6.14 Review exercises for Chapter 6

Transcription

1 6.4 Review exercises for Chapter Review exercises for Chapter 6 In Exercise 6., B is an n n matrix and ϕ and ψ are both - forms on R 3 ; v and w are vectors 6. Which of the following are numbers? Identify those that are not. a. v w b. dx dx 2( v, w) c. v w d. det B e. rank B f. tr B g. dim A k (R n ) h. v i. A k (R k ) j. ϕ ψ( v, w) k. R f(x) dx l. sgn(σ) in R 3 ; f is integrable. 6.2 Let F be a vector field in R n, let v,..., v n be vectors in R n, and let ϕ be the (n )-form on R n given by ϕ( v,..., v n ) = det[ F (x), v,..., v n ]. Write ϕ as a linear combination of elementary (n )-forms on R n, in terms of the coordinates of F. 6.3 Use Definition 6..2 to write the wedge product ϕ ψ( v, v 2, v 3, v 4), where ϕ is a -form and ψ is a 3-form, as a combination of values of ϕ and ψ evaluated on appropriate vectors (as in equations and 6..32). 6.4 Set up each of the following integrals of form fields over parametried domains as an ordinary multiple integral. a. [γ(i)] y2 dy + x 2 d, where I = [,a] and γ(t) = t 2 + t 2 b. ( ) sin [γ(u)] y2 dx d, where U = [,a] [,b], and γ uv = t3 ( u 2 ) v uv v 4 6.5( Find ) a -form field on R 2 whose sign orients the circle of radius centered 3 at in the clockwise direction. 6.6 Find an orientation Ω for the surface S R 3 given by x 2 + y =. 6.7 Consider the manifold S 3 R 4 of equation x 2 + x x x 2 4 =. a. Show that sgn dx dx 2 dx 3 is not an orientation of S 3. b. Show that Ω x( v, v 2, v 3) = sgn det[x, v, v 2, v 3] is an orientation. 6.8 a. Show that the locus M R 4 given by x 2 + x x x 2 4 = is a smooth manifold. b. Find a 3-form field whose sign orients M. ( ) ( ) cos θ sin θ 6.9 In Example we saw that γ (θ) = and γ 2(θ) = give sin θ cos θ opposite orientations. Confirm (Proposition 6.4.8) that det[d(γ 2 γ )] <. 6. Let = x + iy, 2 = x 2 + iy 2 be coordinates in C 2. Compute the integral of dx dy + dx 2 dy 2 over the part of the locus of equation 2 = k where <, oriented by sgn dx dy. 6. For the following -forms, write down the corresponding vector field. Sketch the vector field. Describe a path over which the work of the -form would be small. Describe a path over which the work would be large. a. (x 2 + y 2 ) d, on R 3 b. y dx x dy d, on R 3

2 7 Chapter 6. Forms and vector calculus F x y = y y Vector field for Exercise a. In R 2, a vector field defines two -forms: the work and the flux. Show that they are related by formula ([ ] ) W F ( v) =Φ F v. [ ] b. What does the transformation correspond to geometrically? c. Can you explain why the work and the flux on R 2 are related by the formula in part a? 6.3 Find the flux of the vector field F shown in the margin, through S, where S is the part of the cone = x 2 + y 2 where x, y, x 2 + y 2 R, and S is oriented by the inward-pointing normal (i.e., the flux measures flow into the cone). 6.4 Let S be the part of the surface of equation = sin xy + 2 where x 2 + y 2 and x, oriented by the upward-pointing normal; let F = F through S?. What is the flux of x + y 6.5 Consider the manifold S 3 R 4 of equation x 2 +x 2 2 +x 2 3 +x 2 4 =, oriented by Ω x( v, v 2, v 3) = sgn det[x, v, v 2, v 3]. Let X be the subset of S 3 where x 4. a. Show that X is a piece-with-boundary of S 3. b. Find a basis for the tangent space T x( X) at the point x = X which is direct for the boundary orientation. 6.6 a. Compute the derivative of xy d from the definition. b. Compute the same derivative using the formulas given in Theorem 6.7.4, stating clearly at each stage what property you are using. 6.7 a. Let ϕ = xy dy. Compute from the definition the number dϕ P ( e 2, e 3). 2 3 Exercise 6.8 gives another way to derive equation b. What is dϕ? Use your result to check the computation in part a. x 6.8 Let r =.. x n be the radial vector field in R n. a. Show that d(φ r )=n(dx dx n). b. Let B n () and S n be the unit ball and the unit sphere in R n, the ball with the standard orientation and the sphere with the boundary orientation. Use Stokes s theorem to prove ) vol n (B n () = n voln (Sn ).

3 6.9 Using Theorem 6.8.3, prove the equations 6.4 Review exercises for Chapter 6 7 curl (grad f) = and div(curl F )= for any function f and any vector field F (of class at least C 2 ). 6.2 a. For what vector field F is the -form on R 3 x 2 dx + y 2 dy + xy d the work form field W F? b. Compute the exterior derivative of x 2 dx + y 2 dy + xy d using Theorem Show that it is the same as Φ F. Exercise 6.2, part b: The subscript on Φ may be hard to read. It is r 2m r. 6.2 a. There is an exponent m such that (x 2 + y ) m x y = ; find it. *b. More generally, there is an exponent m (depending on n) such that the x (n )-form Φ r 2m r has exterior derivative, when r is the vector field.., and x n r = r. Can you find it? (Start with n = and n = 2. ) 6.22 a. Find the unique polynomial p such that p() = and such that if then dω = dx dy d. ω = x dy d 2p(y) dx dy + yp(y) d dx, b. For this polynomial p, find the integral ω, where S is that part of the S sphere x 2 +y = where 2/2, oriented by the outward-pointing normal a. Compute the exterior derivative of the 2-form ( ) ϕ = x dy d + y d dx + dx dy. (x 2 + y ) 3/2 b. Compute the integral of ϕ over the unit sphere x 2 + y =, oriented by the outward-pointing normal. c. Compute the integral of ϕ over the boundary of the cube of side 4, centered at the origin, and oriented by the outward-pointing normal. d. Can ϕ be written dψ for some -form ψ on R 3 {}? 6.24 Let S be the surface of equation =9 y 2, oriented by the upwardpointing normal. a. Sketch the piece X S where x,, and y x, indicating carefully the boundary orientation. b. Give a parametriation of X, being careful about the domain of the parametriing map and whether it is orientation preserving. c. Find the work of the vector field x around the boundary of X Let U R 3 be a subset bounded by a surface S, to which we will give the boundary orientation. What relation is there between the volume of U and the flux S Φ? x y

4 Exercise 6.27: Equation relates the coordinates x, y,, t of a frame of reference moving at speed v in the direction of the x- axis with respect to a frame with coordinates x,y,,t. Equation gives the electromagnetic field of a charge at rest. It is often said that the magnetic field is a relativistic side effect of the electric field of moving charges. Exercise 6.27 illustrates this: with the particle at rest the magnetic field is, but a moving charge will deflect a magnetic needle. Note the v/c in the magnetic field of the moving charge. At ordinary (human) speeds, the magnetic field will be extremely small. Exercise 6.3: The surface S might be very complicated, for instance, the boundary of a cloud. Understanding the charge on the boundary of a cloud is essential for understanding lightning and thunderstorms. 72 Chapter 6. Forms and vector calculus 6.26 Let F be the vector field F x F(x, y) y = F 2(x, y), where F and F 2 are defined on all of R 2. Suppose D 2F = D F 2. Show that there exists a function f : R 3 R such that F = f Show that the electromagnetic field of a charge q moving in the direction of the x-axis at constant speed v is qγ E = ) x vt 3/2 y 4π ((γx γvt) 2 + y B = v c qγ ) 3/2, where γ = 4π ((γx γvt) 2 + y y v2 /c Find a -form ϕ such that dϕ = y d dx x dy d Let U R 3 be a 3-dimensional piece-with-boundary. a. What does Stokes s theorem say about df? b. What does Stokes s theorem say about U U dm? 6.3 Let S R 3 be a smooth oriented surface, and X S a 2-dimensional piece-with-boundary. Let I =[t,t ] be a time interval. a. Show that V = S I is a 3-dimensional piece-with-boundary of R 4. b. What does Stokes s theorem say about df? Show that if we divide the integral by t t and let t tend to t, we find Faraday s law. c. What does Stokes s theorem say about dm? Show that if we divide the integral by t t and let t tend to t, we find Ampère s law. 6.3 Let F n = x be the vector field defined in Example 6.7.7, and let S n x n R be the sphere x = R, oriented by the outward-pointing normal. Show that S n Φ Fn does not depend on R, and that it equals vol n (S n ). R 6.32 Compute the integral Φ F, where F x y = x2 y y, and S is S ( 2 )xy the part of the parabolic cylinder of equation y = 9 x 2 where y and, oriented by the transverse vector field e Let V, W, be two finite-dimensional real vector spaces, oriented by V V Ω V : B(V ) {±} and Ω W : B(W ) {±}. Show that Ω V W ({v}, {w}) def = Ω V ({v})ω W ({w}) orients V W.

5 6.4 Review exercises for Chapter 6 73 Exercise 6.34 asks you to show that df = and to compute dm. In both cases, these exterior derivatives are in the sense of distributions ; see Remark 6.2. (page 676). **6.34 Let r = x y and r = r = x 2 + y Show that F = qw r/r 3 cdt is an electromagnetic field. What are the corresponding charges and currents? Remarks.. Exercise 6.34 isn t a straightforward computation. For F to be an electromagnetic field, we must have df = everywhere, and it isn t clear what this means when r =. In Example 6.3. we saw that trouble may be hiding at points where a form is not defined. We deal with this as follows. A 3-dimensional piece-with-boundary X R 4 will be called r-adapted if locally near points where r = it represents t as a function of x, y, and ; in that case we write X ɛ = X {r<ɛ} and we define inn X ɛ X ɛ to be the subset where r = ɛ, with the boundary orientation. If ϕ is a 2-form on R 4 {r =} and X R 4 is an r-adapted oriented 3-dimensional piece-with-boundary, we define ( ) dϕ def = lim dϕ ϕ. X ɛ X ɛ innx ɛ If ϕ is well defined where r =, then, by Stokes s theorem, dϕ = dϕ + dϕ = dϕ ϕ, X X ɛ X X ɛ X ɛ innx ɛ so the formula is correct in that case, and otherwise the boundary term captures whatever is hiding on {r =}. 2. Note that d ( ) = W r/r 3. r This contrasts with Example There we had d arctan y x = x dy y dx x 2 + y 2. But arctan y x is not a well-defined function on R2 {}, whereas /r is a well-defined function on R 3 {}. Forms were defined to be integrands, so it is quite reasonable to define df via its integrals over oriented pieces-with-boundary. It is also quite reasonable to restrict to r-adapted pieces: pieces-with-boundary for which, locally near points where r =, the piece X is the graph of a map expressing t as a function of x y. Pieces that are not r-adapted intersect the locus r = in exceptional ways, and it may be difficult to say what share of whatever nastiness is hiding there is carried by X. Such pieces are exceptional; by budging them arbitrarily little we can make them r-adapted.