Question #1: Tensor Relations Prove the following...

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Mostafa Najafiyazdi Advanced Fluid Dynamics (MECH 562) Department of Mechanical Engineering Fundamentals of Fluid Mechanics (MECH 610) McGill University Homework #1 Deadline: Sunday January 28, 2017 Question #1: Tensor Relations Prove the following identities using index notation 1. Relation commonly known as abc equals back minus cab . ⃗a × ( b × c ) = b ( ⃗a · c ) - c ( ⃗a · b ) . (1) 2. Curl of gradient: ∇ × ∇ f = 0. 3. Divergence of a curl: ∇ · ( ∇ × ⃗u ) = 0. 4. Curl of a curl: ∇ × ( ∇ × ⃗u ) = ( ∇ · ⃗u ) - ∇ 2 ⃗u Question #2: Cylindrical coordinates The separation or displacement vector between two particles A and B is defined as r i = r ( A ) i - r ( B ) i . (2) The velocity vector , u i is defined as the total time derivative of a position vector r i : u i = dr i d t . (3) Therefore, the relative velocity of particle B with respect to particle A is defined as u ( BA ) i = r i dt = dr ( B ) i dt - dr ( B ) i dt = u ( B ) i - u ( A ) i . (4) In Cartesian coordinates, the basis vectors are fixed and time independent , ( ˆ i, ˆ j, ˆ k ). It means that d ˆ i/dt = d ˆ j/dt = d ˆ k/dt

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