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 sc
ENGI 4430 Advanced Calculus for Engineering
Mid Term Test - Solutions
2017 06 14
1.
The graph of a curve is defined in plane polar coordinates by
r tan
(a) Show that 1 x 1 everywhere on this curve.
(
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 solu
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, with
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 usi
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 comp
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
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 x
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) E
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 1
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
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 )
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
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/~ggeorg
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 p
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
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
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 mul
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 repre
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 or
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 d
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 i
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