Download Vector Analysis Example Sheet 2: Divergence, Curl, and Stokes's Theorem and more Slides Mathematical Methods in PDF only on Docsity!
MA231 Vector Analysis
Example Sheet 2
2010, term 1 Stefan Adams
Hand in solutions to questions B1, B2, B3 and B4 by 3pm Monday of week 6.
A1 Calculating divergences Calculate the divergence div v = ∇·v for the following vector fields v : R^3 → R^3 with:
(a) v(x, y, z) = (x^2 , xy, xz) (b) v(x, y, z) = (cos(xy),
1 + x^2 y^2 z^2 , zy sin(xy))
(c) v(x, y, z) = ∇f (x, y, z) where f : R^3 → R, (x, y, z) 7 → f (x, y, z) = xyez^.
A2 Examples of the divergence theorem For each of the following vector fields v : R^3 → R^3 calculate the flux integral
S v·^ Nˆ dS out of the sphere S given by x^2 + y^2 + z^2 = R^2. Calculate them first as surface integrals and then confirm that your answer agrees with the volume integral given by the divergence theorem.
(a) v(x, y, z) = (x, y, z) (b) v(x, y, z) = (−y, x, 0) (c) v(x, y, z) = (−x, y, z).
A3 Using the divergence theorem Let f : R^3 → R, (x, y, z) 7 → f (x, y, z) = x^4 + y^4 + z^4. Use the divergence theorem to calculate the outward flux of ∇f through the following surfaces:
(a) The boundary of the cube 0 ≤ x, y, z ≤ 1 , (b) The sphere x^2 + y^2 + z^2 = R^2.
A4 Calculating curls For each of the following vector fields v : R^3 → R^3 calculate the curl ∇ × v:
(a) v(x, y, z) = (−y, x, 1) (b) v(x, y, z) = (xy + z,
x^2 + 2yz, y^2 + x)
(c) v(x, y, z) = (xz, yz, 0)
A5 Identities
(a) For v : R^3 → R^3 show that ∇·(∇ × v) = 0. (b) For f : R^3 → R and v : R^3 → R^3 show that ∇ × (f v) = f ∇ × v + ∇f × v.
A6 Stokes’s theorem in the plane Let v : R^2 → R^2 , (x, y) 7 → v(x, y) = (0, x). Calculate curl(v). Apply Stokes’s theorem to v on the region Ω bounded by the ellipse x
2 α^2 +^
y^2 β^2 = 1. Hence prove that the area of^ Ω^ is^ παβ.
A7 Stokes’s theorem in R^3 Sketch the surface S given by the portion of a paraboloid z = 2 − x^2 − y^2 where z ≥ 0. Using an inward facing normal vector field, explain how to parameterise the boundary ∂S so that the unit tangent vector Tˆ and the unit normal vector Nˆ are correctly oriented for Stokes’ theorem. For the vector field v : R^3 → R^3 , (x, y, z) 7 → v(x, y, z) = (y, z, x) calculate both the surface flux
S ∇ ×^ v·^ Nˆ dS and the tangential line integral
∂S v·^ Tˆ ds and verify that Stokes’s Theorem holds.
B1 The flux integral
(a) Calculate the flux of the vector field f : R^3 → R^3 , (x, y, z) 7 → f (x, y, z) = (z, x, − 3 y^2 z) across the surface of the cylinder {(x, y, z) ∈ R^3
∣x^2 + y^2 = 16, z ∈ [0, 5]}. (b) Calculate the flux of the vector field
f : R^3 → R^3 , (x, y, z) 7 → f (x, y, z) = (4xz, −y^2 , yz)
across the surface of the unit cube bounded by the planes x = 0, x = 1, y = 0, y = 1, z = 0 and z = 1. (the unit cube is the set {(x, y, z) ∈ R^3
∣ (^0) ≤ x ≤ 1 , 0 ≤ y ≤ 1 , 0 ≤ z ≤ 1 })
B2 The divergence theorem
Use the divergence theorem to calculate the following flux integrals. (a) The outward flux of the two dimensional vector field
f : R^2 → R^2 , (x, y) 7 → f (x, y) = (x/2 + y
x^2 + y^2 , y/ 2 − x
x^2 + y^2 )
through the boundary of the ball Ω = {(x, y) ∈ R^2
∣ (^) x^2 + y^2 ≤ R^2 } ⊂ R^2 , R > 0. (b) The outward flux of the vector field f : R^3 → R^3 , (x, y, z) 7 → f (x, y, z) = (2x, −y, 3 z)
through the boundary of the pyramid Ω bounded by the planes x + 2y + 3z = 6, x = 0, y = 0 and z = 0. (Hint: you may quote a formula for the volume of a pyramid).
B3 Identities and harmonic functions
(a) For three dimensional vector fields u : R^3 → R^3 and v : R^3 → R^3 show that
∇·(u × v) = v·(∇ × u) − u·(∇ × v).
(b) For three dimensional scalar fields f : R^3 → R and g : R^3 → R show that ∆(f g) = f ∆g + 2∇f ·∇g + g ∆f.
(c) For the scalar field ϕ : R^3 → R and the vector field u : R^3 → R^3 show that
∇·(ϕu) = (∇ϕ)·u + ϕ(∇·u).
(d) Give a sketch of the proof of the following statement: Let f : D → R, D ⊂ R^3 , be harmonic (that is ∆f (x) = 0 for all x ∈ D). Then for any closed ball B(a, r) ⊂ D (recall B(a, r) = {x = (x 1 , x 2 , x 3 ) ∈ R^3 |‖x − a‖ ≤ r}) having radius r > 0 and origin a ∈ D with surface S = ∂B(a, r) = {x = (x 1 , x 2 , x 3 ) ∈ R^3 |‖x − a‖ = r} it holds that the value of the function at the origin of the ball is the mean value of the function over the surface of that ball, i.e. f (a) =
4 πr^2
S
f.
B4 Integration by parts formulae
For a function f : Rn^ → R and a vector field v : Rn^ → Rn^ show that
∇·(f v) = ∇f ·v + f ∇·v. Deduce the integration by parts formula, for a region Ω ⊆ R^3 , ∫
Ω
f ∇·v dV = −
Ω
∇f ·v dV +
∂Ω
f v· Nˆ dS.
Write out the formula in the special case where v = ∇g for some g : R^3 → R. Deduce that ∫
Ω
f ∆g dV =
Ω
g∆f dV +
∂Ω
f ∇g· Nˆ dS −
∂Ω
g ∇f · Nˆ dS.
This final identity is called Green’s identity, named after George Green who was the son of a Nottinghamshire baker and a self taught mathematician. See question C4 for an application of this identity.
