Review of vector and tensor mathematics

CHEE 3363 Spring 2013 Handout 02

1

CHEE 3363 Handout 02

Learning objectives for lecture

1. Draw the unit vectors in each of the three coordinate systems.�

2. Explain the geometric meaning of the dot and cross products.�

3. Calculate the components of a vector in each coordinate system.

2

CHEE 3363 Handout 02

Fields of scalars, vectors, tensorsWe define continuum quantities as continuous fields over variables:

- Scalars:

Properties (number of components):

Examples:

- Vectors:

Properties (number of components):

Examples:

- Tensors:

Properties (number of components):

Examples:

3

CHEE 3363 Handout 02CHEE 3363 Handout 02

General representation of a vector

i1i2

i3

x2

x3

x1a

Given: a basis set of vectors (x1, x2, x3) Construct: normalized basis vectors (i1, i2, i3) ���Write: components vector a along (i1, i2, i3):

Write: vector a as a sum over the basis vectors:

4

CHEE 3363 Handout 02

Rectangular coordinate system

i j

k

Normalized basis vectors:

Time-varying velocity field:

5

Coordinates of generalized vector:

CHEE 3363 Handout 02

Cylindrical coordinate system I

er

eθk

6

Basis vectors in terms of rectangular basis:

CHEE 3363 Handout 02

Normalized basis vectors:

Cylindrical coordinate system II

er

eθk

7

Coordinates of generalized vector:

CHEE 3363 Handout 02

Spherical coordinate system I

er

eθ

eφ

8

CHEE 3363 Handout 02

Basis vectors in terms of rectangular basis:

Normalized basis vectors:

Spherical coordinate system II

er

eθ

eφ

9

CHEE 3363 Handout 02

Coordinates of generalized vector:

Important vector operations: dot productDefinition of the dot product:

a

b

θ

10

Geometric meaning of the dot product:

Components of a ⋅ b:

CHEE 3363 Handout 02

Kronecker deltaKronecker delta:

Use of the Kronecker delta to express orthonormality of basis vectors:

CHEE 3363 Handout 02

Using the dot product I(a) Determining relationships between vector components in different coordinate systems:

12

CHEE 3363 Handout 02

Using the dot product II(b) Determining unit normals to surfaces:

x

y

13

CHEE 3363 Handout 02

Using the dot product III(c) Calculating flux through surface of area A:

14

CHEE 3363 Handout 02

Important vector operations: cross productDefinition of the cross product:

a

bθ

e

List two important quantities in fluid mechanics that are defined in terms of the cross product:

15

Geometric meaning of the cross product:

CHEE 3363 Handout 02

Vector calculus operations I(a) The gradient operator

(rectangular)

(cylindrical)

(spherical)

(b) The gradient of a scalar function is:

16

CHEE 3363 Handout 02

Vector calculus operations II

divergence of a vector is:

17

CHEE 3363 Handout 02

(c) The gradient of a vector

Vector calculus operations III(e) The definition of the divergence of a tensor is:

(f) All of these expressions will be used later in the course in the Euler equation of motion:

18

CHEE 3363 Handout 02

v ·∇v =?

Exercise in vector calculusWrite out the one term in the Euler equation not explicitly written out yet:

Answer:

19

CHEE 3363 Handout 02

a = 3i + 7j b = 3j − 6k

Example problem: finding unit normals I(a) Find: the unit normal e to the plane defined by the vectors

and

20

CHEE 3363 Handout 02

a = 3i + 7j b = 3j − 6k

Example problem: finding unit normals II(b) Find: the angle between a and b

and

21

CHEE 3363 Handout 02

Example problem: projection vector(c) Find the projection vector P of the projection of c = i + j + k on the

plane formed by a and b

e

a

b

cc ⋅ e

P

Exercise: crank out the numbers

For general a and b:

22

CHEE 3363 Handout 02

vr = −V∞

(

1 −a2

r2

)

cos θ

vθ = −V∞

(

1 +a2

r2

)

sin θ

Example: changing coordinate systems I(d) The velocity profile for the flow of a low viscosity fluid around a cylinder away from the cylinder’s surface can be approximated by

Find vx = vx(x, y) and vy = vy(x, y) for this flow.

23

CHEE 3363 Handout 02

vr = −V∞

(

1 −a2

r2

)

cos θ

vθ = −V∞

(

1 +a2

r2

)

sin θ

Example: changing coordinate systems II(d) The velocity profile for the flow of a low viscosity fluid around a cylinder away from the cylinder’s surface can be approximated by

Find vx = vx(x, y) and vy = vy(x, y) for this flow.

Exercise: repeat procedure to find vy = vy(x, y).24

CHEE 3363 Handout 02