Vector Algebra and Calculus- Class 1-

AOE 5104Advanced Aero- and Hydrodynamics

Dr. William Devenport andLeifur Thor Leifsson

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Outline

• Vector algebra and calculus divided into three classes

• Class 1– Vector basics and coordinate systems

• Class 2– Differentation in 3-D

• Class 3– 1st and 2nd order integral theorems

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Vector basicsVector: A, AMagnitude: |A|, AScalar: p, φ

Types– Polar vector

•

– Axial vector •

– Unit vector•

MAGDIR

P

Q

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Vector Algebra

• AdditionA + B =

• Dot, or scalar, product

A.B = ABcosθ• E.g. Work=F.s• Flow rate through dA=V.dA or V.ndA

• A.B=B.A A.A=A2 A.B=0 if perpendicular

A B

A

Bθ

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Vector Algebra

• Cross, or vector, productAxB=ABsinθe

• AxB=-BxA• AxA=0• AxB=0 if A and B parallel

A

Bθ

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Vector Algebra – Triple Products

1. (A.B)C = (B.A)C2. Mixed product A.BxC

• Volume of parallelepiped bordered by A, B, C• May be cyclically permutedA.BxC=C.AxB=B.CxA• Acyclic permutation changessign A.BxC=-B.AxC etc.

3. Vector triple product• Ax(BxC) = Vector in plane of B and C

= …

A

B

C

BxC

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PIV of Flow Downstream of a Circular Cylinder

Chiang Shih , Florida State University

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Cartesian Coordinates

r

ji

k

• Coordinates x, y , z

• Unit vectors i, j, k (in directions of increasing coordinates) are constant

• Position vectorr =

• Vector components F =

=

Components same regardless of location of vector

z

x

y

z

y x

F

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Cylindrical Coordinates

R

er

eθez

• Coordinates r, θ , z

• Unit vectors er, eθ, ez (in directions of increasing coordinates)

• Position vector R =

• Vector components F =

Components not constant, even if vector is constant

rθ

z

F

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Spherical Coordinates

r

ereθ

eφ

• Coordinates r, θ , φ

• Unit vectors er, eθ, eφ (in directions of increasing coordinates)

• Position vectorr =

• Vector components F =φ

θ

r

F

Note. Figure in presentation shows eθ and θ incorrectly. This figure is correct.

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Vector Algebra in Components

321

321

321

332211.

BBBAAA

BABABAeee

BA

BA

=×

++=

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CFD of Flow Around a FighterFinFlo Ltd

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Concept of Differential Change In a Vector. The Vector Field.

V

-2

-1

0

1

2y

/ L

-2

0

2-T / U L0

1

2z / L

V=V(r,t)

φ=φ(r,t)Scalar field

Vector field

Differential change in vector• Change in • Change in

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PP'

er

eθ

ez

θ dθ

r

z

Change in Unit Vectors –Cylindrical System

er

eθ

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Change in Unit Vectors –Spherical System

θφ

φθ

φθ

θφθφθφθ

θφθ

eeeeee

eee

cossincos

sin

dddddd

ddd

r

r

r

−−=+−=

+=

r

er

eθ

eφ

φ

θ

r

See “Formulae for Vector Algebra and

Calculus”

Note. Figure in presentation shows eθ and θ incorrectly. This figure is correct.

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Example

R=R(t)

Fluid particleDifferentially small piece of the fluid material

V=V(t) The position of fluid particle moving in a flow varies with time. Working in different coordinate systems write down expressions for the position and, by differentiation, the velocity vectors.

O

... This is an example of the calculus of vectors with respect to time.

Cartesian System

Cylindrical System

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Vector Calculus w.r.t. Time

• Since any vector may be decomposed into scalar components, calculus w.r.t. time, only involves scalar calculus of the components