Lent 24 ECTOR CALCULU - EXAMPLE G. Taylor A star means extra practice or can save until later, and not necessarily is harder.. The curve given parametrically by (acos 3 t,asin 3 t) with t 2π is called an astroid. ketch it, and find its length. 2. 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 the length of this segment. *3. In R 2 a path is defined in polar coordinates by r = f(θ), α θ β, with a function f C [α,β]. how that the length of the path is L = β α (f(θ) ) 2 + ( f (θ) ) 2 dθ. ketch the paths r = aθ and r = a(+cosθ), where for both a > and θ 2π. Calculate their lengths. 4. Given a function f(r) in two dimensions, use the chain rule to express its partial derivatives with respect to Cartesian coordinates (x, y) in terms of its partial derivatives with respect to polar coordinates (ρ, φ). From the relationship between the basis vectors in these coordinate systems, deduce that f = f ρ e ρ + f ρ φ e φ. 5. In three dimensions, use suffix notation and the summation convention to show that where a is any 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. In spherical polars, for a function f of r and θ only, f = f r e r + f r θ e θ. 6. Evaluate explicitly each of the line integrals (a) (xdx+ydy +zdz), (b) (ydx+xdy +dz), (c) (ydx xdy +e x+y dz) along (i) the straight line path joining the origin to x = y = z =, and (ii) the parabolic path given parametrically by x = t,y = t,z = t 2 with t. For which of these integrals do the two paths give the same results, and why? 7. Obtain the equation of the plane which is tangent to the surface z = 3x 2 ysin(πx/2) at the point x = y =. Take East to be in the direction (,,) and North to be (,,). In which direction will a marble roll if placed on the surface at x =, y = 2? 8. Let F = (3x 2 yz 2,2x 3 yz,x 3 z 2 ) and G = (3x 2 y 2 z,2x 3 yz,x 3 y 2 ) be vector fields. Compute explicitly the line integrals F dx and G dx from (,,) to (,,) along (i) the straight line joining the points, and (ii) the path x(t) = (t,t 2,t 2 ). how that only one of F and G is a conservative field and find a scalar potential for this one. Comment on the answers to your integrals in the light of this.
9. A curve C is given parametrically in Cartesian coordinates by x(t) = ( cos(sinnt)cost, cos(sinnt)sint, sin(sinnt) ), t 2π, where n is some fixed integer. Using spherical polar coordinates, sketch and describe C. how that ( ) C F dx = 2π, where F(x) = y x,, and C is traversed in the direction of increasing t. x 2 +y 2 x 2 +y 2 Can F be written as the gradient of a scalar? Comment on your results.. Use the substitution x = rcosθ, y = 2rsinθ to evaluate x 2 x 2 +4y 2 da, where A is the region between the two ellipses x 2 +4y 2 =, x 2 +4y 2 = 4. A. 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 area enclosed by C is A. By calculating the integrals explicitly, show that where C is described anticlockwise. C (x 2 ydx+xy 2 dy) = A (y 2 x 2 )da = 4 5 a4, 2. The region A is bounded by the 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 da (x 2 +y 2 ) /2. *3. 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. A 4. Let T be the tetrahedron with vertices at (,,), (,,), (,,) and (,,). Find the volume of T, and hence find the centre of volume, given by xd. 5. A solid cone is bounded by the surface θ = α (in spherical polar coordinates) and the surface z = a. Its mass density is ρ cosθ. how that its mass is 2π 3 ρ a 3 (secα ). 6. [Tripos, 25/III/2] Express the integral I = dx dy T x dz xe Ax/y Bxy Cyz in terms of the new variables α = x/y, β = xy, γ = yz. Hence show that I = Assume that A, B and C are positive. 2A(A+B)(A+B +C). 2
Lent 24 ECTOR CALCULU - EXAMPLE 2 G. Taylor. A circular helix is given by x = (acost,asint,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. *2. Explain why the tangent, principal normal and binormal form an orthonormal system. how that the torsion can be written as τ = [ κ 2 t, dt ] ds, d2 t ds 2, where [a,b,c] denotes the scalar triple product a b c. erify this identity for the helix in question. 3. *(a) how that a curve in the plane given by x(t) = (x(t),y(t)) has curvature κ = ẋÿ ẍẏ (ẋ 2 +ẏ 2 ) 3/2. (b) Find the minimum and maximum curvature of the ellipse x 2 /a 2 +y 2 /b 2 =. Comment on the case when a = b. If you have done (a), feel free to quote the formula you derived for κ. 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) Evaluate the divergence and curl of the following: rx, a(x b), a x, where r = x, and a and b are fixed vectors. x a x a 3, 5. Let u and v be vector fields. how using suffix notation 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). 6. how that the vector field H = ( 3x 2 tanz y 2 e xy2 siny,(cosy 2xysiny)e xy2,x 3 sec 2 z ) is conservative, and find the most general scalar potential for H. Hence calculate the line integral P 2 P H dx from the point P = (,,) to the point P 2 = (,π/2,π/4). 7. how that the vector field u = e x (xcosy +cosy ysiny)i+e x ( xsiny siny ycosy)j is irrotational and express it as the gradient of a scalar field φ. how that u is also solenoidal and find a vector potential for it in the form ψk, for some function ψ. 3
8. (a) A vector field B(x) is parallel to the normals of a family of surfaces f(x) = constant. how that B ( B) =. (b) The vector fields v(x) and B(x) are everywhere parallel and are both solenoidal. how that B (v/b) =, where v = v and B = B. *(c) The tangent vector at each point on a curve is parallel to a non-vanishing vector field H(x). how that the curvature of the curve is given by H 3 H (H )H. *9. Consider A(x) = how that A = B if B = everywhere. x B(xt)t dt.. A fluid flow has the velocity vector v = (,,z + a) in Cartesian coordinates, where a is a constant. Calculate the volume flux of fluid flowing across the open hemispherical surface r = a, z, and also that flowing across the disc r a, z =. erify the divergence theorem holds. [olume flux of fluid = v d.]. [Tripos, 22/III/4] tate the divergence theorem. Consider the integral I = rn r d, where n > and 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 evaluate erify your result by calculating the integral directly. [You should find that d = (2ρcosϕ,2ρsinϕ,)ρdρdϕ.] F d. 3. erify tokes theorem for the open hemispherical surface r =, z, and the vector field F(x) = (y, x,z). 4. By applying the divergence theorem to the vector field k B, where k is an arbitrary constant vector and B(x) is a vector field, show that B d = B da, where the surface A encloses the volume. erify this result when A is the sphere x = R and B = (z,,) in Cartesian coordinates. 5. By applying tokes theorem to the vector field k B, where k is an arbitrary constant vector and B(x) is a vector field, show that dx B = (da ) B, where the curve C bounds the open surface A. C A erify this result when C is the unit square in the (x,y) plane with opposite vertices at (,,) and (,,), and B = x. 4 A
6. [Tripos, 25/III/] Write down tokes theorem for a vector field B(x) on R 3. Consider the bounded surface defined by z = x 2 + y 2, 4 z. ketch the surface and calculate the surface element d. For the vector field B = ( y 3,x 3,z 3 ), calculate I = ( B) d directly. how using tokes theorem that I may be rewritten as a line integral and verify this yields the same result. *7. [Tripos, 2/III/] tate the divergence theorem for a vector field u(r) in a closed region bounded by a smooth surface. Let Ω(r) be a scalar field. By choosing u = cω for arbitrary constant vector c, show that Ω d = Ω d. ( ) Let be the bounded region enclosed by the surface which consists of the cone (x,y,z) = (rcosθ,rsinθ,r/ 3) with r 3 and the plane z =, where r,θ,z are cylindrical polar coordinates. erify that ( ) holds for the scalar field Ω = (a z), where a is a constant. *8. [Tripos, 22/III/] The first part of the question was to prove question 5(ii) above, so I have omitted it here. is an open orientable surface in R 3 with unit normal n, and v(x) is any continuously differentiable vector field such that n v = on. Let m be a continuously differentiable unit vector field which coincides with n on. By applying tokes theorem to m v, show that (δ ij n i n j ) v i x j d = C u v ds, 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. 5
Lent 24 ECTOR CALCULU - EXAMPLE 3 G. Taylor. (a) Using the operator in Cartesian coordinates and in spherical polar coordinates, calculate the gradient of ψ = Ez = Ercosθ in both systems, where E is a constant. By considering the relationship between the basis vectors, check that your answers agree. (b) Calculate, in three ways, the curl of the vector field B = 2 B( ye x +xe y ) = 2 Bρe φ = 2 Brsinθe φ, by applying the standard formulas in Cartesian, cylindrical, and spherical coordinates. 2. how that the unit basis vectors of cylindrical polar coordinates satisfy e r θ = e θ and e θ θ = e r, with all other derivatives of the three basis vectors being zero. Using these relationships, and given that the operator in cylindrical polars is = e r r +e θ r θ +e z z, derive expressions for A and A, where A = A r e r +A θ e θ +A z e z. 3. The vector field B(x) is given in cylindrical polar coordinates (r,θ,z) by B(x) = r e θ. Using the formula derived in question 2, show that B = when r. Calculate C B dx with C the circle given by r = R, θ 2π, z =. Why does tokes theorem not apply? 4. Let F satisfy F = in the volume and F n = on the boundary. By considering x j (x i F j ), show that F d =. *5. Let the surface enclose the volume, and let P and Q be two solenoidal vector fields (i.e., P = Q = ). how that (Q 2 P P 2 ( Q) d = Q ( P) P ( Q) ) d. 6. (a) The scalar function ϕ 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, 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 two dimensions have the form ϕ = α+βlogr, 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() =. 6
7. 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. Either look up the formula in the notes for 2 f in plane polars, or derive it as f in a manner similar to question 2. Hence solve, for r < a, the following boundary value problem, assuming that f(r,θ) is not singular at the origin: 2 f =, f(a,θ) = sinθ Find also the solution for r > a that satisfies f(r,θ) as r. 8. A spherical shell has density given by for < r < a ρ(r) = ρ r/a for a < r < b for b < r < Find the gravitational field everywhere by three different methods, namely (a) direct solution of Poisson s equation, (b) Gauss s flux theorem, *(c) the integral form of the general solution of Poisson, ϕ(x ) = 4π ρ(x) x x d. 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. 9. Maxwell s equations for electric and magnetic fields E(x, t) and B(x, t) are E = ρ ε, E = B t, B =, B = µ j+ε µ E t. how that these imply that the charge density ρ(x,t) and current density j(x,t) satisfy the conservation equation j = ρ/ t. how also that if j is zero then U = 2 (ε E 2 +µ B2 ) and P = µ E B satisfy P = U/ t. *. [Tripos, 24/III/] 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? Justify your answer.. Using an integral theorem, derive the following (one of Green s Identities): ψ 2 φ φ 2 ψ d = ψ φ n φ ψ n d Recall that ϕ/ n = n ϕ. 7
2. 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. 3. how that the solution to Laplace s equation in a volume with boundary condition is unique if g(x) on. g ϕ +ϕ = f on n Find a non-zero (and so non-unique) solution of Laplace s equation 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? 4. The functions u(x) and v(x) on satisfy 2 u = on and v = on. how that u v d =. Let w be a function on which satisfies w = u on. By considering v = w u, show that w 2 d u 2 d, i.e. the solution of the Laplace problem minimises w 2 d. 5. The scalar field ϕ is harmonic in a volume bounded by a closed surface. Given that does not contain the origin (r = ), show that ( ( ) ( ) ) ϕ ϕ d =. r r Now let be the volume given by ε r a and let 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. Deduce that if ϕ is harmonic in a general volume, then it attains its maximum and minimum values on. Note: harmonic means satisfies Laplace s equation. *6. [Tripos, 25/III/9] The first part of the question was essentially question 4 above. o I have omitted it here, but bear that context in mind in what follows. Consider the partial differential equation w t = 2 w, for w = w(t,x), with initial condition w(,x) = w (x) in, and boundary condition w(t,x) = f(x) on for all t. how that w 2 d, ( ) t with equality holding only when w(t,x) = u(x). (This is the u from the question 4 part.) how that ( ) remains true with the boundary condition on, provided α(x). w t +α(x) w n = 8
Lent 24 ECTOR CALCULU - EXAMPLE 4 G. Taylor. 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. 2. Given vectors u = (,,), v = (,, ) and w = (,,), find all components of the secondrank and third-rank tensors defined by T ij = u i v j +v i w j ; 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. 3. 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. Explain why α, ω, and B together provide a space of the correct dimension to parameterise an arbitrary 3 3 matrix. 2 3 Check your calculations are correct by finding α, ω and B for the matrix A = 4 5 6. 2 3 *4. [Tripos, 25/I/8 Algebra & Geometry] 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. 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. 6. A body has the symmetry that its shape 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 B, 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. 7. AsecondranktensorisdefinedintermsofthepositionvectorxbyT ij = δ ij +ε ijk x k. Calculate the following integrals, with being the surface of the unit sphere. (i) x i d, (ii) T ij d, (iii) T ij T jk d. 8. Evaluate the following integrals over the whole of R 3 for positive γ and r 2 = x p x p : (i) r 3 e γr2 x i x j d, (ii) r 5 e γr2 x i x j x k d. 9
9. 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 cubic equation for the eigenvalues is λ 3 αλ 2 + 2 (α2 β)λ 6 (α3 3αβ +2γ) =.. Given that the most general isotropic rank 4 tensor is λδ ij δ kl +µδ ik δ jl +νδ il δ jk for λ,µ,ν R, show that ε ijk ε ilm = δ jl δ km δ jm δ kl.. Given a non-zero vector v i, the orthogonal projection tensor P ij is defined by P ij = δ ij v iv j v k v k. (i) erify that P ij satisfies (a) P ij v j = and (b) P ij u j = u i for any vector u i which is orthogonal to v i. (ii) how that P ij is unique: that is, if another tensor T ij satisfies both (a) and (b), then (P ij T ij )w j = for any vector w i. (iii) For A ij = ε ijk v k, show that P ij A jk A km = v k v k P im. 2. (i) A tensor of rank 3 satisfies T ijk = T jik and T ijk = T ikj. how that T ijk =. (ii) 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. *3. Three Cartesian frames of reference in R 3 are such that the i th axis of the first frame coincides with the (i+n) th axis (modulo 3) of the (n+) th frame (n =,,2). A physical entity has components,, in the three frames, respectively. how that this entity cannot be a tensor. how that the entity with respective components,, could be a tensor, and on the assumption that it is, find its components in an arbitrary Cartesian frame whose axes are at angles θ, θ 2 and θ 3 to the -axis of the first frame, with cosθ = λ, cosθ 2 = µ and cosθ 3 = ν (i.e., the axes have direction cosines λ,µ,ν with respect to the -axis of the first frame). *4. The array D ikm with 3 3 elements is not known to represent a tensor. If, for every symmetric tensor represented by a km, b i = D ikm a km represents a vector, what can be said about the transformation properties under rotations of the coordinates axes of (i) D ikm, (ii) D ikm +D imk?
5. A conductor positioned in a magnetic field H carries a steady current density J = H, and the magnetic flux intensity B = µh satisfies B =. The mechanical force per unit volume acting on the conductor can be written as J B. If the permeability µ is a constant, show that this force per unit volume can be written as s ik / x k in terms of a tensor s ik = µ(h i H k 2 H mh m δ ik ). 6. In linear elasticity, the symmetric second-rank stress tensor σ ij is linear in the symmetric second-rank strain tensor e kl. how that in an isotropic material, σ ij = λδ ij e kk +2µe ij, with two scalars λ and µ. (You may quote the form of the general isotropic fourth-rank tensor.) Invert this equation to find an expression for e ij in terms of σ kl, assuming that µ and 3λ 2µ. how that the eigenvectors of σ are parallel to the eigenvectors of e. *7. A vector field u i has the following components in a particular system of Cartesian coordinates x i : u = x x 2 2, u 2 = x 2 x 2 3, u 3 = x 3 x 2. Express the tensor u i / x k at the point x = 2, x 2 = 3, x 3 = as a linear combination of ε ijk w j (where w j is a vector to be determined) and a symmetric tensor e ik. erify that the principal axes of e ik are in the directions 5 (, 2,), 5 (2,,) and (,,). *8. [Tripos, 29/III/] A second-rank tensor T(y) is defined by T ij (y) = (y i x i )(y j x j ) y x 2n 2 da(x), where y is a fixed vector with y = a, n >, and the integration is over all points x lying on the surface of the sphere of radius a, centred on the origin. Explain briefly why T might be expected to have the form T ij = αδ ij +βy i y j, where α and β are scalar constants. how that y (y x) = a 2 ( cosθ), 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 Hence, by evaluating another scalar integral, determine α and β, and find the value of n for which T is isotropic.. If you left any unstarred questions on any of these sheets, attempt them over the Easter break.. Please send any corrections or comments to me at glt@cam.ac.uk