ENGI 4430
4.
Lines of Force
Page 4-01
Lines of Force
A vector function of n variables in n is a vector field.
f1 ( x, y, z )
F ( x, y , z ) = f 2 ( x, y , z )
f 3 ( x, y , z )
(and the fi form scalar fields).
The domain must be defined. If not explic
ENGI 4430
Parametric Vector Functions
Page 2-01
Parametric Vector Functions (continued)
2.
Any non-zero vector r can be decomposed into its magnitude r and its direction:
r r r , where r r > 0
Tangent Vector:
dx dy dz
T =
dt dt dt
T
=
dr
dt
If the pa
ENGI 3424 Engineering Mathematics
Problem Set 2 Questions
(Chapter 1 all sections)
1.
Find the complete solution of the initial value problem
dy
4x
y ( 0) 1
2 x e=
,
=
+ 4y
dx
2.
Find the general solution of the ordinary differential equation
3
dy
4 x 3e
ENGI 3424 Engineering Mathematics
Problem Set 3 Questions
(Section 2.1 Second order homogeneous linear ordinary differential equations)
Determine the general or complete solution, as appropriate, without using Laplace transforms.
Note: If a complete solut
ENGI 3424 Engineering Mathematics
Problem Set 4 Questions
(Sections 2.2-2.4 Second order linear ordinary differential equations)
Determine the general or complete solution, as appropriate, without using Laplace transforms.
Note: If a complete solution is
ENGI 3424 Engineering Mathematics
Problem Set 5 Questions
(Chapter 3 Laplace transforms applied to linear ODEs; complex numbers)
Where possible, use Laplace transforms to determine the general or complete solution, as
appropriate. Questions 1-13 are ident
ENGI 3424 Engineering Mathematics
Problem Set 6 Questions
(Chapter 3 Laplace transforms)
1.
Find the Laplace transform F (s) for the function
f ( t ) = t 4e 2t
2.
Find the Laplace transform F (s) for the function
t
( 0 t < 1)
f ( t ) = 1t
( t 1)
e
3.
Fi
ENGI 3424 Engineering Mathematics
Problem Set 7 Questions
(Sections 4.1 - 4.4 Partial Derivatives, Differentials, Jacobians)
1.
2u
2u
Find
and
for the function u x, y, z, t defined by
x2
z t
u 2 x2 y 2 z 2 t 2
2.
3.
2 z
2 z
and
.
x2
t2
Hence show th
ENGI 3424 Engineering Mathematics
Problem Set 8 Questions
(Sections 4.5 - 4.8 Gradient Vector, Maxima & Minima)
1.
A scalar field V ( x, y, z ) in 3 is defined by
x2 + y 2
V ( x, y , z ) =
1+ z2
(a) Evaluate the gradient vector V at the point P (3, 4, 0)
ENGI 3424 Engineering Mathematics
Problem Set 9 Questions
(Sections 5.01 5.06 Sequences, Series, Tests for Convergence)
1.
Write down the next two terms and the general term an of the sequence
1 3 7 15
cfw_ an = , , , ,
2 4 8 16
Determine whether or no
ENGI 4430
9.
Surface Integrals
Page 9.01
Surface Integrals - Projection Method
Surfaces in 3
In
3
a surface can be represented by a vector parametric equation
r = x ( u, v ) + y ( u, v ) + z ( u, v ) k
i
j
where u, v are parameters.
Example 9.01
The unit
ENGI 4430
12.
PDEs - Fourier Solutions
Page 12.01
The Wave Equation Vibrating Finite String
The wave equation is
2u
= c 2 2u
2
t
If u(x, t) is the vertical displacement of a point at location x on a vibrating string at time t,
then the governing PDE is
Lecture Notes for
ENGI 4430
Advanced Calculus
for Engineering
by
Dr. G.H. George
Associate Professor,
Faculty of Engineering and Applied Science
fifth edition
2014 Spring
http:/www.engr.mun.ca/~ggeorge/4430/
i
ii
ENGI 4430
Advanced Calculus for Engineerin
ENGI 4430
8.
Line Integrals; Greens Theorem
Page 8.01
Line Integrals
Two applications of line integrals are treated here: the evaluation of work done on a
particle as it travels along a curve in the presence of a [vector field] force; and the
evaluation o
ENGI 4430
5.
Numerical Integration
Page 5-01
Numerical Integration
In some of our previous work, (most notably the evaluation of arc length), it has been
difficult or impossible to find the indefinite integral. Various symbolic algebra and
calculus softwa
ENGI 4430
1.
Parametric & Polar Curve Sketching
Page 1-01
Parametric Vector Functions
The general position vector r ( x, y, z ) has its tail at the origin (0, 0, 0) and its tip at any point
( x, y, z ) in
3
.
One constraint (that is, one equation) connect
ENGI 4430
3.
Multiple Integration Cartesian Double Integrals
Page 3-01
Multiple Integration
This chapter provides only a very brief introduction to the major topic of multiple
integration.
Uses of multiple integration include the evaluation of areas, volu
ENGI 4430
6.
Gradient, Divergence and Curl
The Gradient Vector - Review
If a curve in
2
is represented by y = f (x) , then
f ( x + x ) f ( x )
dy
y
= lim
= lim
Q P x
x0
dx
x
If a surface in
3
is represented by z = f (x, y) , then in a slice y = constant,
ENGI 4430
7.
Non-Cartesian Coordinates
Page 7-01
Conversions between Coordinate Systems
In general, the conversion of a vector F = Fx + Fy + Fz k from Cartesian coordinates
i
j
(x, y, z) to another orthonormal coordinate system (u, v, w) in 3 (where ortho
ENGI 4430
11.
PDEs - dAlembert Solutions
Page 11.01
Partial Differential Equations
Partial differential equations (PDEs) are equations involving functions of more than one
variable and their partial derivatives with respect to those variables.
Most (but n
ENGI 4430
10.
Gauss & Stokes Theorems; Potentials
Page 10.01
Gauss Divergence Theorem
Let S be a piecewise-smooth closed surface enclosing a volume V in
vector field. Then
the net flux of F out of V is =
FidS = F
N
S
3
and let F be a
dS ,
S
where FN is t
ENGI 3424 Engineering Mathematics
Problem Set 10 Questions
(Sections 5.07 5.11 Power Series, Fourier Series, Series Solutions of ODEs)
1.
Find the radius R and interval I of convergence for the power series
f ( x) =
( 2x 6)
n
n =1
2.
n
Find the radius R a