CALCULUS III (MATH 243): REVIEW SHEET I. Chapter 12. (a) Limits, derivatives and antiderivatives of vector-valued functions. (b) Initial value problems for vector functions. (c) Arc length in three dimensions. Arc length parametrizations of curves. (d) Curvature. Circle of curvature. Center of curvature. Radius of curvature. (e) Frenet-Serret frames. T; N; B; κ, τ. Frenet-Serret equations 2T3 2 0 κ 03 2T3 d 4N5 = 4−κ 0 τ5 4N5 ds B 0 −τ 0 B (f) Osculating plane, normal plane, rectifying plane. (g) Tangential and normal components of acceleration. (h) Velocity and acceleration in polar and cylindrical coordinates. (i) Kepler's Laws. General advice on preparing for a math test. Be prepared to demonstrate understanding in the following ways. (i) Know the definitions of new concepts, and the meanings of the definitions. (ii) Know the statements and meanings of the major theorems. (iii) Know examples/counterexamples. (The purpose of an example is to illustrate the extent of a definition or theorem. The purpose of a counterexample is to indicate the limits of a definition or theorem.) (iv) Know how to perform the different kinds of calculations discussed in class. (v) Be prepared to prove elementary statements. (Understanding the proofs done in class is the best preparation for this.) (vi) Know how to correct mistakes made on old HW. Some formulas. (1) x = r cos(θ); y = r sin(θ); r = px2 + y2; θ = tan−1(y=x) (2) L = R jr0j dt = R t1 p(x0)2 + (y0)2 + (z0)2 dt t0 dT dT ds 1 dT (3) κ = = = = . ds dt dt jr0j dt dT (4) = κN. ds (5) ρ = 1/κ. jr0 × r00j (6) κ = . jr0j3 (r0 × r00) • r000 (7) τ = . jr0 × r00j2 (8) B = T × N. d2s ds2 (9) a = , a = κ. T dt2 N dt GmM r (10) F = − jrj2 jrj , F = ma. dr dθ dz d2r dθ 2 d2θ dr dθ d2z (11) r = rur + zk, v = dt ur + r dt uθ + dt k, a = ( dt2 − r( dt ) )ur + (r dt2 + 2 dt dt )uθ + dt2 k. 1 2 Practice problems. (1) Solve the Initial Value Problems. (a) d2r = −(i + j + k) dt2 dr = 0 dt t=0 r(0) = 10i + 10j + 10k (b) dr 1 1 1 = ; ; p dt 1 + t2 1 − t2 1 − t2 r(0) = h1; 2; 3i (2) Suppose that a particle travels along a space curve in such a way that its acceleration vector is constant. Can its path be a circle? an ellipse? a parabola? a square? Explain. (3) Find the length of the curves parametrized by (a) ht; t2; 2t3=3i as t ranges from 0 to 1. (b) he2t; e4t=2; ti as t ranges from 0 to 1. (c) hln(t); 2t; t2i as t ranges from 1 to 2. (d) hcos(t); sin(t); t2=2i as t ranges from 0 to 1. (4) Find the radius of curvature and the center of the circle of curvature of (a) the sine curve ht; sin(t); 0i at the top of its first arch in the xy-plane. (b) the cycloid ht − sin(t); 1 − cos(t); 0i at the top of its first arch in the xy-plane. 3 (5) Suppose that (u; v; w) is a right handed system of vectors. Put circles around the right handed systems below and squares around the left handed systems. (a) (u, w, v) (d) (w, v, u) (g) (k, i, j) (j) (j, i, k) (b) (u, v, w) (e) (i, k, j) (h) (w, u, v) (k) (w, v, u) (c) (v, u, w) (f) (v, w, u) (i) (2j, 2k, 2i) (l) (2i, 3j, 4k) (6) (a) Can the curvature of a space curve be negative? If so, give an example. If not, explain why not. (b) Can the torsion of a space curve be negative? If so, give an example. If not, explain why not. (7) This problem concerns the question: How does reversal of direction affect T; N; B; κ, τ? Suppose that the position of a particle at time t is given by r(t), and that the Frenet- Serret data are T(t); N(t); B(t); κ(t); τ(t). Assume that all quantities exist for all t. Now suppose that the position of a second particle is given by r∗(t) = r(−t), and that the Frenet-Serret data for the second particle are T∗(t); N∗(t); B∗(t); κ∗(t); τ∗(t). How are T(0); N(0); B(0); κ(0); τ(0) related to T∗(0); N∗(0); B∗(0); κ∗(0); τ∗(0)? (8) (a) Find T; N; B; κ, τ at time t = 0 for the space curve 1 1 1 r(t) = (1 + t)3=2; (1 − t)3=2; p t ; −1 < t < 1: 3 3 2 (b) Find the tangential and normal components of acceleration at time t = 0 for the curve in part (a). (c) Find the osculating plane, the normal plane, and the rectifying plane at time t = 0 for the curve in part (a). (9) State Kepler's Laws. (10) For the plane curve with the polar description r = 2 cos(4t) and θ = 2t, find the velocity and acceleration vectors in terms of ur and uθ..