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Class 3 - Vector Calculus

# Class 3 - Vector Calculus - AOE 5104 Class 3 Online...

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AOE 5104 Class 3 9/2/08 Online presentations for today’s class: Vector Algebra and Calculus 1 and 2 Vector Algebra and Calculus Crib Homework 1 due 9/4 Study group assignments have been made and are online. Recitations will be Mondays @ 5:30pm (with Nathan Alexander) Tuesdays @ 5pm (with Chris Rock) Locations TBA Which recitation you attend depends on which study group you belong to and is listed with the study group assignments

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Unnumbered slides contain comments that I inserted and are not part of Professor’s Devenport’s original presentation.
3 Last Class… Vectors, inherent property of direction Algebra Volumetric flow rate through an area Taking components, eqn. of a streamline Triple products, A.BxC, Ax(BxC) Coordinate systems

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4 Cylindrical Coordinates R e r e θ e z Coordinates r, θ , z • Unit vectors e r , e θ , e z (in directions of increasing coordinates) Position vector R = r e r + z e z Vector components F = F r e r + F θ e θ + F z e z Components not constant, even if vector is constant r θ z F x y z
5 Spherical Coordinates r e r e θ e φ φ θ r F Coordinates r, θ , φ • Unit vectors e r , e θ , e φ (in directions of increasing coordinates) Position vector r = r e r Vector components F = F r e r + F θ e θ + F φ e φ Errors on this slide in online presentation y x z

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1 1 1 r r x y z x r r r r r r F F F F F F F F F F x r y r r r r r r h h r r h h r θ θ φ φ θ θ φ φ θ θ = + + = + + = = + + = θ φ = θ φ = θ = θ φ + θ φ + θ = = θ φ + θ φ + θ = = = = θ φ + θ φ - θ = g g g g sin cos , sin sin , cos sin cos sin sin cos sin cos sin sin cos cos cos cos sin sin F e e e i j k F i e i e i e i r i j k r r e i j k r e i j k 1 x r r h r h F F F F φ φ φ θ φ = ∂θ = = - φ + φ = = θ ∂φ ∂φ = θ φ + θ φ - φ sin cos sin sin cos cos cos sin r r r e i j
7 J. KURIMA, N. KASAGI and M. HIRATA (1983) Turbulence and Heat Transfer Laboratory, University of Tokyo LOW REYNOLDS NUMBER AXISYMMETRIC JET

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8 Class Exercise Using cylindrical coordinates ( r , θ , z ) Gravity exerts a force per unit mass of 9.8m/s 2 on the flow which at (1,0,1) is in the radial direction. Write down the component representation of this force at a) (1,0,1) b) (1, π ,1) c) (1, π /2,0) d) (0, π /2,0) R e r e θ e z r θ z 9.8m/s 2 a) (9.8,0,0) b) (-9.8,0,0) c) (0,-9.8,0) d) (9.8,0,0) x y z
Vector Algebra in Components ( 29 ( 29 ( 29 1 1 2 2 3 3 1 2 3 1 2 3 1 2 3 2 3 3 2 1 3 1 1 3 2 1 2 2 1 3 cos where is the smaller of the two angles between and sin where is determined by the right-han = + + = ϕ ϕ × = = - + - + - × = ϕ A B A B A B e e e A B e e e A B A B n n g A B A B A B A A A B B B A B A B A B A B A B A B d rule A B ϕ n

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3. Vector Calculus Fluid particle: Differentially Small Piece of the Fluid Material

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