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Math 103 – Fall 2007 Chapter 12 and 13 Alex Beutel Section 12.2 - Three-Dimensional Vectors |v|= x2 + y 2 + z 2 (1) Dot Product Properties a · b = a1 b1 + a2 b2 + a3 c3 a · a =| a | 2 (2) (3) (4) (5) (6) (7) (8) (9) a·b=b·a a(b + c) = a · b + a · c (ra) · b = r(a · b) = a · (rb) a · b =| a || b | (cos)θ Perpendicular if a · b = 0 compb a = a·b | a || b | cosθ = |b| |b| 12.3 - The Cross Product of Vectors i a1 b1 j a2 b2 k a3 b3 a×b= (10) (11) (12) | a × b | =| a || b | sinθ Parallel if a × b = 0 Propeties of Cross Product a × b = −(b × a) (ka) × b = a × (kb) = k(a × b) a × (b + c) = (a × b) + (a × c) a · (b × c) = (a · c)b − (a · b)c Volume of Parallelepiped =| a · (b × c) | 1 Volume of Tetrahedron (Pyramid) = | a · (b × c) | 6 a1 a2 a3 a · (b × c) = b1 b2 b3 c1 c2 c3 (13) (14) (15) (16) (17) (18) (19) 12.4 - Lines and Planes in Space Lines r = r0 + tv x = x0 + at y = y0 + bt z = z0 + ct x − x0 y − y0 z − z0 = = a b c (20) (21) (22) (23) (24) Planes n(r − r0 ) = 0 a(x − x0 ) + b(y − y0 ) + c(z − z0 ) = 0 P0 = (x0 , y0 , z0 ) Normal Vector = n = a, b, c (25) (26) (27) (28) 12.5 - Curves and Motion in Space Differentiation Formulas Dt [u(t) + v(t)] = u (t) + v (t) Dt [cu(t)] = cu (t) Dt [h(t)u(t)] = h (t)u(t) + h(t)u (t) 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) Speed = v(t) =| v(t) | = dx dt 2 2 2 (29) (30) (31) (32) (33) (34) + dy dt + dz dt 2 12.6 - Curvature and Acceleration b b Arc Length = s = a v(t)dt = a dx dt 2 + dy dt 2 + dz dt 2 dt (35) (36) (37) (38) (39) (40) (41) (42) (43) Unit Tangent Vector = T(t) = v(t) | v(t) | |xy −x y | dφ |xy −x y | = κ= 3= ds v2 [(x )2 + (y )2 ] 2 dT = κN ds dv a= T + κv 2 N dt v·a r (t) · r (t) dv = = aT = dt v | r (t) | | r (t) × r (t) | aN = κv 2 = | r (t) | a = aT T + aN N κ= | r (t) × r (t) | |v×a| = v3 | r (t) |3 12.7 - Cylinders and Quadric Surfaces Ellipsoid x2 y2 z2 + 2 + 2 =1 a2 b c (44) Elliptic Paraboloid x2 y2 z + 2= 2 a b c (45) Elliptical Cone x2 y2 z2 + 2= 2 a2 b c (46) Hyperboloid of One Sheet x2 y2 z2 + 2 − 2 =1 a2 b c (47) Hyperboloid of Two Sheets z2 x2 y2 − 2 − 2 =1 c2 a b 3 (48) Hyperbolic Paraboloid x2 z y2 − 2= 2 b a c (49) 13.5 - Multivariable Optimization Problems Necessary for Local Extrema: fx (a, b) = 0 = fy (a, b) (50) Types of Absolute Extrema • An interior point at which ∂f ∂x = ∂f ∂y =0 • An interior piont of R where not both partial derivatives exist • A point on the boundary of R 13.6 - Increments and Linear Approximation f (x + h) ≈ f (x) + f (a + h) = f (a) + f (x) · h f (a) · h + (h) · h (51) (52) 13.7 - Multivariable Chain Rule ∂w dx ∂w dy ∂w dz dw = · + · + · dt ∂x dt ∂y dt ∂z dt (53) 13.8 - Directional Derivatives and The Gradient Vector Du f (x) = Max direction derivative when u = | f (x) · u f (x) f (x) | (54) (55) 13.9 - Lagrange Multiplier Constraint g(x, y) = 0 Function f (x, y) = 0 f (P ) = λ g(P ) Second Constraint h(x, y) = 0 f (P ) = λ g(P ) + µ h(P ) (56) (57) (58) (59) (60) 13.10 - Critical Points of Functions of Two Variables A = fxx (a, b) B = fxy (a, b) C = fyy (a, b) ∆ = AC − B Two Varialbe Second Derivative Test: • f (a, b) is a local minimum value of f if A > 0 and ∆ > 0 • f (a, b) is a local maximum value of f if A < 0 and ∆ > 0 • f (a, b) is neither a local minimum value nor local maximum of f if ∆ < 0 • If A = 0 then ∆ < 0 and therefore is neither a local min nor max of f • If ∆ = 0 then the test is indeterminant and it can be eiter local min, local max, or neither. 2 (61) (62) (63) (64) 5 

Document Info

Posted By:

Alex Beutel

Date:

Thursday, December 27, 2007

School:

Duke University

Class:

Math 103

Tags:

• Study Guide
• Formula Sheet
• Integration
• Integrals
• Differentiation
• Math
• Calculus
• Calc3
• Vectors

About this Document

This study guide contains all of the formulas from chapters 12 and 13 from our Math 103 textbooks. This includes: vectors, dot products, cross products, curves in spaces, partial derivatives, cylinder and quadratic surfaces, multivariable chain rule, directional derivatives, gradient, Lagrange multipliers, and critical points.