EXERCISES FOR MATHEMATICS 138AWINTER 2010The references denote sections of the text for the course:B. O’Neill, Elementary Differential Geometry (Second Edition). Academic Press, SanDiego, CA, 1997. ISBN: 0–125–26745–2.I . Classical Differential Geometry of CurvesI.1 : Cross products(O’Neill, § 2.2)Additional exercises1. Verify that the cross product of vectors in R3satisfies the Jacobi identity:a × (b × c) + b × (c × a) + c × (a × b) = 0 .2. Let u, v, and w be orthonormal vectors in R3such that w = u × v (cross product).Compute v × w and w × u.Note. The preceding result has the following consequence: Suppose that T is a lineartransformation on R3which takes the standard unit vectors e1, e2, and e3to the orthonormalvectors u, v, and w respectively. Then we have T (x × y) = T (x) × T (y) for all vectors x,y in R3.— The basic idea is merely that if a linear transformation preserves cross products on a basis, thenby the Distributive Law of Multiplication it must preserve all cross products.I.2 : Parametrized curves(O’Neill, § 1.4)O’Neill, § 1.4 (2ndEd. pp. 21–22): 2, 8Additional exercises1. Find a parametrized curve α(t) which traces out the unit circle about the origin in thecoordinate plane and has initial point α(0) = (0, 1).2. Let α(t) be a parametrized cure which does not pass through the origin. If α(t0) isthe point in the image that is closest to the origin and α0(t0) 6= 0, show that α(t0) and α0(t0) areperpendicular.3. If Γ is the figure 8 curve with parametrization γ(t) = (3 cos t, 2 sin 2t), where 0 ≤ t ≤ 2π,find a nontrivial polynomial P(x, y) such that the image of γ is contained in the set of points where1P (x, y) = 0. [Hint: Recall that sin 2t = 2 sin t cos t and sin2t + cos2t = 1; the latter implies thatcos2t = sin2t cos2t + cos4t.]4. Two objects are moving in the coordinate plane with parametric equations x(t) =(t2− 2,12t2− 1) and y(t) = (t, 5 − t2). Determine when, where, and the angle at which the objectsmeet.5.∗Prove that a regular smooth curve lies on a straight line if and only if there is a pointthat lies on all its tangent lines.I.3 : Arc length and reparametrization(O’Neill, §§ 1.4, 2.2)O’Neill, § 2.2 (2ndEd. pp. 55–56): 3–5, 10, 11Additional exercises1. Prove that a necessary and sufficient condition for the plane N · x = 0 to be parallel tothe line x = x0+ t · u is for N and u to be perpendicular.2.∗Suppose that F (x, y) is a function of two variables with continuous partial derivativessuch that F (a, b) = 0 but∂∂yF (a, b) 6= 0, and also suppose that g(x) is a function such that the setF (a, b) = 0 has the parametrization y = g(x) over the interval [a − h, a + h]. Prove that the lengthof this curve is given by the integralZa+ha−h|∇F (x, g(x) )||F2(x, g(x) )|dxwhere F2denotes the partial derivative with respect to the second variable. [Hint: Use the implicitdifferentiation formula for g in terms of the partial derivatives of F .]3.∗(a) Given a > 0, consider the set of all continuously differentiable real valued functionsf on [0, 1] such that f(0) = 0 and f (1) = a > 0. Define L(f ) by the formula L(f) =Ra0|f0(t)| dt .Show that the minimum value of L(f ) is a, and if equality holds then f0is everywhere nonnegative.[Hints: Since f0≤ |f0| a similar inequality holds for their definite integrals. This inequality ofintegrals is strict if and only if f0(t) < |f0(t)| for some t, which happens if and only if f0(t) < 0 forthat choice of t.](b) Let γ(t) be a regular smooth curve in R2or R3such that γ(0) = 0 and γ(1) is the first unitvector e1with first coordinate equal to 1 and the other coordinate(s) equal to zero. Prove that thelength of γ is at least 1, and equality holds if and only if γ is a reparametrization of the straightline segment joining γ(0) to γ(1). [Hint: Write γ = (x, y, z) in coordinates, let β = (x, 0, 0) andexplain why the length of β is less than or equal to the length of γ, with equality if and only ify = z = 0. Apply the first part of the problem to show that x(t) defines a reparametrization of theline segment joining the endpoints.Note. The file greatcircles.pdf in the course directory proves the corresponding result forcurves of shortest length on the sphere; namely, these shortest curves are given by great circle arcs.As noted in the cited document, the argument uses material from later units in this course, and atseveral points it is “somewhat advanced.” An more elementary proof for the distance minimizingproperty of great circles can be derived fairly quickly from the first theorem in the online document2http://math.ucr.edu/∼res/math133/polyangles.pdfand the standard formula which states that the length of a minor circular arc is equal to the productof the radius of its circle times the measure of its central angle expressed in radians.4. (a) If an object is attached to the edge of a circular wheel and the wheel is rolled alonga straight line on a flat surface at a uniform speed, then the curve traced out by the object is acycloid (there is an illustration in the file cyc-curves.pdf). If the circle has radius a > 0 and itscenter starts at the point with coordinates (0, a), then the object starts at (0, 0) and its parametricequations are given by the classical formula x(t) = a · (t − sin t, 1 − cos t).Find the length of the cycloid over the parameter values 0 ≤ t ≤ 2π.(b) In the classical geocentric theory of planetary motion which appears in the Almagestof Claudius Ptolemy (c. 85–165), there is an assumption that planets travel in curves given byepicycles. The simplest examples of these involve circular motion where the center of the circle ismoving in a circular path around a second circle (this is similar to the motion of the moon aroundthe earth, which is given by an ellipse while the earth itself is moving around the sun by a largerellipse; an illustration appears in cyc-curves.pdf; in the full theory one also allowed the secondcircle to move around a third cycle, and so on). A typical example is given by the following formula,in which the first circle has radius14, the second one is the unit circle about the origin, and thebody rotates four times around the small circle as the large circle makes one revolution around itscenter:x(t) = (cos t, sin t) +14(cos 4t, sin 4t)Find the length of this curve over the parameter values 0 ≤ t ≤ 2π.Notes. For both parts of these exercises the standard formulas for | sin12θ | and |