Consider a system of n particles with the Toda potential
U (x1 , . . . , xn ) =
n−1
X
i=1
1
eα(xi+1 −xi ) .
i.
ii.
iii.
iv.
v.
Find the equations of motion of the first particle (x1 ).
Find the equations of motion of the second particle (x2 ).
Find the equations of motion of the last particle (xn ).
Find the equations of motion of the i-th particle (xi ) for 1 < i < n.
Write the equations of motion of the i-th particle (xi ) for 1 ≤ i ≤ n by using
the Kronecker delta δij .
(c) Find the equations of motion of a system of n particles with the rational CalogeroMoser potential
n
X
α
.
U (x1 , . . . , xn ) =
2
(x
i − xj )
i,j=1,i6=j
3. Compute the differential df of
1
1
1
f (x1 , x2 , . . . , xn ) = ( + x1 )α1 ( + x2 )α2 · · · ( + xn )αn ,
2
2
2
and find its local linear approximation at ( 12 , 12 , . . . , 21 ).
4. The Taylor series is given by
∞
X
f (~x) =
k1 ,...,kn
∂1k1 · · · ∂nkn f (x~o ) k1
∆x1 · · · ∆xknn ,
k1 ! · · · kn !
=0
(1)
where we denote
f (xo1 , . . . , xon ) ≡ f (x~o ) ,
f (x1 , . . . , xn ) ≡ f (~x) ,
and ∂i0 f ≡ f ; ∂ik f ≡
∂k f
∂xki
xi − xoi ≡ ∆xi
(2)
is the k-th partial derivative of f with respect to xi .
The Taylor series can be equivalently written as
∞
n
X
1 X
∂ q f (x~o )
f (~x) =
∆xi1 · · · ∆xiq .
q! i ,...,i =1 ∂xi1 · · · ∂xiq
q=0
1
(3)
q
Check the equality by computing the Taylor series expansion up to the third order.
5. Consider the “Higgs” potential
U (x1 , . . . , xn ) = −
κ2 2 λ2 4
r + r ,
2
4
n
X
r=
xi e i ,
κ > 0, λ > 0 .
i=1
(a) Plot the potential for n = 1, and for κ = λ = 2.
(b) Plot the potential for n = 2, and for κ = λ = 2.
(c) Find the Taylor series expansion of the “Higgs” potential about the point xo1 = λκ ,
xoi = 0, i = 2, . . . , n up to the fourth order in yi ≡ xi − xoi . Use Mathematica to
check your answer.
2
6. Find the Taylor series expansion of the periodic Toda potential
U (x1 , . . . , xn ) =
n
X
eα(xi+1 −xi ) ,
xk+n ≡ xk ,
k ∈ Z,
i=1
about the point xi = a, i = 1, . . . , n up to the fourth order in xi − a. Use Mathematica
to check your answer.
Bonus questions (each bonus question is worth extra 25 marks)
1. Show that the Taylor series can be equivalently written as in (3).
3