Lent 29 ECTOR CALCULU EXAMPLE HEET G. Taylor. (a) The curve defined parametrically by x(t) = (a cos 3 t, a sin 3 t) with t 2π is called an astroid. ketch it, and find its length. (b) The curve defined by y 2 = x 3 is called Neile s parabola. ketch the segment of Neile s parabola with x 4, and find its length. 2. (a) A path in R 2 is defined in polar coordinates by r = f(θ) for α θ β. how that the length of the path is β (f(θ) ) 2 ( L = + f (θ) ) 2 dθ. α (b) The curves r = aθ, r = ae bθ and r = a( + cos θ) are called an Archimedean spiral, a logarithmic spiral and a cardioid, respectively. For a, b > and θ 2π, sketch the curves and find their lengths. 3. A circular helix is given by x(t) = (a cos t, a sin t, ct). Calculate the tangent t, principal normal n, curvature κ, binormal b, and torsion τ. ketch the helix for a, c >, showing t, n, b at some point on the helix. 4. (a) how that a planar curve given by x(t) = (x(t), y(t)) has curvature κ(t) = ẋÿ ẍẏ (ẋ 2 + ẏ 2 ) 3/2. (b) Find the maximum and minimum curvature of the ellipse x2 a 2 + y2 =, for a > b >. b2 5. Let f be a scalar function on R 2. By expressing the Cartesian basis vectors i, j in terms of the polar basis vectors e ρ, e φ, and using the chain rule to express the partial derivatives with respect to Cartesian coordinates x, y in terms of those with respect to polar coordinates ρ, φ, show that f = f ρ e ρ + f ρ φ e φ. 6. In three dimensions, use suffix notation and the summation convention to show that where a is a constant vector and r = x. (i) (a x) = a and (ii) (r n ) = nr n 2 x, Obtain the same results using spherical polar coordinates. For (i) in spherical polars, choose suitable axes to write a x without using the polar angle φ. 7. Find the equation of the plane tangent to the surface z = 3x 2 y sin( π 2 x) at the point x = y =. Taking East to be in the direction (,, ) and North to be (,, ), in which compass direction will a marble roll if placed on the surface at x =, y = 2? 8. Evaluate explicitly each of the line integrals (a) (x dx + y dy + z dz), (b) (y dx + x dy + dz), (c) (y dx x dy + e x+y dz) along (i) the straight line path joining (,, ) to (,, ), and (ii) the parabolic path given by x(t) = (t, t, t 2 ) with t. For which of these integrals do the two paths give the same results, and why?
9. Let F(x) = (3x 2 y 2 z, 2x 3 yz, x 3 y 2 ) and G(x) = (3x 2 yz 2, 2x 3 yz, x 3 z 2 ) be vector fields. how that F is conservative and G is not, and find the most general scalar potential for F. By exploiting similarities between F and G, or otherwise, evaluate the line integral G dx from (,, ) to (,, ) along the path x(t) = (t, sin π 2 t, sin π 2 t) with t.. (i) A curve C is given parametrically in Cartesian coordinates by x(t) = ( cos(sin nt) cos t, cos(sin nt) sin t, sin(sin nt) ), t 2π, where n is some fixed integer. Using spherical polar coordinates, sketch C. ( ) (ii) Let F(x) = y x,,. By evaluating the line integral explicitly, show that x 2 +y 2 x 2 +y 2 F dx = 2π, where C is traversed in the direction of increasing t. C (iii) Find a scalar function f such that f = F. Comment on this, given (ii).. Use the substitution x = ρ cos φ, y = 2ρ sin φ to evaluate x 2 x 2 + 4y 2 da, where A is the region between the two ellipses x 2 + 4y 2 = and x 2 + 4y 2 = 4. A 2. The closed curve C in the (x, y) plane consists of the arc of the parabola y 2 = 4ax (a > ) between the points (a, ±2a) and the straight line joining (a, 2a). The region enclosed by C is A. By calculating both integrals explicitly, show that (x 2 y dx + xy 2 dy) = (y 2 x 2 ) da = 4 5 a4, where C is traversed anticlockwise. C 3. The region A is bounded by the straight line segments {x =, y }, {y =, x }, {y =, x 3 4 }, and by an arc of the parabola y2 = 4( x). Consider a mapping into the (x, y) plane from the (u, v) plane defined by the transformation x = u 2 v 2, y = 2uv. ketch A and also the two regions in the (u, v) plane which are mapped into it. Hence calculate A A da (x 2 + y 2 ) /2. The remaining questions are optional. 4. A curve in the plane is given in polar coordinates as r = f(θ). Find an expression for its curvature as a function of θ. Find the curvature of the curve given by r = sin θ, and sketch it for θ π. 5. The field lines of a non-vanishing vector field F are the curves parallel to F(x) at each point x. how that the curvature of the field lines of F is given by F 3 F (F )F. 6. Let f : R 3 R 3 be a homogeneous function of degree n, i.e., such that f(kx) = k n f(x) for all k. By differentiating with respect to k, show that x f = nf. 2
7. Without changing the order of integration, show that [ x y ]dx (x + y) 3 dy = 2 [ ], and x y (x + y) 3 dx dy = 2. Comment on these results. 8. The curve x 3 + y 3 3axy = with a > is called the Cartesian Leaf. ketch the Cartesian Leaf, and find the area bounded by it in the first quadrant. 3
Lent 29 ECTOR CALCULU EXAMPLE HEET 2 G. Taylor. Let T be the tetrahedron with vertices at (,, ), (,, ), (,, ) and (,, ). Find the volume of T, and also the centre of volume, given by x d. T 2. A solid cone is bounded by the surface θ = α (in spherical polar coordinates) and the surface z = a, and its mass density is ρ cos θ. how that its mass is 2 3 πρ a 3 (sec α ). You can use either spherical or cylindrical polars for this calculation. (You could do both!) It is also possible to use a mixture of spherical and cylindrical polar variables to make the limits of the integrals particularly nice. 3. Let a, b, c be positive. By using the new variables α = x/y, β = xy, γ = yz, show that dx dy x dz x e ax/y bxy cyz = 2a(a + b)(a + b + c). 4. (a) Let ψ(x) be a scalar field and v(x) a vector field. Using suffix notation, show that (ψv) = ( ψ) v + ψ v and (ψv) = ( ψ) v + ψ v. (b) Find the divergence and curl of the following vector fields on R 3 : where r = x, and a, b are fixed vectors. rx, a(b x), a x, x/r 3, Use part (a) where you can. Thinking about the definitions of divergence and curl, could you have guessed any of the answers in advance? 5. Let u and v be vector fields. Using suffix notation, show that (i) (u v) = v ( u) u ( v) (ii) (u v) = u( v) + (v )u v( u) (u )v (iii) (u v) = u ( v) + (u )v + v ( u) + (v )u Deduce from (iii) that (u )u = ( 2 u2 ) u ( u), where u = u. Feel free to omit any of these that were proved in lectures. 6. erify that the vector field v(x) = ( e x (x cos y + cos y y sin y), e x ( x sin y sin y y cos y), ) is irrotational and express it as the gradient of a scalar field ϕ. erify also that v is also solenoidal and express it as the curl of a vector field (,, ψ). 7. (a) A vector field F is parallel to the normals of a family of surfaces f(x) = constant. how that F ( F) =. (b) The vector fields F and G are everywhere parallel and are both solenoidal. how that F (F/G) =, where F = F and G = G.
8. (a) Using the operator in Cartesian coordinates and in spherical polar coordinates, calculate the gradient of the scalar field ψ = z = r cos θ in both systems. By considering the relationship between the basis vectors, check that your answers agree. (b) Calculate, in three ways, the curl of the vector field F(x) = ye x + xe y = ρe φ = r sin θ e φ, by applying the standard formulae in Cartesian, cylindrical, and spherical coordinates. 9. how that the unit basis vectors of cylindrical polar coordinates satisfy e ρ φ = e φ and e φ φ = e ρ and that all other derivatives of the three basis vectors are zero. Given that the operator in cylindrical polars is = e ρ ρ + e φ ρ φ + e z z, derive expressions for F and F, where F = F ρ e ρ + F φ e φ + F z e z. Also derive an expression for 2 f, for a scalar function f. Don t just quote div/curl formulae from notes the intention is for you to derive them here.. The vector field F is given in cylindrical polar coordinates (ρ, φ, z) by F = ρ e φ, for ρ. how that F = for ρ. Calculate C F dx with C the circle given by ρ = R, φ 2π, z =. Why does tokes theorem not apply?. erify tokes theorem for the vector field F(x) = (y, x, xyz) and the open surface defined in cylindrical polar coordinates by ρ + z = a and z, where a >. 2. erify tokes theorem for the vector field F(x) = ( y 3, x 3, z 3 ) and the open surface defined by z = x 2 + y 2 and 4 z. 3. By applying tokes theorem to the vector field c F, where c is an arbitrary constant vector and F is a vector field, show that dx F = (da ) F, where the curve C bounds the open surface A. C A erify this result when F(x) = x and A is the unit square in the (x, y) plane with opposite vertices at (,, ) and (,, ). The remaining questions are optional. 4. A tricylinder is the body formed by intersecting the three solid cylinders given by the equations x 2 + y 2 a 2, y 2 + z 2 a 2 and z 2 + x 2 a 2, with a >. Using cylindrical polar coordinates, show that the volume of a tricylinder is 8(2 2)a 3. 2
5. Using the formula for in the appropriate coordinates, prove the following two results. (i) Let L be the line through the origin in the direction of the fixed non-zero vector a. Then the field f(x) has cylindrical symmetry about L if and only if a (x f) =. (ii) The field f(x) has spherical symmetry about the origin if and only if x f =. Explain geometrically why these results make sense. 6. Let be a surface with unit normal n, and v(x) a vector field such that v n = on. Let m be a unit vector field such that m = n on. By applying tokes theorem to m v, show that (δ ij n i n j ) v i d = u v ds, x j where s denotes arc-length along the boundary C of, and u is such that uds = ds n. erify this result by taking v = r and to be the disc r R in the z = plane. 7. (a) By considering G(x) = show that if F = then G = F. (b) By considering ϕ(x) = show that if x ( F) = then ϕ = F. C F(tx) tx dt, F(tx) x dt, how that F(x) = r 2 c x, where c is a non-zero constant vector, satisfies F and x ( F) =. Why does this not contradict the usual necessary and sufficient condition for F to be the gradient of a scalar? 3
Lent 29 ECTOR CALCULU EXAMPLE HEET 3 G. Taylor. Let I = rn x d, for n >, where is the sphere of radius R centred at the origin. Evaluate I directly, and by means of the divergence theorem. 2. Let F(x) = (x 3 + 3y + z 2, y 3, x 2 + y 2 + 3z 2 ), and let be the open surface x 2 + y 2 = z, z. Use the divergence theorem (and cylindrical polar coordinates) to show that F d = 2π. erify this by calculating the integral directly. 3. By applying the divergence theorem to the vector field c F, where c is an arbitrary constant vector and F(x) is a vector field, show that F d = da F, where the surface A encloses the volume. erify this result when F(x) = (z,, ) and is the cuboid x a, y b, z c. 4. Let be a volume with boundary, let f be a scalar field and F a vector field on. Prove the following two results by writing the i th component of the integrand as the divergence of a suitable vector field, then using the divergence theorem. A (a) If f is constant on, then f d =. (b) If F = in and F n = on, then F d =. Explain why these results make sense, given the conditions on f and F. 5. Let F, G be vector fields satisfying F = G = in the volume. how that F 2 G G 2 ( ) F d = F ( G) G ( F) d. 6. (a) The scalar field ϕ(r) depends only on the radial distance r = x in R 3. Use Cartesian coordinates and the chain rule to show that ϕ = ϕ (r) x r and 2 ϕ = ϕ (r) + 2 r ϕ (r). What are the corresponding results when working in R 2 rather than R 3? (b) how that the radially symmetric solutions of Laplace s equation in R 2 have the form ϕ = α + β log r, where α and β are constants. (c) Find the solution of 2 ϕ = in the region r in R 3 which is not singular at the origin and satisfies ϕ() =. 7. (a) Find all solutions of Laplace s equation, 2 f =, in two dimensions that can be written in the separable form f(r, θ) = R(r)Φ(θ), where r and θ are plane polar coordinates. (b) Consider the following boundary value problem: 2 f =, f(a, θ) = sin θ. Find the solution for r a which is not singular at the origin. Find the solution for r a that satisfies f(r, θ) as r. Find the solution for a r b that satisfies f f n (b, θ) =. Recall that n = n f.
8. A spherically symmetric charge density is given by for r < a ρ(r) = ρ r/a for a r b for b < r < Find the electric field everywhere in two ways, namely: (i) direct solution of Poisson s equation, using formulae from question 6, (ii) Gauss s flux method. You should assume that the potential is a function only of r, is not singular at the origin and that the potential and its first derivative are continuous at r = a and r = b. (Note that you are asked to find the electric field, not the potential.) 9. The surface encloses a volume in which the scalar field ϕ satisfies the Klein-Gordon equation 2 ϕ = m 2 ϕ, where m is a real non-zero constant. Prove that ϕ is uniquely determined if either ϕ or ϕ/ n is given on.. how that the solution to Laplace s equation in a volume with boundary condition g ϕ n + ϕ = f on is unique if g(x) on. Find a non-zero (and so non-unique) solution of Laplace s equation defined on r which satisfies the boundary condition above with f = and g = on r =. Don t assume that the solution is spherically symmetric. (Why not?). The scalar fields u(x) and v(x) satisfy 2 u = on and v = on. how that u v d =. Let w be a scalar field which satisfies w = u on. how that w 2 d u 2 d, i.e. the solution of the Laplace problem minimises w 2 d. 2. The scalar field ϕ is harmonic (i.e., solves Laplace s equation) in a volume bounded by a closed surface. Given that does not contain the origin, show that ( ( ) ( ) ) ϕ ϕ d =. r r Now let be the volume given by ε r a and let a be the surface r = a. Given that ϕ(x) is harmonic for r a, use this result, in the limit ε, to show that ϕ() = 4πa 2 ϕ(x) d. a Deduce that if ϕ is harmonic in a general volume, then it attains its maximum and minimum values on. 2
The remaining questions are optional. 3. By applying the divergence theorem to a suitable vector field, show that f d = f d for a scalar field f(x). erify this result when f = xz and is the paraboloidal region given by x 2 + y 2 + cz a 2 and z, where a and c are positive constants. 4. Find the electric field in question 8 by a third method, namely the integral solution of Poisson s equation: if 2 ϕ = ρ then ϕ(y) = ρ(x) 4π x y d. 5. Let be the 3-dimensional sphere of radius centred at (,, ), 2 be the sphere of radius 2 centred at ( 2,, ) and 3 be the sphere of radius 4 centred at ( 4,, ). The eccentrically-shaped planet Zog is composed of rock of uniform density ρ occupying the region within and outside 2 and 3. The regions inside 2 and 3 are empty. Give an expression for Zog s gravitational potential at a general coordinate x that is outside. Is there a point in the interior of 3 where a test particle would remain stably at rest? 6. Consider the partial differential equation u t = 2 u, for u = u(t, x), with initial condition u(, x) = u (x) in, and boundary condition u(t, x) = f(x) on for all t. how that When does equality hold? u 2 d, t how that ( ) remains true with the boundary condition on, provided α(x). u t + α(x) u n = ( ) 3
Lent 29 ECTOR CALCULU EXAMPLE HEET 4 G. Taylor Where needed, you may quote that the most general isotropic tensor of rank 4 has the form λδ ij δ kl + µδ ik δ jl + νδ il δ jk for λ, µ, ν R.. Let n be a unit vector in R 3, and let P ij = δ ij n i n j. (a) Find the eigenvalues and eigenvectors of P ij. (b) For A ij = ε ijk n k, find A ij A jk A kl A lm in terms of P im. Explain this result geometrically. 2. If u(x) is a vector field, show that u i / x j transforms as a second-rank tensor. If σ(x) is a second-rank tensor field, show that σ ij / x j transforms as a vector. 3. Given vectors u = (,, ), v = (,, ) and w = (,, ), find all components of the secondrank and third-rank tensors defined by (i) ij = u i v j + v i w j (ii) T ijk = u i v j w k v i u j w k + v i w j u k w i v j u k + w i u j v k u i w j v k. 4. Any 3 3 matrix A can be decomposed in the form Ax = αx + ω x + Bx, where α is a scalar, ω is a vector, and B is a traceless symmetric matrix. erify this claim by finding α, ω k and B ij explicitly in terms of A ij. 2 3 Check your calculations are correct by finding α, ω and B for the matrix A = 4 5 6. 2 3 5. Let u(x) be a vector field on R 3. how that for some traceless symmetric tensor ij. u i x j = 3 ( u)δ ij + ij 2 ε ijk( u) k, Find ij in the case u(x) = (x x 2 2, x 2x 2 3, x 3x 2 ). erify that (,, ) is one of the principal axes of ij at the point x = (2, 3, ), and find the others. 6. The electrical conductivity tensor σ ij has components 2 σ ij = 2. 2 how that there is a direction along which no current flows, and find the direction(s) along which the current flow is largest, for an electric field of fixed magnitude. 7. A body has symmetry such that it is unchanged by rotations of π about three perpendicular axes which form a basis B. how that any second-rank tensor calculated for the body will be diagonal in this basis, although the diagonal elements need not be equal. Find the inertia tensor of a cuboid of uniform density with sides of length 2a, 2b and 2c about the centre of the cuboid.
8. For any second-rank tensor T ij, prove using the transformation law that the quantities are the same in all bases. α = T ii, β = T ij T ji, and γ = T ij T jk T ki If T ij is a symmetric tensor, express these invariants in terms of the eigenvalues. Deduce that the determinant of T ij is 6 (α3 3αβ + 2γ). 9. Let be the surface of the unit sphere. (a) Calculate the following integrals using properties of isotropic tensors: (i) x i d, (ii) x i x j d. erify your answers using the tensor divergence theorem. (b) For the second-rank tensor T ij = δ ij + ε ijk x k, calculate the following integrals: (i) T ij d, (ii) T ij T jk d. If your lectures only considered isotropic integrals over volumes rather than surfaces, then you should assume that the corresponding results for surfaces hold.. Evaluate the following integrals over the whole of R 3. (i) r 3 e r2 x i x j d, (ii) r 4 e r2 x i x j x k d, (iii) r 5 e r2 x i x j x k x l d.. (a) how that ε ijk ε ijl is isotropic of rank 2, and deduce that it equals 2δ kl. (b) Using that ε ijk ε ilm is isotropic of rank 4, show that it equals δ jl δ km δ jm δ kl. δ ip δ iq δ ir (c) Prove that ε ijk ε pqr = δ jp δ jq δ jr δ kp δ kq δ kr. 2. The array d ijk with 3 3 elements is such that d ijk s jk is a vector for every symmetric second-rank tensor s jk. how that d ijk need not be a tensor, but that d ijk + d ikj must be. 3. In linear elasticity, the symmetric second-rank stress tensor σ ij depends on the symmetric second-rank strain tensor e ij according to σ ij = c ijkl e kl. Assuming that c ijkl = c ijlk, explain why c ijkl must be a fourth-rank tensor. how that, in an isotropic material, σ ij = λδ ij e kk + 2µe ij ( ) for two scalars λ and µ. Assume below that µ > and λ > 2 3 µ. Use ( ) to find an expression for e ij in terms of σ ij, and explain why the principal axes of of σ ij and e ij coincide. The elastic energy density resulting from a deformation of the material is given by E = 2 e ijσ ij. how that E > for any non-zero strain e ij. 2
The remaining questions are optional. 4. Given a non-zero vector v i, any 3 3 symmetric matrix T ij can be expressed in the form T ij = Aδ ij + Bv i v j + (C i v j + C j v i ) + D ij for some numbers A and B, some vector C i and a symmetric matrix D ij, where C i v i =, D ii =, D ij v j =, and the summation convention is implicit. how that the above statement is true by finding A, B, C i and D ij explicitly in terms of T ij and v j, or otherwise. Explain why A, B, C i and D ij together provide a space of the correct dimension to parameterise an arbitrary symmetric 3 3 matrix T ij. 5. (a) A tensor of rank 3 satisfies T ijk = T jik and T ijk = T ikj. how that T ijk =. (b) A tensor of rank 4 satisfies T jikl = T ijkl = T ijlk and T ijij =. how that T ijkl = ε ijp ε klq pq, where pq = T rqrp. 6. In question 8, you were told to assume that the results from lectures regarding isotropic integrals over volumes also hold for surfaces. Prove that this is valid for a spherical surface, using the tensor divergence theorem. 7. A second-rank tensor T (y) is defined for n > by T ij (y) = (y i x i )(y j x j ) y x 2n 2 da, where y is a fixed vector with y = a, and the surface is the sphere of radius a centred on the origin. Explain briefly why T might be expected to have the form where α and β are scalar constants. T ij = αδ ij + βy i y j, how that y (y x) = a 2 ( cos θ), where θ is the angle between y and x, and find a similar expression for y x 2. Using suitably chosen spherical polar coordinates, show that y i T ij y j = πa2 (2a) 2n+2 n + 2 Using this and another scalar integral, determine α and β, and find the value of n for which T is isotropic.. 3