Student Handout 02 2014

Review of vector and tensor mathematics

CHEE 3363 Spring 2013 Handout 02

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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.

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

- Scalars:

Properties (number of components):

Examples:

- Vectors:


Examples:

- Tensors:


Examples:

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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:

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Rectangular coordinate system

i j

k

Normalized basis vectors:

Time-varying velocity field:

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Coordinates of generalized vector:


Cylindrical coordinate system I

er

eθk

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Basis vectors in terms of rectangular basis:



Cylindrical coordinate system II

er

eθk

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Spherical coordinate system I

er

eθ

eφ

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Basis vectors in terms of rectangular basis:


Spherical coordinate system II

er

eθ

eφ

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Important vector operations: dot productDefinition of the dot product:

a

b

θ

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Geometric meaning of the dot product:

Components of a ⋅ b:


Kronecker deltaKronecker delta:

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


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

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

x

y

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

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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:

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Geometric meaning of the cross product:


Vector calculus operations I(a) The gradient operator

(rectangular)

(cylindrical)

(spherical)

(b) The gradient of a scalar function is:

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Vector calculus operations II

divergence of a vector is:

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(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:

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v ·∇v =?

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

Answer:

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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

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a = 3i + 7j b = 3j − 6k

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

and

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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:

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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.

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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


Student Handout 02 2014

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