Similar presentations:

# Scalars, vectors and tensors

## 1.

CEE 451G ENVIRONMENTAL FLUID MECHANICSLECTURE 1: SCALARS, VECTORS AND TENSORS

A scalar has magnitude but no direction.

An example is pressure p.

The coordinates x, y and z of Cartesian space are scalars.

A vector has both magnitude and direction

Let ˆi, ˆj, kˆ denote unit vectors in the x, y and z direction. The hat

denotes a magnitude of unity

The position vector x (the arrow denotes a vector that is not a unit

vector) is given as

x x ˆi yˆj zkˆ

z

x

kˆ

ˆi

x

ˆj

y

1

## 2.

LECTURE 1: SCALARS, VECTORS AND TENSORSThe velocity vector u is given as

dx dx ˆ dy ˆ dz ˆ

u

i

j

k

dt dt

dt

dt

The acceleration vector a is given as

du du ˆ dv ˆ dw ˆ d2x d2x ˆ d2y ˆ d2z ˆ

a

i

j

k 2 2 i 2 j 2 k

dt dt

dt

dt

dt

dt

dt

dt

The units that we will use in class are length L, time T, mass M and

temperature °. The units of a parameter are denoted in brackets. Thus

[x ] L

[u] LT 1

[a ] ?

LT 2

Newton’s second law is a vectorial statement: where F denotes the

force vector and m denotes the mass (which is a scalar)

F ma

2

## 3.

LECTURE 1: SCALARS, VECTORS AND TENSORSThe components of the force vector can be written as follows:

F F ˆi F ˆj F kˆ

x

y

z

The dimensions of the force vector are the dimension of mass times

the dimension acceleration

[F] [Fx ] MLT 2

Pressure p, which is a scalar, has dimensions of force per unit area.

The dimensions of pressure are thus

[p] MLT 2 /(L2 ) ML 1T 2

The acceleration of gravity g is a scalar with the dimensions of (of

course) acceleration:

[g] LT 2

3

## 4.

LECTURE 1: SCALARS, VECTORS AND TENSORSA 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:

p p(x, t) , u u(x, t)

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 .

xx xy xz

y x y y y z

zx

zy

zz

The stress tensor has 9 components. What do they mean? Use the

following mnemonic device: first face, second stress

4

## 5.

LECTURE 1: SCALARS, VECTORS AND TENSORSConsider the volume element below.

z

y

x

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.

5

## 6.

LECTURE 1: SCALARS, VECTORS AND TENSORSConsider the face below.

z

yy yz

yx

y

x

The face is in the direction y.

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 yy and yz.

Here yy is a normal stress (acts normal, or perpendicular to the face)

6

and yx and yz are shear stresses (act parallel to the face)

## 7.

LECTURE 1: SCALARS, VECTORS AND TENSORSSome conventions are in order

z

yy yz

yx

yx

yz

yy

y

x

Normal stresses are defined to be positive outward, so the orientation

is reversed on the face located y from the origin

Shear stresses similarly reverse sign on the opposite face face are the

stresses yy and yz.

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.`

7

## 8.

LECTURE 1: SCALARS, VECTORS AND TENSORSAnother way to write a vector is in Cartesian form:

x x ˆi yˆj zkˆ (x, y, z)

The coordinates x, y and z can also be written as x1, x2, x3. Thus the

vector can be written as

x (x1, x 2, x 3 )

or as

x (xi ) , i 1..3

or in index notation, simply as

x xi

where i is understood to be a dummy variable running from 1 to 3.

Thus xi, xj and xp all refer to the same vector (x1, x2 and x3) , as the

index (subscript) always runs from 1 to 3.

8

## 9.

LECTURE 1: SCALARS, VECTORS AND TENSORSScalar multiplication: let be a scalar and A = Ai

Then

A Ai ( Ai, A2, A3 )

be a vector.

is a vector.

Dot or scalar product of two vectors results in a scalar:

A B A1B1 A2B2 A3B3 scalar

In index notation, the dot product takes the form

3

3

3

A B AiBi AkBk ArBr

i 1

k 1

r 1

Einstein summation convention: if the same index occurs twice, always

sum over that index. So we abbreviate to

A B AiBi AkBk ArBr

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

9

is thus a scalar.

## 10.

LECTURE 1: SCALARS, VECTORS AND TENSORSMagnitude of a vector:

2

A A A Ai Ai

A tensor can be constructed by multiplying two vectors (not scalar

product):

A1B1

AiB j ( AiB j ) ,i 1..3, j 1..3 A1B2

A B

1 3

A 2B1

A 2B2

A 2B3

A 3B1

A 3B3

A 3B3

Two free indices (i, j) means the result is a second-order tensor

Now consider the expression

A i A jB j

This is a first-order tensor, or vector because there is only one free

index, i (the j’s are paired, implying summation).

Ai A jB j ( A1B1 A 2B2 A3B2 )( A1, A 2 , A3 )

That is, scalar times vector = vector.

10

## 11.

LECTURE 1: SCALARS, VECTORS AND TENSORSKronecker delta ij

1 0 0

1 if i j

ij

0 1 0

0 if i j

0

0

1

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-Civita tensor.

1 if i, j,k cycle clockwise: 1,2,3, 2,3,1 or 3,1,2

ijl 1 if i, j,k cycle counterclockwise: 1,3,2, 3,2,2 or 2,1,3

otherwise

0

Vectorial cross product:

AxB ijk A jBk

One free index, so the result must be a vector.

11

## 12.

LECTURE 1: SCALARS, VECTORS AND TENSORSVectorial cross product: Let C be given as

C AxB

Then

ˆj

ˆi

ˆj

ˆi

kˆ

kˆ

C det A1 A 2 A 3 A1 A 2 A 3

B

B

B

B1 B2 B3

2

3

1

ˆj

ˆj ˆi

ˆj

ˆi

kˆ ˆi

kˆ ˆi

A 1 A 2 A 3 A1 A 2 A1 A 2 A 3 A1

B

B

B

B

B

B

B

B

2

3 1

2

2

3 B1

1

1

ˆj

A2

B2

A2B3 A3B2 ˆi A3B1 A1B3 ˆj A1B2 A2B1 kˆ

12

## 13.

LECTURE 1: SCALARS, VECTORS AND TENSORSVectorial cross product in tensor notation:

Ci ijk A jBk

Thus for example

=1

= -1

=0

C1 1jk A jBk 123 A 2B3 132 A3B2 111A1B1

A2B3 A3B2

a lot of other terms that

all = 0

i.e. the same result as the other slide. The same results are also

obtained for C2 and C3.

The nabla vector operator :

ˆ

ˆ

ˆ

i

j

k

x1

x 2

x 3

or in index notation

x i

13

## 14.

LECTURE 1: SCALARS, VECTORS AND TENSORSThe gradient converts a scalar to a vector. For example, where p is

pressure,

p ˆ p ˆ p ˆ

grad(p) p

i

j

k

x1

x 2

x 3

or in index notation

p

grad(p)

x i

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 divergence converts a vector into a scalar. For example, where u

is the velocity vector,

u u

u

u u

div(u) 1 2 3 i k

x1 x 2 x 3 x i x k

Note that there is no free index (two i’s or two k’s), so the result is a

scalar.

14

## 15.

LECTURE 1: SCALARS, VECTORS AND TENSORSThe curl converts a vector to a vector. For example, where u is the

velocity vector,

ˆi

curl(u) xu

x1

u1

ˆj

x 2

u2

kˆ

x 3

u3

u3 u2 ˆ u1 u3 ˆ u2 u1 ˆ

i

j

k

x 2 x 3 x 3 x1 x1 x 2

or in index notation,

uk

curl(u) ijk

x j

One free index i (the j’s and the k’s are paired) means that the result is a

vector

15

## 16.

LECTURE 1: SCALARS, VECTORS AND TENSORSA useful manipulation in tensor notation can be used to change an index

in an expression:

iju j ui

This manipulation works because the Kronecker delta ij = 0 except when

i = j, in which case it equals 1.

16