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Vector Algebra and Calculus- Class 1-
AOE 5104Advanced Aero- and Hydrodynamics
Dr. William Devenport andLeifur Thor Leifsson
2
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
3
Vector basicsVector: A, AMagnitude: |A|, AScalar: p, φ
Types– Polar vector
•
– Axial vector •
– Unit vector•
MAGDIR
P
Q
4
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