5. (Superposition principle) If u1 , u2 , . . . , uk are solutions of a homogeneous linear PDE, then
the linear combination
u = c 1 u1 + c 2 u2 + + c k uk ,
where c1 , c2 , . . . , ck are constants, is also a solution of this equation .
Note
In this chapt
(i)
(ii)
d[A(t)]
d[B (t)]
d[A(t) + B (t)]
=
+
dt
dt
dt
for all scalars and .
d[A(t)B (t)]
d[B (t)] d[A(t)]
= A(t)
+
B (t)
dt
dt
dt
(product rule ).
In particular,
(i) if (t) is a vector-valued function in Rn , then
y
d [ (t) + (t)]
x
y
d [ (t)]
x
d [ (t)]
MA3150 Chapter 3
1
The Laplace Transform
(Part 1)
Definition and the Inverse Laplace Transform
Let f ( t ) be a function defined for t 0 . We define the Laplace transform of f ( t ) , denoted by F ( s ) or
L [ f ] , to be the integral F ( s ) = L [ f ] =
Chapter 2 Vector Integral Calculus
1. Line Integrals
1.1 Line Integral of the Second Kind
Consider a vector field F = F1 ( x, y , z )i + F2 ( x, y , z ) j + F3 ( x, y , z )k and a smooth curve C with end-points
A and B.
Divide AB into n small segments. Le
Chapter 1 Vector Differential Calculus
1.
Differentiation of a Vector Function
If a ( t ) = a1 ( t ) i + a2 ( t ) j + a3 ( t ) k is a vector function of a scalar variable t, then the derivative of a ( t ) with
respect to t is:
a (t + t ) a (t )
d a (t )