A First Course in Fluid Dynamics

Chapter I

1. Resolve in fig. I.1.

2. Sketches need to be three-dimensional versions of fig. 1.2: you need either a length dz or a length corresponding to dλ and ds will no longer be in the plane of r and θ.

Otherwise you need to find formulae corresponding to

dx = dr cos θ − r sin θ dθ,

and calculate ds2 from

ds2 = dx2 + dy2 + dz2.

3. The formulae for are in §2(c) You must differentiate these, and then express the result in terms of these unit vectors.

4. Taylor's theorem in one-dimension is

when (φ is smooth enough (twice differentiable with continuous second derivative is certainly enough), and where 0 < θ < 1. The last term on the right is less than some constant times h2 if φ″ is continuous on (x, x + h), because φ″ is then bounded in that interval.

Remember that h2 = hihi = hkhk.

5. (i) ∇ × (φA) has ith component

(ii) When is ∂2u/∂x∂t = ∂2u/∂t∂x?

(iii) ∇ · A = ∂Ai/∂xi, and here A = φ∇ψ, and ∇ψ has ith component ∂ψ/∂xi.

(iv), (v) ε123 = + 1, ε213 = −1: look for terms which have these ε in them. Then use ∂2/∂x1∂x2 = ∂2/∂x2∂x1 for smooth functions.

(vi) This is and use a theorem in §4(c).

6. ∇ × ∇ = 0, and write what remains as

∇ × (A × B),

which works out rather like Q5(vi). Finally put the proper value back for B.

7. Use Q5(iii), and the divergence theorem from §5(b).

8. For ∇ · A proceed as in §6(a), and use formulae for derivatives of from §2(c) and Q3. For ∇ × A evaluate the determinant in §6(c).