M2AA2 - Multivariable Calculus. Problem Sheet 1 Solutions
Professor D.T. Papageorgiou, January 2010.
1.
(i) A r = A1 x1 + A2 x2 + A3 x3 , hence
(A r ) = (A1 , A2 , A3 ) = A.
(ii) rn = |r |n = (x2 + y 2 + z 2 )n/2 . First component of
(rn ) is
rn
r
x
= nrn
M2AA2 - Multivariable Calculus. Problem Sheet 1
January 17, 2010. Prof. D.T. Papageorgiou
1. If A is a constant vector eld, calculate the gradients of the following
scalar elds:
(ii) rn
(i) A r
(iii) r
(x + y + z )
[Here r = |r | where r R3 .]
2. (a) Pro
M2AA2 - Multivariable Calculus. Problem Sheet 2 January 28, 2010. Prof. D.T. Papageorgiou 1. If v = (2xy + z 2 , 2yz + x2 , 2xz + y 2 ) show that v = , with = 0 at the origin. v = 0 and find the potential such that
2. If and are harmonic scalar fields (i.
M2AA2 - Multivariable Calculus. Problem Sheet 3
February 5, 2010. Prof. D.T. Papageorgiou
1. (a) Show that the centroid (x, y ) of a closed simply connected region S in R2 is given by
x=
1
2A
x2 dy,
y=
C
1
2A
y 2 dx,
C
where A is the area of S and C is it
M2AA2 - Multivariable Calculus. Problem Sheet 4 February 18, 2010. Prof. D.T. Papageorgiou 1. Verify the Stokes theorem for the vector field 1 1 F = (3x - y, - yz 2 , - y 2 z), 2 2 where S is the upper half surface of the sphere x2 + y 2 + z 2 = 1, so tha
M2AA2 - Multivariable Calculus. Problem Sheet 5 March, 2010. Prof. D.T. Papageorgiou 1. Consider the following curvilinear coordinate system defined in terms of the cartesian system: 1 x1 = (u2 - v 2 ), 2 with u 0. Find the scale factors h1 , h2 and h3 of
M2AA2 - Multivariable Calculus. Problem Sheet 6
March, 2010. Prof. D.T. Papageorgiou
1. Let
I=
S
x ndS
.
|x|3
Show that I = 4 if S is the sphere |x| = R and that I = 0 if S bounds a volume that does
not contain the origin (x = 0).
Show that the electric e
M2AA2 - Multivariable Calculus. Problem Sheet 7
March, 2010. Prof. D.T. Papageorgiou
1. (a) Find the Greens function
2
G = (x x0 ),
x > 0, y > 0
G
(x, 0) = 0.
y
G(0, y ) = 0,
(b) Use part (a) to solve the problem
2
= f (x, y ),
x > 0, y > 0
(x, 0) = p(x)
M2AA2 - Multivariable Calculus. Problem Sheet 8
March, 2010. Prof. D.T. Papageorgiou
1. In each of the problems that follow, nd a stationary function for the integral satisfying the
given conditions at the end points of the interval:
(a)
(b)
21
2
1 x3 (y
M2AA2 - Multivariable Calculus. Assessed Coursework II
March 15, 2010. Prof. D.T. Papageorgiou
DUE March 22, 2010, BEFORE 2PM
1. (a) Use the method of images to nd the two-dimensional Dirichlet Greens function for the
upper half plane, i.e. solve
2
G = (x
M2AA2 - Multivariable Calculus. Assessed Coursework I February 18, 2010. Prof. D.T. Papageorgiou DUE February 26, 2009, BEFORE 2PM 1. Newton's law of gravitation states that the force between that a fixed particle Q with coordinates (, , ) and mass m exer
M2AA2 - Multivariable Calculus. Problem Sheet 2. Solutions. February 2010. Prof. D.T. Papageorgiou
1. Consider the first component y (2xz + y 2 ) - z (2yz + x2 ) = 2y - 2y = 0; similar calculations give 0 for the other two components.
If v = then = 2xy +
M2AA2 - Multivariable Calculus. Problem Sheet 3. Solutions.
Professor D.T. Papageorgiou
1
1
1. (a) The centroid of a region S is given by x = A S xdxdy , y = A S ydxdy , where A is
the area of the region. Now use Greens theorem in the plane,
S (F2x F1y )d
M2AA2 - Multivariable Calculus. Problem Sheet 4. Solutions.
Professor D.T. Papageorgiou
F = (0, 0, 1) and on
1. Need to show that
S ( F ) ndS = C F ds. First, calculate
S the normal is n = (sin cos , sin sin , cos ) (this is the outward pointing normal t
M2AA2 - Multivariable Calculus. Problem Sheet 8. Solutions. Professor D.T. Papageorgiou Recall that the Euler equation that extremises I[y] = f d f = 0. - y dx y f (y, y ) = const. y
x1 x0
f (x, y, y )dx is (1)
If the function f is independent of x, then
M2AA2 - Multivariable Calculus. Problem Sheet 9. Solutions.
Professor D.T. Papageorgiou
1. (a)
ij
xi
xi
=
=3
xj
xi
(b)
ij ik xj xk = jk xj xk = xj xj = x2 + x2 + x2 = r2
1
2
3
(c)
ij
2
2
=
=
xi xj
xi xi
2
(d)
ij jk ki = ik ki = ii = 3
2. We have from the
M2AA2 - Multivariable Calculus. Problem Sheet 9
March, 2010. Prof. D.T. Papageorgiou
1. If ij is the Kronecker delta and xi is a vector, evaluate
xi
(a) ij xj .
(b) ij ik xj xk .
2
(c) ij xi xj .
(d) ij jk ki .
2. A particle of mass m and position xi is i