Lent 2018 VECTOR CALCULUS EXAMPLES 1 G. Taylor

Transcription

1 Lent 28 ECTOR CALCULU EXAMPLE G. Taylor A star means optional and not necessarily harder.. The curve given parametrically by x(t) = (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) for α θ β, with a function f C [α,β]. how that the length of the path is β (f(θ) ) 2 ( L = + f (θ) ) 2 dθ. α The curvesr = aθ, r = ae bθ and r = a(+cosθ)arecalledan Archimedean spiral, alogarithmic spiral and a cardioid, respectively. For a,b > and θ 2π, sketch the curves and find their lengths. 4. Given a function f(x) 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 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. Note/hint: in spherical polars, for a function f of r and θ only, f = f r e r + f r θ e θ. 6. Find the equation of the plane which is tangent to the surface z = 3x 2 ysin( π 2x) at the point x = y =. Taking 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? 7. 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 (,,) 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? 8. Let F = (3x 2 y 2 z,2x 3 yz,x 3 y 2 ) and G = (3x 2 yz 2,2x 3 yz,x 3 z 2 ) be vector fields. how that one of these is a conservative field and that the other is not, and find a scalar potential for the one that is. 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 2t) with t.

2 9. (i) 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 C. ( ) (ii) Let F(x) = y x,,. how explicitly that x 2 +y 2 x 2 +y 2 C F dx = 2π, where C is traversed in the direction of increasing t. (iii) Find a scalar function f such that f = F. Comment on this, given (ii). *. 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.. Use the substitution x = ρcosφ, y = 2ρsinφ 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 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 areaenclosed by C is A. By calculating both integrals explicitly, show that where C is traversed anticlockwise. C (x 2 ydx+xy 2 dy) = A (y 2 x 2 )da = 4 5 a4, 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 da (x 2 +y 2 ) /2. *4. (a) The curve x 3 +y 3 3axy = with a > is called the Cartesian Leaf. Using polar coordinates, sketch the Cartesian Leaf and find the area bounded by it in the first quadrant. (b) 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 Let T be the tetrahedron with vertices at (,,), (,,), (,,) and (,,). Find the volume of T, and also the centre of volume, given by xd. T 2

3 6. 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 2 3 πρ a 3 (secα ). You can use spherical or cylindrical polars for the calculation, but there are alternative coordinates which make the limits of the integrals nicer. *7. [Tripos, 25/III/2] Express the integral I = dx dy x 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. dz xe Ax/y Bxy Cyz 2A(A+B)(A+B +C). 3

4 Lent 28 ECTOR CALCULU EXAMPLE 2 G. Taylor A star means optional and not necessarily harder.. A circular helix is given by x(t) = (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. (a) how that a curve in the plane 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 *(c) 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. 3. (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 vector fields on R 3 : rx, a(b x), a x, x a x a 3, where r = x, and a and b are fixed vectors. Use part (a) where you can. Thinking about the definitions of divergence and curl, could you have guessed any of the answers in advance? 4. 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). Feel free to omit any of (i), (ii), (iii) that were proved in lectures. 5. 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. Find the most general scalar potential for H, and hence calculate the line integral H dx from the point (,,) to the point (,π/2,π/4). 6. erify 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. erify also that u is also solenoidal and find a vector potential for it in the form ψk, for some function ψ.

5 7. (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 B(x). how that the curvature of the curve is given by B 3 B (B )B. 8. 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 that the divergence theorem holds. Recall that volume flux of fluid = v d. 9. [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.. 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 F d. * erify your result by calculating the integral directly.. erify tokes theorem for the vector field F(x) = (y, x,xyz) and the open surface given in cylindrical polar coordinates by ρ+z = a, z, where a >. 2. By applying the divergence theorem to the vector field c B, where c 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 B(x) = (z,,) and is the cuboid x a, y b, z c. 3. By applying tokes theorem to the vector field c B, where c 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 B(x) = x and A is the unit square in the (x,y) plane with opposite vertices at (,,) and (,,). 4. [Tripos, 22/III/9] tate tokes theorem for a vector field B(x) on R 3. Consider the surface defined by z = x 2 +y 2, 9 z. ketch the surface and calculate the area element d in terms of suitable coordinates or parameters. For the vector field B = ( y 3,x 3,z 3 ), compute B and calculate I = ( B) d. Use tokes Theorem to express I as an integral over and verify that this gives the same result. 2 A

6 *5. [Tripos, 2/III/] tate the divergence theorem for a vector field u(x) in a region bounded by a piecewise smooth surface with outward normal n. how, by suitable choice of u, that for a scalar field f(x). f d = Let be the paraboloidal region given by z and x 2 +y 2 +cz a 2, where a and c are positive constants. erify that ( ) holds for the scalar field f = xz. *6. [Tripos, 22/III/] The first part of this question was to prove question 4(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 f d (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. *7. (a) By considering A(x) = show that if B = then A = B. (b) By considering ϕ(x) = show that if x ( B) = then ϕ = B. tx B(tx) dt, x B(tx) dt, how that B(x) = r 2 c x, where c is a non-zero constant vector, satisfies B and x ( B) =. Why does this not contradict the usual necessary and sufficient condition for B to be the gradient of a scalar? 3

7 Lent 28 ECTOR CALCULU EXAMPLE 3 G. Taylor A star means optional and not necessarily harder.. (a) Using the operator in Cartesian coordinates and in spherical polar coordinates, calculate the gradient of the scalar field ψ = z = rcosθ 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 B = ye x +xe y = ρe φ = rsinθe φ, by applying the standard formulae in Cartesian, cylindrical, and spherical coordinates. 2. 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 A and A, where A = A ρ e ρ +A φ e φ +A 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. *3. Using the formula for in appropriate coordinates, prove the following two results. (a) 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) =. (b) The field f(x) has spherical symmetry about the origin if and only if x f =. Explain geometrically why these results make sense. 4. The vector field B is given in cylindrical polar coordinates (ρ,φ,z) by B = ρ e φ, for ρ. Using the formula derived in question 2, show that B = for ρ. Calculate C B dx with C the circle given by ρ = R, φ 2π, z =. Why does tokes theorem not apply? 5. 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 =. Explain why this result makes sense, given the two conditions on F. 6. 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.

8 *7. Let be a volume with boundary, and let P,Q be two solenoidal vector fields. how that (Q 2 P P 2 ( Q) d = Q ( P) P ( Q) ) d. 8. (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, 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() =. 9. (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) = sinθ. 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.. 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 (i) direct solution of Poisson s equation, using formulae from question 8, (ii) Gauss s flux method, *(iii) the integral form of the general solution of Poisson s equation, ϕ(y) = 4π ρ(x) x y 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. (Note that you are asked to find the gravitational field, not the potential.) *. [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. 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. 2

9 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 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?) 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 (i.e., solves Laplace s equation) 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. *6. [Tripos, 25/III/9] The first part of this question was essentially question 4 above, so I have omitted it here. 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 = ( ) 3

10 Lent 28 ECTOR CALCULU EXAMPLE 4 G. Taylor A star means optional and not necessarily harder.. 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. 2 3 Check your calculations are correct by finding α, ω and B for the matrix A = In a particular system of Cartesian coordinates x i, the vector field u i has the components u = x x 2 2, u 2 = x 2 x 2 3, u 3 = x 3 x 2. Express the tensor u i / x j as a linear combination of ε ijk w k (where w k is a vector to be determined) and a symmetric tensor e ij. erify that (,,) is one of the principal axes of e ij at the point x = 2, x 2 = 3, x 3 =, and find the others. *5. [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. 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.

11 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.. 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 4 e γr2 x i x j x k d, *(iii) r 5 e γr2 x i x j x k x l d.. (a) 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. *(b) Prove the following result: δ ip δ iq δ ir 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. (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. 4. 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

12 *5. [Tripos, 2/III/9] Write down the most general isotropic tensors of rank 2 and 3. Use the tensor transformation law to show that they are, indeed, isotropic. Let be the sphere r a. Explain briefly why T i...i n = x i...x in d is an isotropic tensor for any n. Hence show that x i x j d = αδ ij, x i x j x k d =, x i x j x k x l d = β(δ ij δ kl +δ ik δ jl +δ il δ jk ) for some scalars α and β, which should be determined using suitable contractions of the indices or otherwise. Deduce the value of of x (Ω x)d, where Ω is a constant vector. *6. [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.. 3