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ENGINEERING MECHANICS
ENG 2P04
Dieter Stolle, Professor
Department of Civil Engineering
McMaster University
Mathematical Preliminaries
–
Definitions  1
Ordered set (coordinate) representation where the
unit vector directions
implied
Scalar:
Temperature, pressure, density
– frame invariant
Vector:
First order tensor
–
characterized by direction and
magnitude –
frame invariant
Position Vector:
r
xi
yj
zk
To solve problems, we must attach vector to a
frame of reference, usually orthogonal
,,
x y z
r
i j k
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Index Notation:
Given that we have a vector
we may write
Mathematical Preliminaries
–
Definitions  2
r
1 1
2 2
3 3
ii
r
x e
x e
x e
r
x e
where summation is implied over repeated indices
Dot Product:
Given Cartesian vectors written as
the dot product is defined as
1
2
3
1
2
3
,
,
and
,
,
F
F F F
u
u u u
W =
W =
cos
F
u
F u
Mathematical Preliminaries
–
Definitions  3
Dot Product:
Using index notation, we may write
i i
j
j
i
j i
j
i
j
ij
i
i
W
Fe
u e
W
Fu e e
W
Fu
W
Kronecker delta
Cross Product:
Right Hand Rule
sin
M
r
F
r
F
n
3
Mathematical Preliminaries
–
Definitions  4
Cross Product:
sin
M
r
F
r
F
n
1
2
3
1
2
3
1
2
3
= det
e
e
e
M
r
F
r
r
r
F
F
F
x
y
F
r
F
y
F
x
For two dimensional:
yx
M
F x
F y k
Often the direction of rotation is known, a priori
and vectors are implied, but not shown;
i.e.
,
M
F x
F y
Newton’
Laws (1)
1.
A particle originally at rest, or moving in a straight line
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This note was uploaded on 09/21/2011 for the course ENGINEERIN 2P04 taught by Professor Ss during the Spring '11 term at McMaster University.
 Spring '11
 ss
 Civil Engineering

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