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1. 1
CEE 451G ENVIRONMENTAL FLUID MECHANICS
LECTURE 1: SCALARS, VECTORS AND TENSORS
A scalarhas magnitude but no direction.
An example is pressure p.
The coordinates x, y and z of Cartesian space are scalars.
A vectorhas both magnitude and direction
Let denote unitvectors in the x, y and z direction. The hat
denotes a magnitude of unity
The position vector(the arrow denotes a vector that is not a unit
vector) is given ask
ˆ
,j
ˆ
,i
ˆ x
 k
ˆ
zj
ˆ
yi
ˆ
xx 
 x
y
zi
ˆ j
ˆ k
ˆ x


2. 2
LECTURE 1: SCALARS, VECTORS AND TENSORS
The velocity vectoris given as
The accelerationvector is given as
The unitsthat we will use in class are length L, time T, mass M and
temperature °. The units of a parameter are denoted in brackets. Thusu
 k
ˆ
dt
dz
j
ˆ
dt
dy
i
ˆ
dt
dx
dt
xd
u 

 a
 k
ˆ
dt
zd
j
ˆ
dt
yd
i
ˆ
dt
xd
dt
xd
k
ˆ
dt
dw
j
ˆ
dt
dv
i
ˆ
dt
du
dt
ud
a
2
2
2
2
2
2
2
2


 ?]a[
LT]u[
L]x[
1







Newton’s second lawis a vectorial statement: where denotes the
force vector and m denotes the mass (which is a scalar) amF

 F
 2
LT


3. 3
LECTURE 1: SCALARS, VECTORS AND TENSORS
The components of the force vector can be written as follows:
The dimensionsof the force vector are the dimension of mass times
the dimension acceleration
Pressure p, which is a scalar, has dimensions of force per unit area.
The dimensions of pressure are thusk
ˆ
Fj
ˆ
Fi
ˆ
FF
zyx

 2122
TML)L/(MLT]p[


The acceleration of gravity g is a scalar with the dimensions of (of
course) acceleration: 2
xMLT]F[]F[


 2
LT]g[



4. 4
LECTURE 1: SCALARS, VECTORS AND TENSORS
A scalar can be a function of a vector, a vector of a scalar, etc. For
example, in fluid flows pressure and velocity are both functions of
position and time: )t,x(uu,)t,x(pp


A scalar is a zero-order tensor. A vector is a first-order tensor. A
matrix is a second order tensor. For example, consider the stress
tensor .













zzzyzx
yzyyyx
xzxyxx

The stress tensor has 9 components. What do they mean? Use the
following mnemonic device: first face, second stress

5. 5
LECTURE 1: SCALARS, VECTORS AND TENSORS
Consider the volume element below. x
y
z
Each of the six faces has a direction.
For example, this face
and this face
are normal to the y direction
A force acting on any face can act in the x, y and z directions.

6. 6x
y
z

yy
yz
LECTURE 1: SCALARS, VECTORS AND TENSORS
Consider the face below.
The face is in the direction y.

yx
The force per unit face area acting in the x direction on that face is the
stress 
yx(first face, second stress).
The forces per unit face area acting in the y and z directions on that
face are the stresses 
yyand 
yz.
Here 
yyis a normal stress(acts normal, or perpendicular to the face)
and 
yxand 
yzare shear stresses(act parallel to the face)

7. 7x
y
z

yy
yz
LECTURE 1: SCALARS, VECTORS AND TENSORS
Some conventions are in order
Normal stresses are defined to be positive outward, so the orientation
is reversed on the face located y from the origin

yx
Shear stresses similarly reverse sign on the opposite face face are the
stresses 
yyand 
yz.

yy

yz

yx
Thus a positive normal stress puts a body in tension, and a negative
normal stress puts the body in compression. Shear stresses always put
the body in shear.`

8. 8
Another way to write a vector is in Cartesianform:)z,y,x(k
ˆ
zj
ˆ
yi
ˆ
xx 

The coordinates x, y and z can also be written as x
1, x
2, x
3. Thus the
vector can be written as)x,x,x(x
321

or as3..1i,)x(x
i

or in index notation, simply asixx

where i is understood to be a dummy variable running from 1 to 3.
Thus x
i, x
jand x
pall refer to the same vector (x
1, x
2and x
3) , as the
index (subscript) always runs from 1 to 3.
LECTURE 1: SCALARS, VECTORS AND TENSORS

9. 9
LECTURE 1: SCALARS, VECTORS AND TENSORS
Scalar multiplication: let be a scalar and = A
ibe a vector.
ThenA
 )A,A,A(AA
32ii


is a vector.
Dot or scalar productof two vectors results in a scalar:scalarBABABABA
332211


In index notation, the dot product takes the form


3
1r
rr
3
1k
kk
3
1i
ii
BABABABA

Einstein summation convention: if the same index occurs twice, always
sum over that index. So we abbreviate torrkkii BABABABA 

There is no free index in the above expressions. Instead the indices are
paired (e.g. two i’s), implying summation. The result of the dot product
is thus a scalar.

10. 10
LECTURE 1: SCALARS, VECTORS AND TENSORS
Magnitude of a vector:ii
2
AAAAA 

A tensorcan be constructed by multiplying two vectors (not scalar
product):










333231
332221
131211
jiji
BABABA
BABABA
BABABA
3..1j,3..1i,)BA(BA
Two free indices (i, j) means the result is a second-ordertensor
Now consider the expressionjji
BAA
This is a first-order tensor, or vectorbecause there is only one free
index, i (the j’s are paired, implying summation).)A,A,A)(BABABA(BAA
321232211jji

That is, scalar times vector = vector.

11. 11
LECTURE 1: SCALARS, VECTORS AND TENSORS
Kronecker delta
ij
















100
010
001
jiif0
jiif1
ij
Since there are two free indices, the result is a second-order tensor, or
matrix. The Kronecker delta corresponds to the identity matrix.
Third-order Levi-Civitatensor.





0
k,j,iif1
k,j,iif1
ijl
cycle clockwise: 1,2,3, 2,3,1 or 3,1,2
cycle counterclockwise: 1,3,2, 3,2,2 or 2,1,3
otherwise
Vectorial cross product:kjijk
BABxA 

One free index, so the result must be a vector.

12. 12
LECTURE 1: SCALARS, VECTORS AND TENSORS
Vectorial cross product: Let be given as BxAC


ThenC
 































































2
2
1
1
321
321
2
2
1
1
321
321
321
321
321
321
B
A
j
ˆ
B
A
i
ˆ
BBB
AAA
k
ˆ
j
ˆ
i
ˆ
B
A
j
ˆ
B
A
i
ˆ
BBB
AAA
k
ˆ
j
ˆ
i
ˆ
BBB
AAA
k
ˆ
j
ˆ
i
ˆ
BBB
AAA
k
ˆ
j
ˆ
i
ˆ
detC
    k
ˆ
BABAj
ˆ
BABAi
ˆ
BABA
122131132332


13. 13
LECTURE 1: SCALARS, VECTORS AND TENSORS
Vectorial cross product in tensor notation: kjijki
BAC
Thus for example 
111112313232123kjjk11
BABABABAC
= 1 = -1 = 0
a lot of other terms that
all = 02332 BABA
i.e. the same result as the other slide. The same results are also
obtained for C
2and C
3.
The nabla vector operator: 321
x
k
ˆ
x
j
ˆ
x
i
ˆ










or in index notation 
 ix


14. 14
LECTURE 1: SCALARS, VECTORS AND TENSORS
The gradientconverts a scalar to a vector. For example, where p is
pressure, k
ˆ
x
p
j
ˆ
x
p
i
ˆ
x
p
p)p(grad
321










or in index notation ix
p
)p(grad



The single free index i (free in that it is not paired with another i) in the
above expression means that grad(p) is a vector.
The divergenceconverts a vector into a scalar. For example, where
is the velocity vector, k
k
i
i
3
3
2
2
1
1
x
u
x
u
x
u
x
u
x
u
)u(div















 u

Note that there is no free index (two i’s or two k’s), so the result is a
scalar.

15. 15
LECTURE 1: SCALARS, VECTORS AND TENSORS
The curlconverts a vector to a vector. For example, where is the
velocity vector,u
 k
ˆ
x
u
x
u
j
ˆ
x
u
x
u
i
ˆ
x
u
x
u
uuu
xxx
k
ˆ
j
ˆ
i
ˆ
ux)u(curl
2
1
1
2
1
3
3
1
3
2
2
3
321
321


















































or in index notation,j
k
ijk
x
u
)u(curl




One free index i (the j’s and the k’s are paired) means that the result is a
vector

16. 16
A useful manipulation in tensor notation can be used to change an index
in an expression:ijij
uu
This manipulation works because the Kronecker delta 
ij= 0 except when
i = j, in which case it equals 1.
LECTURE 1: SCALARS, VECTORS AND TENSORS