Problems 1. Let F(x , y) = sin(y2) + 2x , e3x + 3x y be a vector field in R2. Let C be the curve in R2 described in polar form by r = cos(2 ) for 0 /4. A particle moves along the curve C from (1,0) to (0,0). You will try two different ways to compute R C F · n ds, the normal flow of F across C . (1a) Express the normal flow as a single variable integral using a parametrization of C . Do not compute it. MAT237 Problem Set 8 – Page 2 of 10 April 1, 2022 (1b) Use Green’s theorem to compute the normal flow. Youmay useWolframAlpha to calculate single variable integrals; indicate when you have done so. Hint: Close the loop. MAT237 Problem Set 8 – Page 3 of 10 April 1, 2022 2. Multivariable calculus has shown how you can do calculus with all of your linear algebra. Now, near the end of your journey, it is time to do linear algebra with all of your calculus (in two dimensions). Let U R2 be an open set. Let C1(U) be the set of real-valued functions f : U ! R with infinitely many partial derivatives; that is, @ f exists and is continuous on U for all multi-indices 2 N2. The space of C1 scalar functions V = C1(U) and space of C1 vector fields V 2 = V V can each be thought of as a space of vectors. For example, the zero function belongs to V and acts like the zero vector. Moreover, any linear combination in V also belongs to V . Similar statements hold true for V 2. (2a) You can view the differential operators ‘grad’ and ‘curl’ as linear transformations on these spaces. Gradient is a linear map of C1 scalar functions to C1 vector fields. That is, grad : V ! V 2 is a linear map. Hence, if f 2 V then grad( f ) 2 V 2. Two-dimensional curl is a linear map of C1 vector fields to C1 scalar functions. That is, curl : V 2! V is a linear map. Hence, if F 2 V 2 then curl(F) 2 V . Prove that curl is a linear map from V 2 to V . You may assume that the partial derivative operators @1 : V ! V and @2 : V ! V are linear maps. (2b) The image of the gradient is contained in the kernel of curl. That is, img(grad) ker(curl). There are two ways to prove this fact: by direct boring calculation or by the “one true proof”. Prove that img(grad) ker(curl) by a direct boring calculation with partial derivatives. MAT237 Problem Set 8 – Page 4 of 10 April 1, 2022 (2c) The “one true proof” of img(grad) ker(curl) relies upon the fundamental theorem of line integrals and Green’s theorem. Here is such an attempt to prove this containment. 1. Let F 2 img(grad) and p 2 U. Then 8″ > 0, I @ B”(p) (F · T ) ds = 0. 2. =) 8″ > 0, ZZ B”(p) (curl F) dA= 0 =)
lim “!0+ 1 area(B”(p)) ZZ B”(p) (curl F) dA = 0 3. =) (curl F)(p) = 0 =) F 2 ker(curl) There are no serious errors but it is terribly written. Rewrite this into a well-written justified proof. Do not use PS7 Q4 or Lemma 8.6.7. Hint: Use FTLI, Green’s, and integral MVT. (Revised 2022-03-28) MAT237 Problem Set 8 – Page 5 of 10 April 1, 2022 (2d) The images may or may not equal the kernels in (2b). It depends on the topology of U R2. Give an example of a set U = U1 where img(grad) = ker(curl). Give an example of a set U = U2 where img(grad) 6= ker(curl). Briefly justify each of your examples. Hint: See Problem Set 7. (2e) All of your observations about grad and curl can be beautifully encapsulated in this elegant diagram. V V 2 V grad curl At first glance, this appears to just be a composition of maps but if you dig a bit deeper, you will notice it actually captures all of vector calculus in R2. Take an arbitrary element at the leftmost V in the diagram. Map it once to V 2 and then map it again to the next V . The element has moved two stages to the right. What happened to this element How do the two core theorems of vector calculus in R2 relate to this phenomenon Explain in two to three full sentences using the previous parts of this question. MAT237 Problem Set 8 – Page 6 of 10 April 1, 2022 3. Fix 0< b < a. Define the map G : [0,2 ] [0,2 ]! R3 by G(s, t) = ((a+ b cos t) cos s, (a+ b cos t) sin s, b sin t). Define the set T = imgG R3 so T is a torus. See this Math3D demo for an illustration. (3a) Show that T is a surface parametrized by G. MAT237 Problem Set 8 - Page 7 of 10 April 1, 2022 (3b) Find the surface area of T . MAT237 Problem Set 8 - Page 8 of 10 April 1, 2022 4. Let G : U ! R3 and H : V ! R3 be parametrizations of a surface S R3. Assume G and H are C1 and {@1G,@2G} are linearly independent on the interior of their domains. Assume there exists a diffeomorphism : A! B such that U A and V B and G = H ', where ' = |U : U ! V . If detD > 0, then show 8u 2 Uo, v 2 V o,G(u) = H(v) =) @1G @2G(u)||@1G @2G(u)|| = @1H @2H(v) ||@1H @2H(v)|| . In other words, the unit normal is invariant under reparametrization. (Revised 2022-03-26) MAT237 Problem Set 8 – Page 9 of 10 April 1, 2022 5. An electric field generated by a wire along the z-axis is given by F(x , y, z) = x x2+y2 , y x2+y2 , 0 . Fix H,R > 0. Compute the flux of F across the cylinder S = {(x , y, z) 2 R3 : x2 + y2 = R2, 0 z H} oriented with unit normal pointing radially away from the z-axis. (Revised 2022-03-25) MAT237 Problem Set 8 – Page 10 of 10 April 1, 2022