formulasheet_exam1

formulasheet_exam1 - FORMULA SHEET FOR MIDTERM I(1(2(3(4(5...

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Unformatted text preview: FORMULA SHEET FOR MIDTERM I (1) (2) (3) (4) (5) P1 = (x1 , y1 , z1 ), P2 = (x2 , y2 , z2 ) ⇒ |P1 P2 | = (x2 − x1 )2 + (y2 − y1 )2 + (z2 − z1 )2 . Sphere : (x − a)2 + (y − b)2 + (z − c)2 = r2 . u · v = u1 v1 + u2 v2 + u3 v3 = |u||v | cos θ. a · a = |a|2 , a · (b + c) = a · b + a · c, a · b = b · a, (ca) · b = c(a · b) = a · (cb), 0 · a = 0. F ·D D Scalar projection: compD F = F cos θ = |D| . Vector projection: projD F = compD F |D| . (6) u × v = u2 v3 − u3 v2 , −u1 v3 + u3 v1 , u1 v2 − u2 v1 . (7) |u × v | = |u| |v | sin θ (8) u × v = −v × u, u × (v + w) = u × v + u × w, u · (v × w) = (u × v ) · w, (u · w)v − (u · v )w. (9) Straight lines: (x0 + at, y0 + bt, z0 + ct) or y − y0 z − z0 x − x0 = = . a b c (10) If n = a, b, c is a normal vector, then n · x − x0 , y − y0 , z − z0 = 0 (11) dy = dx (12) b a dy dt dx dt u × (v × w ) = or d dt a(x − x0 ) + b(y − y0 ) + c(z − z0 ) = 0. dy dx dx dt 2 , d2 y = dx2 dx dt as long as dx = 0. dt + dy dt 2 dt. r (t) (13) Tangent vector: r (t) = f (t), g (t), h (t) . Unit tangent vector: T (t) = |r (t)| . d d d (14) dt [u(t) + v (t)] = u (t) + v (t), dt [c u(t)] = c u (t), dt [f (t) u(t)] = f (t)u(t) + f (t)u (t), d d dt [u(t) · v (t)] = u (t) · v (t) + u(t) · v (t), dt [u(t) × v (t)] = u (t) × v (t) + u(t) × v (t), d dt [u(f (t))] = f (t)u (f (t)). (15) Integration: b b b b r(t) dt = a a f (t) dt, a b g (t) dt, a h(t) dt . (16) FTC: r(t) dt = R(t) a = R(b) − R(a), where R (t) = r(t). (17) b dx dy dz + + dt = |r (t)| dt. dt dt dt a a (18) The arclength for the segment of the curve measured from the point where t = a is b t 2 2 2 s(t) = a |r (u)| du. (19) Curvature κ = (20) dT ds = |T (t)| |r (t)| = |r (t)×r (t)| |r (t)|3 N (t) = T (t) |T (t)| , B (t) = T (t) × N (t). (21) Normal: N & B . Osculating: T & N . (22) v (t) = r (t), a(t) = v (t). (23) a = v T + κv 2 N 1 ...
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