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MA231 Vector Analysis
Example Sheet 3
2010, term 1 Stefan Adams
This sheet contains some questions on the tail end of the vector analysis part of the course and some on the warm up sections of the complex variable part of the course. Hand in solutions to questions B1, B2, B3 and B4 by 3pm Monday of week 8.
A1 Calculations in spherical coordinates At the back of this example sheet are the formulae for spherical coordinates. Let v = r^2 sin ϑ ˆer + (^) r cos^1 ϑ ˆeϕ + (^1) r ˆeϑ. Calculate div(v), giving the answer in in spherical coordinates (r, ϕ, ϑ). Calculate ∇ × v giving the answer in the curvilinear basis ˆer , ˆeϕ, ˆeϑ.
A2 Algebra on C Algebraic manipulation of complex numbers is used in all the remaining parts of the course and you should practice if you feel at all rusty.
(a) Express the following numbers in polar form r(cos ϑ + i sin ϑ): i, −i, 1 + i, − 12 +
√ 3 2 i.
(b) Express the following in the form x + iy: (^) 6+2^1 i , i^7 ,
− 12 + i
√ 3 2
(c) Find both square roots of −8+6i. Then solve the quadratic equation z^2 +i
32 z − 6 i = 0. (d) Find all four solutions to z^4 = − 1. Plot them on the complex plane.
A3 Complex conjugates
(a) Give the complex conjugate z¯ for a complex number z both in cartesian form x + iy and in polar form r(cos ϑ + i sin ϑ). (b) Show vw = ¯v w¯ for complex numbers v, w.
A4 Curves in C Sketch the set {z ∈ C | z z¯ = 1} in the complex plane. Find a parametrisation ϕ : [0, 1] → C for this set.
A5 Convergence of complex series
(a) Calculate the radius of convergence of the following power series: (i)
n=0 2 −n^2 zn, (ii) ∑∞ n=
n! nn^ z
n.
(b) Show that the series
(3 + 4i)n(z − 4 i)n^ converges inside a certain ball in the complex plane. (c) Show that
n=
in i+n^2 is convergent and that^
n=
1 i+n is divergent. (d) Find the power series
cnzn^ for: (i) (^) i+^1 z , (ii) sinz z, (iii) cos(z + i). (e) Find the power series
cn(z − i)n^ for: (i) (^1) z , (ii) ez^.
A6 Complex special functions
(a) Show that sin z = e
iz (^) −e−iz 2 i. (b) Show that sin(2z) = 2 sin z cos z. (c) Show that sin(x + iy) = sin x cosh y + i cos x sinh y. (d) Find z so that sin z = 2. Can you find all such z?
B1 Stokes’s theorem, Spherical coordinates
(a) Let C be the curve formed by the intersection of the cylinder x^2 + y^2 = 1 in R^3 and the plane x + y + z = 1. Orient the curve C so that the tangent vector at (1, 0 , 0) has a negative z-component. Use Stokes’ theorem to calculate, for the vector field v(x, y, z) = (−y^3 , x^3 , z^3 ), the tangential line integral
C 〈v,^
Tˆ 〉.
(b) Check that the vectors ˆer , ˆeϕ and ˆeϑ are orthonormal with ˆer × ˆeϕ = ˆeϑ. (c) Calculate ∇T and ∆T for a radial symmetric scalar field T (r, ϕ, ϑ) = T (r).
Hint: You may use the formulae from the back of the assignment sheet for your solution.
B2 Complex numbers
(a) Determine the set of all complex numbers z ∈ C for which |z + 1| < 2 |z − i| holds, and draw a figure of that set. (b) Write the number z = 1 + i in polar form and determine z^7 with Moivre’s formula. (c) Determine all solutions of z^4 = 1 + i
3 in polar form. (d) Determine
3 i.
B3 Complex series
(a) Show that the series
n=
(1+i)^2 n n^4 (3z^ +^ i)
n (^) converges inside a certain ball in the complex plane. At what points on the boundary of the ball does the series converge? (b) Show that
n=
1 z+n^2 defines a continuous function on the set^ {^ z^ =^ x^ +^ iy^ |^ x >^0 }. (c) Find the power series
cn(z − a)n, and their radius of convergence, for each of the following functions around the given points: (i) z^2 with a = −i (ii) (^) z−^11 with a = i. (d) Show that cosh z is surjective.
B4 Special functions
(a) Show that sin(z + 2π) = sin(z) and sin(z + π 2 ) = cos(z) holds for all z ∈ C. (b) Consider the M¨obius transform f defined by
f (z) =
iz + 1 z + i
(i) Determine f (i), f (−i), f (0), and f (∞). (ii) Determine the image of the imaginary axis z = iy, y ∈ R, including ∞ under the mapping f. (iii) Show that the following equivalence holds:
|z − i| ≤ 1 ⇔ |f (z) +
i 3
C1 Gradient, divergence and curl in cylindrical coordinates
(a) Consider the value of the tangential line integral
C 〈∇f ,^ Tˆ 〉 ds for f (r, ϑ, z), written in cylindrical coordinates, along the line segment C joining the points (r, ϑ, z) to (r, ϑ + ∆ϑ, z). Apply the fundamental theorem of calculus for gradient vector fields to find ∇f · ˆeϑ. (The other components of ∇f can be found in a similar way.) (b) Consider the value of the surface integral
S 〈∇ ×^ v,^ Nˆ 〉 dS for v = vr ˆer + vϑ ˆeϑ + vz ˆer , written in the cylindrical curvilinear basis, over the quadrilateral joining the points (r, ϑ, z), (r, ϑ + ∆ϑ, z), (r + ∆r, ϑ, z) and (r + ∆r, ϑ + ∆ϑ, z). Apply Stokes theorem and by considering the circulation
∂S 〈v,^ Tˆ 〉 ds find the value of ∇×v·ˆez. (The other components of ∇ × v can be found in a similar way.)
spherical coordinates coordinate map:
(r, ϕ, ϑ) 7 →
r cos(ϕ) cos(ϑ) r sin(ϕ) cos(ϑ) r sin(ϑ)
basis vectors:
ˆer =
cos(ϕ) cos(ϑ) sin(ϕ) cos(ϑ) sin(ϑ)
(^) , ˆeϕ =
− sin(ϕ) cos(ϕ) 0
(^) , ˆeϑ =
− cos(ϕ) sin(ϑ) − sin(ϕ) sin(ϑ) cos(ϑ)
differential operators:
grad(f ) =
∂f ∂r
ˆer +
r cos ϑ
∂f ∂ϕ
ˆeϕ +
r
∂f ∂ϑ
ˆeϑ
div(v) =
r^2
∂r
r^2 vr
r cos ϑ
∂vϕ ∂ϕ
r cos ϑ
∂ϑ
cos(ϑ)vϑ
curl(v) =
r cos ϑ
( (^) ∂v ϑ ∂ϕ
∂ϑ
cos(ϑ)vϕ
ˆer +
r
( (^) ∂v r ∂ϑ
∂r
rvϑ)
ˆeϕ
r
∂r
rvϕ
cos ϑ
∂vr ∂ϕ
ˆeϑ
cylindrical coordinates coordinate map:
(r, ϕ, z) 7 →
r cos(ϕ) r sin(ϕ) z
basis vectors:
ˆer =
cos(ϕ) sin(ϕ) 0
(^) , ˆeϕ =
− sin(ϕ) cos(ϕ) 0
(^) , ˆez =
differential operators:
grad(f ) =
∂f ∂r
ˆer +
r
∂f ∂ϕ
ˆeϕ +
∂f ∂z
ˆez
div(v) =
r
∂r
rvr
r
∂vϕ ∂ϕ
∂vz ∂z
curl(v) =
r
∂vz ∂ϕ
∂vϕ ∂z
ˆer +
( (^) ∂v r ∂z
∂vz ∂r
ˆeϕ +
r
∂r
rvϕ
∂vr ∂ϕ
ˆez
