mpc2w2.tex
Week 2. 19.10.2011
GREEN FUNCTIONS
These go back to George GREEN (1793-1841) in 1828, in his Essay:
G. GREEN, Essay on the application of mathematical analysis to the theories of electricity and magnetism.
Lu := p0 u + p1 u + p2 u = 0,
0 x .
(H
mpc2w4.tex
Week 4. 2.11.2011
2. THE HEAT EQUATION (Joseph FOURIER (1768-1830) in 1807;
Thorie analytique de la chaleur, 1822).
e
One dimension.
Consider a uniform bar (of some material, say metal, that conducts heat),
of cross-sectional area S , with side
mpc2w5.tex
Week 5. 9.11.2011
Schrdinger Equation (Erwin SCHRODINGER (1887-1961) in 1926).
o
This postulates that a particles motion is described by a wave function
= (x, t) (x = (x1 , x2 , x3 ), or (x, y, z ) satisfying a PDE
h2
ih/t =
+ V
2m
(Schrding
mpc2w6.tex
Week 6. 16.11.2011
Identity Matrix.
Recall the Kronecker delta ij :
ij := 1 if i = j,
0 otherwise
(Leopold KRONECKER (1823-1891); posth. book 1903).
The n n matrix I , or In , with (i, j ) element ij ,
I=
1 0 .
01
.
.
.
.
0
0
0
.
.
.
1
is calle
mpc2w7.tex
Week 7. 23.11.2011
Eigenvalues and Eigenvectors.
For a square matrix A, if
ax = x
for a non-zero vector x and scalar , x is called an eigenvector, with eigenvalue
(eigen = proper, German; other terms are proper value/vector, characteristic val
mpc2w8.tex
Week 9. 7.12.2011
PDEs via the Fourier Transform.
Take e.g. the heat equation
ux x(x, t) = ut (x, t)/k.
If we take Fourier transforms w.r.t. x, u(x, t) u(, t): as by above f (x)
corresponds to 2 f ( ), here 2 u(x, t)/x2 corresponds to 2 u(, t).
mpc2w8.tex
Week 10. 14.12.2011
Line, Surface and Volume Integrals
For a vector eld a and a curve L joining points
0 and
for a small displacement along the curve L, of length ds. Then
a = ax i + ay j + az k,
The line integral
a1 ), written
L
d = dxi + dy j
mpc2w8.tex
Week 8. 30.11.2011
A system of functions n on [0, 2 ], or more generally on an interval [a, b],
is called:
b
orthogonal if (m , n ) := a m n = 0 for m = n;
orthonormal if (m , n) = mn (orthogonal as above, plus the normalization
condition divid
mpc2prob1.tex
SOLUTIONS 1. 17.10.2011
Q1. (i) If y = x , y = x1 , y = ( 1)x2 ,
x2 y 2y = x [( 1) 2],
which is 0 i
2 2 = 0,
( 2)( + 1) = 0,
= 1 or 2.
So independent solutions to (H ) are y1 = x1 , y2 = x2 , with y1 = x2 ,
y2 = 2x.
(ii) Take y = y1 u1 + y2
mpc2soln1.tex
SOLUTIONS 2. 24.10.2011
Q1. (i) G is continuous;
(ii) Using prime for /x (dierentiation w.r.t. the rst argument),
k sin ky cos k ( x)
k sin k
k cos kx sin k ( y )
=
k sin k
G (x, y ) =
Let x y :
(0 y < x),
(x < y ).
G (y +, y ) =
sin ky cos
mpc2soln3.tex
SOLUTIONS 3. 31.10.2011
Q1. Draw an Argand diagram in each case.
(i) i = ei/2 .
(ii) 1 i = 2ei/4 = 2e7i/4 .
(iii) 3 i = 2ei/6 .
(Recall see e.g Problems 2 that cos(/3) = sin(/6) = 1/2, so sin(/3) =
cos(/6) = 3/2, so tan(/6) = 1/ 3.)
(iv) By