Module MA1132 (Frolov), Advanced Calculus
Tutorial Sheet 4
To be solved during the tutorial session Thursday/Friday, 11/12 February 2016
1. Consider the function
π
π
z = 2e 2 −y sin x − 3ex− 4 cos y
(a) Find
i)
∂z π π
( , ),
∂x 4 2
∂z π π
( , ).
∂y 4 2
ii)
π
π
(b) Find the slope of the surface z = 2e 2 −y sin x − 3ex− 4 cos y in the x-direction at the
point ( π2 , π6 );
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) Find the equations of motion of a system of n particles moving in a Coulomb field
n
X
r=
xi e i .
α
U (x1 , . . . , xn ) = ,
r
i=1
(b) Find the equations of motion of a system of n coupled harmonic oscillators
U (x1 , . . . , xn ) =
n−1
X
κ
i=1
2
(xi+1 − xi )2 ,
(c) Find the equations of motion of a system of n particles with pairwise interaction
n
X
U (x1 , . . . , xn ) =
V (xi − xj ) .
i,j=1,i6=j
Here V is an even function of a single variable, and we use the notation
n
X
i,j=1,i6=j
aij ≡
n
n
X
X
j=1 i=1,i6=j
1
aij =
n
n
X
X
i=1 j=1,j6=i
aij .
3. The Taylor series is given by
f (~x) =
∞
X
k1 ,...,kn
∂1k1 · · · ∂nkn f (x~o ) k1
∆x1 · · · ∆xknn ,
k
!
·
·
·
k
!
1
n
=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
(a) Check the equality for functions of three variables by computing the Taylor series
expansion up to the third order.
4. Find the Taylor series expansion up to the quadratic order of the periodic Toda potential
U (x1 , . . . , xn ) =
n
X
eα(xi+1 −xi ) ,
i=1
about the point xi = a, i = 1, . . . , n.
2
xk+n ≡ xk ,
k ∈ Z,