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Math 103 – Fall 2007 Test 3 Notes Alex Beutel Section 10.2 - Polar Coordinates x = rcosθ y = rsinθ r =x +y y tanθ = x 2 2 2 (1) (2) (3) (4) Section 12.8 - Cylindrical and Spherical Coordinates Cylindrical x = rcosθ y = rsinθ z=z r =x +y y tanθ = x z=z 2 2 2 (5) (6) (7) (8) (9) (10) Spherical x = ρsinφcosθ y = ρsinφsinθ z = ρcosφ ρ =x +y +z 2 2 2 2 (11) (12) (13) (14) Section 14.3 f (x, y)dA R (15) 1 Section 14.4 f (rcosθ, rsinθ)rdrdθ R (16) (17) (18) x = rcosθ y = rsinθ Section 14.5 m= x= ¯ 1 m 1 y= ¯ m δ(x, y)dA xδ(x, y)dA R (19) (20) (21) yδ(x, y)dA R Section 14.6 - Triple Integrals m= T δdV dV T (22) (23) V= x= ¯ 1 m 1 y= ¯ m 1 z= ¯ m T xδ(x, y)dV T (24) (25) (26) (27) (28) (29) (30) (31) (32) yδ(x, y)dV T zδ(x, y)dV T Ix = Iy = T (y 2 + z 2 )δdV (x2 + z 2 )δdV (x2 + y 2 )δdV T Iz = IAxis = r(x, y, z)2 δdV T r(x, y, z) = distance from axis of rotation Section 14.7 - Integration in Cylindrical and Spherical Coordinates z2 (x,y) f (x, y, z)dV = T R z1 (x,y) f (x, y, z)dz dA (33) (34) (35) dV = rdzdrdθ dV = ρ sinφdρdφdθ 2 14.8 - Surface Area r(u, v) = x(u, v), y(u, v), z(u, v) N = ru × rv A = a(S) = R (36) (37) ∂r ∂r × | dudv ∂u ∂v | N(u, v) | dudv = R | ∂f ∂y 2 (38) (39) A = a(S) = R 1+ ∂f ∂x 2 + dxdy 14.9 - Change of Variables in Multiple Integrals F (x, y)dxdy = R S F (T (u, v)) | JT (u, v) | dudv ∂(x, y, z) = Determinant ∂(u, v, w) (40) (41) JT (u, v, w) = 15.1 - Vector Fields ∂f ∂f ∂f i+ j+ k ∂x ∂y ∂z (af + bg) = a f + b g f= F = P, Q, X ∂Q ∂r ∂P + + divF = · F = ∂x ∂y ∂z curlF = × F = Determinant (42) (43) (44) (45) (46) 15.2 - Line Integrals b f (x, y, z)ds = C a f (x(t), y(t), z(t)) [x (t)]2 + [y (t)]2 + [z (t)]2 dt (47) (48) dm = δ(x, y, z)ds m= C dm = C δds (49) (50) (51) (52) (53) x= ¯ 1 m 1 y= ¯ m 1 z= ¯ m I= C xdm C ydm C zdm C p2 dm P dx + C C P dx + Qdy + Rdz = C Qdy + C Rdz (54) (55) F · T ds = C C P dx + Qdy + Rdz 15.3 - Fundamental Theorem and Independence of Path f · dr = f (r(b)) − f (r(a)) C (56) (57) (58) (59) F is independent of path if and only if: F = f f is a potential function ∂Q ∂P = Conservative if for every point in R ∂y ∂x 15.4 - Green’s Theorem ∂Q ∂P − ∂x ∂y P dx + Qdy = C R dA ydx = C C (60) xdy (61) (62) (63) A= Φ= 1 2 C −ydx + xdy = − C F · nds | F = ru × rv F| n= 15.5 - Surface Integrals ∂r ∂r × ∂u ∂v f (r(u, v)) | N(u, v) | dudv D N= f (x, y, z)dS = S (64) (65) (66) dudv (67) (68) dS =| N(u, v) | dudv =| F · ndS = S S ∂r ∂r × | dudv ∂u ∂v ∂(y, z) ∂(x, z) ∂(x, y) P +Q +R ∂(u, v) ∂(u, v) ∂(u, v) Φ= S F · ndS 15.6 - Divergence Theorem F · ndS = S T · FdV (69) 15.7 - Stokes’ Theorem F · Tds = C S (curlF) · ndS ∂R ∂Q − ∂y ∂z dydz + ∂P ∂R − ∂z ∂x dzdx + ∂Q ∂P − ∂x ∂y dxdy (70) (71) P dx + Qdy + Rdz = C S 4 

Document Info

Posted By:

Alex Beutel

Date:

Monday, December 24, 2007

School:

Duke University

Class:

Math 103

Tags:

• Vectors
• Calc3
• Calculus
• Stokes
• Greens
• Divergence
• Integration
• Formulas
• Formula Sheet
• Study Guide
• Surface Area
• Polar
• Spherical
• Cylindrical

About this Document

This is a formula sheet for the second half of Calc. 3. Includes formulas for polar, cylindrical and spherical coordinates, double and triple integration, surface integrals, Green's theorem, divergence theorem, and Stokes' Theorem.