Poutrelles universelles
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British
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conformes BS 4 part 1 - 1993
in accordance with BS 4 part 1 - 1993
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Commande minimale et
Minimum order and other
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Civil Engineering Mathematics I (KAEA3201)
Assignment 2
1. A stretched string of length 20 cm is set oscillating by displacing its mid-point a
distance 1 cm from its rest position and releasing it with zero initial velocity. Solve
the wave equation
2u 1
53 sections 202/204 Quiz 8 Solutions
Problem 1 (10 pts). Find the work done by the force field F(x, y, z) = (x y 2 , y z 2 , z x2 ) on a
particle that moves along the line segment from (0, 0, 1) to (2, 1, 0).
Solution: Call the segment C. Using the parame
Laplace equation
2u 2u
0
x 2 y 2
and the boundary conditions
(a)
(b)
(c)
(d)
u = 0 on y = 0
u = -xe-x/2 on y = 1
u = y cos (y/2) on x = 0
u 0 as x
(x 0)
(x 0)
(0 y 1)
Assume that the solution is given by u(x, y) = F(x)G(y)
F " G FG 0
F"
G
p2
F
G
F " p
Example 1 : Wave equation
Solve
2u 2u
2
2
t
x
restricted to boundary conditions
u( 0, t) = 0
u(, t) = 0
u(x, 0) = f(x) = K(x x2)
and
u
0
t t 0
Assume that the solution is
Then
or
u(x, t) = F(x)G(t)
u
F ' G,
x
2u
F "G
2
x
u
FG
t
2u
FG
2
t
(1)
F"G
LINE INTEGRALS
In this section we shall discuss line integrals, which are integrals
over curves in two or three-dimensional space.
1
2
3
If a curve C is traced in a certain direction, then the same curve
traced in the opposite direction is denoted by the
Q1
A stretched string of length 20 cm is set oscillating by displacing its mid-point a
distance 1 cm from its rest position and releasing it with zero initial velocity. Solve the
wave equation
Where
to determine the resulting motion,
Solution:
1
2
3
4
Q2
DOUBLE INTEGRALS
The notion of a definite integral can be extended to function of two or more
variables. In this section we shall discuss the double integral, which is the
extension to functions of two variables.
DEFINITION OF A DOUBLE INTEGRAL
b
Firstly