MA2081 – Vector Calculus and Fluid Dynamics

Manolis Georgoulis

January 2007

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Course Information

Module Webpage

http://www.math.le.ac.uk/PEOPLE/eg64/TEACHING/MA2081/MA2081.html

Contact Details

Dr. Manolis GeorgoulisMichael Atiyah BuildingOffice 123email: eg64@mcs.le.ac.ukwww: http://www.math.le.ac.uk/PEOPLE/eg64/

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Module Information

Prerequisites

� MA1001 Multivariate Calculus

� MA1002 Vectors and ODEs

� MA1051 Newtonian Dynamics

Assessment

� 2 hour examination: 60%

� 10 weekly problem sheets 20%

� 10 computer practicals: 20%

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Problem classes / Computer Practicals

Weekly problem classes

Manolis GeorgoulisVenue: BEN LT5Time: Mondays 1:30-2:30pm (starting in 2 weeks)

Weekly computer practicals

Manolis GeorgoulisVenue: CW305Time: Tuesdays 2:30pm (starting TOMORROW)

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Module Description

� Vector differential and integral Calculus� computing arc lengths, surface areas and volumes� computing tangents and integrals over useful domains� the implicit function theorem

� Applications to continuum mechanics and fluid dynamics� calculate physical quantities, such as velocity, acceleration, energy,

work done by a force, etc.� modelling of physical phenomena, such as fluid flow, waves, heat

transfer, etc.

� Computer simulations� simple numerical computations using (mainly) MATLAB� nice pictures!

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Reading List

Vector Calculus:

� Calculus, Edwards and Penney, 6th edition

Accessible and colourful.

� Advanced Engineering Mathematics, Kreyszig, 8th edition

Less worked examples but covers the material thoroughly; applicationsoriented. (It includes more topics than Calculus.)

� Vector Calculus, Marsden and Tromba

Rigorous and enjoyable but slightly demanding.

Fluid Dynamics:

� Elementary Fluid Dynamics, Acheson

We shall only cover Chapters 1 and 3.

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Let’s get started then!

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... with a bit of Geometry.

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Vectors in Space

Figure: Components of a vector in space

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Vector Algebra

Let a = (a1, a2, a3), b = (b1, b2, b3), c = (c1, c2, c3) vectors, and λ ascalar:

a + b = (a1 + b1, a2 + b2, a3 + b3) = b + a

(a + b) + c = a + (b + c)

λa = (λa1, λa2, λa3)

a + 0 = a, for 0 = (0, 0, 0)

−a = (−1)a

Length of a vector:

|a| =√

a21 + a2

2 + a23

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Unit Vector Notation

Figure: The representation a = a1i + a2j + a3k

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Scalar Product (a.k.a. dot product, inner product ...)

Let a = (a1, a2, a3), b = (b1, b2, b3) vectors, and γ the angle betweenthem

Scalar Product:

a · b = |a||b| cos γ

= a1b1 + a2b2 + a3b3

Properties:a · b = b · a

a · a = a21 + a2

2 + a23 = |a|2

(a + b) · c = a · c + b · cIf a perpendicular to b then a · b = 0

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Scalar Product – Application

Work done by a force:

Consider a constant force F acting on a body that results to adisplacement d of the body. What is the work W done by F?

W = F · d

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Vector Product (a.k.a. cross product, wedge product ...)

Let a = (a1, a2, a3), b = (b1, b2, b3) vectors, and γ the angle betweenthem

Vector Product:

a × b = v

with

v = (a2b3−a3b2, a3b1−b3a1, a1b2−b1a2)

and length

|v | = |a||b|| sin γ|

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Vector Product – Applications

Area of parallelogram:

Consider a = (a1, a2, a3), b =(b1, b2, b3) vectors. Then the area ofthe parallelogram having a and b asedges is given by

Area = |a × b|

BIG IDEA: Area can be described using vector product!

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Vector Product – Applications

Moment of a force:

Let F a force acting on a body. Body at distancer from a point O. The moment vector m of theforce F about the point O is

m = r × F

Moment describes the rotation about the pointO caused by F . F

r

Obody

m

BIG IDEA: Rotation can be quantified using vector product!

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Vector Product – Properties

The formula

a × b = (a2b3 − a3b2, a3b1 − b3a1, a1b2 − b1a2)

= (a2b3 − a3b2)i + (a3b1 − b3a1)j + (a1b2 − b1a2)k

seems hard to remember...

Let us see if we can relate it to something we know already!

Keyword: Determinants!

∣∣∣∣a b

c d

∣∣∣∣ = ad − bc

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Vector Product – Properties

Hence

a × b =

∣∣∣∣∣∣i j k

a1 a2 a3

b1 b2 b3

∣∣∣∣∣∣Other properties:

a × a = 0

a × b = −b × a antisymmetry

(a + b) × c = a × c + b × c

...it is not the same as

c × (a + b) = c × a + c × b

because of antisymmetry!

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Vector Product – Example

Find the moment m of the force F = (3, 2, 4) about the pointO = (0, 0, 0) of a body located at r = (1, 2, 4) from O.

Answer:(3, 2, 4) × (1, 2, 4) = ... = (0, 8,−4)

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Plane perpendicular to a vector

Problem: How to describe a plane that goes through a point (x0, y0, z0) inspace and it perpendicular to a vector a = (a1, a2, a3)?

Solution:a1(x − x0) + a2(y − y0) + a3(z − z0) = 0

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Polar CoordinatesSometimes it is hard to describe curves or surfaces in Cartesiancoordinates. For example, a circle with centre (0, 0) and radius 1 cannotbe represented as a function y = f (x)...

Let us think of an alternative way, that might be useful in this case...

o(x,y)

θ

r

Figure: Polar coordinates

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Polar Coordinates

Hencex = r cos θ, y = r sin θ

So we have the change of variables

(x , y) → (r , θ) with 0 ≤ θ < 2π

Example: Find (x , y) = (1, 1) in polar coordinates.

Solution: We have (r , θ) = (√

2, π

4 ).

In general:

r =√

x2 + y2, θ = arctan(y

x

)

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