Vector calculus MA2VC MA3VC Solutions of the review exercises of Section 1.7

Transcription

1 Vector calculus MAVC MA3VC 4 5 Solutions of the review eercises of Section.7 Consider the following scalar and vector fields: f = z 3, g = cos+sin +, G = e z î+ ĵ, h = e cos, l = r 3, F = +zî++zĵ++ˆk, H = r î, L = r r, m = >, M = z zî+zĵ+3ˆk. Consider also the three following curves, defined in the interval < t < : a = t 3 tî+ t ˆk, b = t3î+t ĵ+ˆk, c = e t cosπtî+e t sinπtĵ. Answer the questions, tring to be smart and to avoid brute force computations whenever possible.. Compute gradient, Hessian and Laplacian of the five scalar fields. We onl need to compute several partial derivatives and combine them appropriatel. f = z 3 î+z 3 ĵ+3 z ˆk = M, Hf = z3 3 z z 3 z 3 6z, f = zz +3, 3 z 6z 6 z g = sin+cos + î+cos +ĵ, cos sin + sin + Hg = sin + 4sin +, g = cos 5sin +, h = e cosî e sinĵ, Hh = h e sin e sin h, h =, l = 3 + +z î+ 3 ĵ+zˆk, 6 z Hl = 6, l = 6, z m = î+ logĵ, +log Hm = +log log, m = + log.. Compute Jacobian, divergence, curl and vector Laplacian of the five vector fields. We use H = r î = zĵ ˆk, M = f, L = 4 r 4 see also Eercise.37. JF =, F =, F =, F =, JG = ze z, G =, G = ze z ĵ, G = z e z î+ĵ, JH =, H =, H = î, H =,

2 A. Moiola, Universit of Reading Vector calculus, solutions of review eercises of Section.7 JL = 3 + +z z +3 +z z, L = 5 r, z z + +z L =, L = r, J M = Hf, M = f, M = f =, M = f = f = z 3 +6 zî+zĵ+6z +6 ˆk. 3. Which of the fields are solenoidal and which are irrotational? Which are harmonic? The onl scalar harmonic field is h. F is solenoidal and irrotational. G is neither solenoidal nor irrotational. H is solenoidal onl. L is irrotational onl. M is irrotational onl. Moreover F and H are harmonic in the sense that all their components are harmonic. 4. Compute a scalar and a vector potential for F, a vector potential for H can 3 help ou? and a scalar potential for M. Can ou guess a scalar potential for L? Proceeding as in Eample.56, we obtain the following potentials: F = +z +z +λ = z î+ z ĵ+ ˆk, H = r î, L = 4 r 4 +λ, M = f +λ for an scalar λ. F z = z G = z H z = L z = Figure : Representation of some sections of the vector fields

3 A. Moiola, Universit of Reading Vector calculus, solutions of review eercises of Section.7 5. Show that G does not admit neither scalar nor vector potential. G is neither irrotational nor solenoidal, therefore the eistence of a scalar or a vector potential would entail a contradiction with the bo in Section.5 conservative irrotational, vector potential solenoidal or with identities and Show that H r and L r are orthogonal to each other at ever point r R 3. H r L r = zĵ ˆk r r = r +z z =. 7. Tr to graphicall represent the fields in 35. E.g. ou can draw a qualitative plot like those of Section.. for G on the plane =, for F, H and L on the plane z =. The suggested sections are plotted in Figure. Note that onl for G, H and L the plots are in the same form as those seen in the notes, since in these cases the fields are tangential to the considered planes. On the contrar, F is not tangent to the plane {z = }, thus it can not be represented as a two-dimensional field. You might prove that F is tangential to the plane {+ +z = }. 8. Demonstrate identities 3 and 5 for G; 4 for f; 6 for f and h; 7 for G and H; 8 and 3 for h and H. You can also demonstrate other identities of Propositions.44 and.46 for the various fields in 35. We demonstrate the suggested identities. In some cases, the best strateg is to first epand the terms on both sides of the differential identities and then manipulate them, aiming at a common epression. 3 G = ze z ĵ = ze z, 5 G = ze z ĵ = z e z î = ĵ ĵ+z e z î = G = G G, 4 f = z 3 î+z 3 ĵ+3 z ˆk = 6z 6z î+3 z 3 z ĵ+z 3 z 3 ˆk =, 6 fh = z 3 e cos = +e z 3 cosî+e cos sinz 3 ĵ+3e cosz ˆk = z 3 e cosî e sinĵ+e cos z 3 î+z 3 ĵ+3 z ˆk = f h+h f, 7 G H = e z î+ ĵ zĵ ˆk = z = zĵ+ ˆk = k+zĵ+ze z î ze z î+ ˆk = e z + zĵ ˆk+ z e z î+ ĵ z +zĵ ˆk ze z ĵ+e z î+ ĵ î = G H+ H G+ H G+ G H, 8 h H = e coszĵ e cosˆk = e sinz + = e cosî e sinĵ zĵ ˆk+ = h H+h H, 3

4 A. Moiola, Universit of Reading Vector calculus, solutions of review eercises of Section.7 3 h H = e coszĵ e cosˆk = e cos sin e cos î+e cosĵ+e coszˆk = e sinî+e cosĵ+ze cosˆk e cosî = e cosî e sinĵ zĵ ˆk e cosî = h H+h H. 9. Compute the total derivatives of the curves i.e. d a, d b d c and and tr to draw them. d a = dt3 tî+ t ˆk = 3t î tˆk, d b = dt3 î+t ĵ+ˆk = 3t î+tĵ, d c = d e t cosπtî+e t sinπtĵ = e t î+ ĵ cosπt πsinπt sinπt+πcosπt = e t i+πĵcosπt+ĵ πîsinπt. The curves and their derivatives are shown in Figure. From the plots of the curves we note that a is a loop, b is singular, c is a part of a logarithmic spiral. The total derivatives of a and b which are also curves have almost the same plot, the can be transformed into each other b a rigid motion. Despite this fact, a and b have quite different shapes. z at bt ct z d at/ d bt/ d ct/ 3 Figure : The images of the three curves and of their total derivatives. 4

5 A. Moiola, Universit of Reading Vector calculus, solutions of review eercises of Section.7. Compute the following total derivatives of the scalar fields along the curves: dh a You can either use the chain rule 34 or first compute the composition., df b Deriving the composition of the field with the curve: d h a t = det3 = e t3 t 3t note that a lies in plane =, d f b = dt7 = 7t 6, d l c = d e 3t cosπt e 3t sin 3 πt = e 3t 3cosπt 3sin 3 πt πsinπt 6πsin πtcosπt. and dl c. Alternativel, using the vector chain rule 34: d h a = h a d a = e cosî e sinĵ 3t î tˆk = e t3 t 3t, d f b = f b d b = z 3 î+z 3 ĵ+3 z ˆk 3t î+tĵ = t 4 î+t 5 ĵ+3t 7ˆk 3t î+tĵ = 7t 6, d l c = l c d c = 3 + +z î+ 3 ĵ+zˆk e t î+ ĵ cosπt πsinπt sinπt+πcosπt = e t 3cos πt+sin πt î+ cosπtsinπt 3sin πt ĵ e t î+ ĵ cosπt πsinπt sinπt+πcosπt = e 3t 3cosπt 3sin 3 πt πsinπt 6πsin πtcosπt. 5