Let γ(t) be the curve given by
Let γ(t) be a differentiable curve in R". If there is some differentiable function F : Rn R with F(γ(t)) C constant, show that DF(γ(t))T is orthogonal to the tangent vector γ(t).
component functions denoted by y(t) ((t), y(t), z(t). The plane curve t) = (x(t), y(t)) represents the projection of γ onto the xy-plane. Assume that γ, is nowhere parallel to (0,0,1), so that γ is regular. Let K and K denote the curvature functions of y and 7 respectively. Let v,v denote the velocity functions of γ and γ respectively. (1) Prove that R 2RV. In particular, at a time t e I for which v(t) lies in the ay-plane,...
17 Proposition. Let γ be a rectifiable curve and suppose that f is a function continuous on (y). Then : 7) sup [lfe): z E (c) If ce C then J,f(z) dz -Jyef(z-c) dz 17 Proposition. Let γ be a rectifiable curve and suppose that f is a function continuous on (y). Then : 7) sup [lfe): z E (c) If ce C then J,f(z) dz -Jyef(z-c) dz
EXERCISE 1.63. The unsigned curvature of a plane curve γ(t)-(x(t), v(t)) can be computed with Proposition 1.46 by considering it to have a vanishing third component function: γ(t) (x(t),y(t),0). Use this method to compute the curvature function of the parabola y(t) (t, t2). How can the signed curvature be determined from this approach? EXERCISE 1.63. The unsigned curvature of a plane curve γ(t)-(x(t), v(t)) can be computed with Proposition 1.46 by considering it to have a vanishing third component function:...
Let M be a surface and 11 a plane that intersects M in a curve γ. Show that y is a geodesic if II is a plane of symmetry of M, i.e., the two sides are mirror images. Let M be a surface and 11 a plane that intersects M in a curve γ. Show that y is a geodesic if II is a plane of symmetry of M, i.e., the two sides are mirror images.
5, (25 points 4 pages max) Suppose that γ(t) = (x(t), y(t)) is a smooth (infinitely differentiable) plane curve. For curves such that lh'(t) 0, the (signed) curvature is defined to be the quantity K(t) (a) Suppose the curve γ(t) is the graph of a function, ie x(t)-t and y(t) f(t) for some function f. Write the formula for the curve in this case. Suppose you were at a critical point of the graph of f. What does the curvature...
Problem 4.9 (e) /(z) = and γ is parametrized by r(t), 0 z + t 1, and satisfies Imr(t)> 0, r(0) -4 + i, and γ(1) 6 + 2i (f) f(s) sin(z) and γ is some piecewise smooth path from 1 to π. 4.2 and the fact that the length of γ does not change under 4.9. Prove Proposi reparametrization. (Hint: Assume γ, σ, and τ are smooth. Start with the definition off, f, apply the chain rule to σ...
sin(s) cos(t)] Let S be the unit sphere, with the usual parameterization γ(st)-|sin(s)sin(t) cos(s) Let w zdz Λ dy. Find w. sin(s) cos(t)] Let S be the unit sphere, with the usual parameterization γ(st)-|sin(s)sin(t) cos(s) Let w zdz Λ dy. Find w.
(sin(π/z) -1dd 2. Compute the integral: sin(π/s)-.--d γ is the cl γ is the curve shown in the 2. where 721-1 following figure: arked points on the coordinate axes correspond to T,-T, 2, 2. (sin(π/z) -1dd 2. Compute the integral: sin(π/s)-.--d γ is the cl γ is the curve shown in the 2. where 721-1 following figure: arked points on the coordinate axes correspond to T,-T, 2, 2.
EXERCISE 1.52. If γ : [a,b]+ R2 is a closed plane curve and r den its rotation index, prove that 16 Ks(t) dt 2π otes lrohra svetem to ơranh the sim EXERCISE 1.52. If γ : [a,b]+ R2 is a closed plane curve and r den its rotation index, prove that 16 Ks(t) dt 2π otes lrohra svetem to ơranh the sim