MA 261 - Spring 2010Study Guide # 31. Cylindrical Coordinates (r, θ, z):From CC to RC :x = r cos θy = r sin θz = zxzzy(x,y,z)0θrGoing from RC to CC use x2+ y2= r2and tan θ =yx(make sure θ is in correct quadrant).2. Spherical Coordinates (ρ, θ, φ), where 0 ≤ φ ≤ π:From SC to RC :x = (ρ sin φ) cos θy = (ρ sin φ) sin θz = ρ cos φxzy(x,y,z)0θφρρ cos φφsinρGoing fr om RC to SC use x2+ y2+ z2= ρ2, tan θ =yxand cos φ =zρ.3. Triple integrals in Cylindrical Coordinates:x = r cos θy = r sin θz = z, dV = r dz dr dθZZZEf(x, y, z) dV =ZZZEf(r cos θ, r sin θ, z) r dz dr dθ↑4. Triple integrals in Spherical Coordinates:x = (ρ sin φ) cos θy = (ρ sin φ) sin θz = ρ cos φ, dV = ρ2sin φ dρ dφ dθZZZEf(x, y, z) dV =ZZZEf(ρ sin φ cos θ, ρ sin φ sin θ, ρ cos φ) ρ2sin φ dρ dφ dθ↑5. Vector fields on R2and R3:~F(x, y) = hP (x, y), Q(x, y)i and~F(x, y, z) = hP (x, y), Q(x, y), R(x, y)i;~F is a conservative vector field if~F = ∇f, for some real-valued function f.16. Line integral of a function f(x, y) along C, parameterized by x = x(t), y = y(t) and a ≤ t ≤ b, isZCf(x, y) ds =Zbaf(x(t), y(t))sdxdt2+dydt2dt .(independent of orientation of C, other pro perties and applications of line integrals of f)Remarks:(a)ZCf(x, y) ds is sometimes called the “lin e integral of f with respect to arc length”(b)ZCf(x, y) dx =Zbaf(x(t), y(t)) x′(t) dt(c)ZCf(x, y) dy =Zbaf(x(t), y(t)) y′(t) dt7. Line integral of vector field~F(x, y) along C, parameterized by~r(t) and a ≤ t ≤ b, is given byZC~F · d~r =Zba~F(~r(t)) ·~r′(t) dt .(depends o n orientation of C, other properties and applications of line integrals of f )8. Connection between line integral of vector fields and line integral of functions:ZC~F · d~r =ZC(~F ·~T) dswhere~T is the unit t angent vector to the curve C.9. If~F(x, y) = P (x, y)~i + Q(x, y)~j, thenZC~F · d~r =ZCP (x, y) dx + Q(x, y) dy ; Work =ZC~F · d~r.10. Fundamental Theorem of Calculus for Line Integrals:ZC∇f·d~r = f(~r(b))−f(~r(a)):Cr(b)r(a)11. A vector field~F(x, y) = P (x, y)~i + Q(x, y)~j is conservative (i.e.~F = ∇f) if∂Q∂x=∂P∂y; how todetermine a potential function f if~F(~x) = ∇f (~x).12. Green’s Theorem:ZCP (x, y) dx + Q(x, y) dy =ZZD∂Q∂x−∂P∂ydA (C = boundary of D):DC213. Del Operator:∂∂x~i +∂∂y~j +∂∂z~k; if~F(x, y, z) = P (x, y, z)~i + Q(x, y, z)~j + R(x, y, z)~k, thencurl~F = ∇ ×~F =~i~j~k∂∂x∂∂y∂∂zP Q Rand div~F = ∇ ·~F =∂P∂x+∂Q∂y+∂R∂zProperties of curl and divergence:(i) If curl~F =~0, then~F is a conservative vector field (i.e.,~F(~x) = ∇f(~x)).(ii) If curl~F =~0, then~F is irrotational; if div~F = 0, then~F is incompressible.(iii) Laplace’s Equation: ∇2f =∂2f∂x2+∂2f∂y2+∂2f∂z2= 0.14. Parametric surface S:~r(u, v) = hx(u, v), y(u, v), z(u, v)i, where (u, v) ∈ D:DSnuvxyzr(u,v)Normal vector to surface S :~n =~ru×~rv; tangent planes and normal lines to parametric surfaces.15. Surface area of a surface S:(i) A(S) =ZZD|~ru×~rv| dA(ii) If S is the graph of z = f (x, y) above D, then A(S) =ZZDs1 +∂z∂x2+∂z∂y2dA;Remark: dS = |~ru×~rv| dA = differential of surface area; while d~S = (~ru×~rv) dA16. The surface integral of f(x, y, z) over the surface S:(i)ZZSf(x, y, z) dS =ZZDf(~r(u, v)) |~ru×~rv| dA.(ii) If S is the graph of z = h(x, y) above D, thenZZSf(x, y, z) dS =ZZDf(x, y, h(x, y))s1 +∂z∂x2+∂z∂y2dA.317. The surface integral of~F over the surface S (recall, d~S = (~ru×~rv) dA):ZZS~F · d~S =ZZD~F · (~ru×~rv) dA.ZZS~F · d~S =ZZS(~F ·~n) dS =ZZD~F · (~ru×~rv) dA.(i) Connection between surface integral of a vector field and a function:ZZS~F · d~S =ZZS(~F ·~n) dS.(The above gives another way to computeRRS~F · d~S)(ii)ZZS~F · d~S =ZZS(~F ·~n) dS = fluxof~F across the surface S.Sn18. Stokes’ Theorem:ZC~F · d~r =ZZScurl~F · d~S (recall, curl~F = ∇ ×~F).nSCZC~F · d~r = circulation of~F around C.19. The Divergence Theorem/Gauss’ Theorem:ZZS~F · d~S =ZZZEdiv~F dV(recall, div~F = ∇ ·~F).SEnnnn420. Summary of Line Integrals a nd Surface Integrals:Line Integrals Surface IntegralsC :~r(t), where a ≤ t ≤ b S :~r(u, v), where (u, v) ∈ Dds = |~r′(t)| dt = differential o f arc length dS = |~ru×~rv| dA = differential of surface areaZCds = length of CZZSdS = surface area of SZCf(x, y, z) ds =Zbaf(~r(t)) |~r′(t)| dtZZSf(x, y, z) dS =ZZDf(~r(u, v)) |~ru×~rv| dA(independent of orientation of C ) (independent of normal vector~n)d~r =~r′(t) dt d~S = (~ru×~rv) dAZC~F · d~r =Zba~F(~r(t)) ·~r′(t) dtZZS~F · d~S =ZZD~F(~r(u, v)) · (~ru×~rv) dA(depends o n orientation of C) (depends o n normal vector~n)ZC~F · d~r =ZC~F ·~TdsZZS~F · d~S =ZZS~F ·~ndSThe circulation of~F around C The flux of~F across S in direction~n521. Integration Theorems:Fundamental Theorem of Calculus:ZbaF′(x) dx = F (b) − F (a)baFundamental Theorem of Calculus For Line Integrals:Zba∇f·d~r = f(~r(b))−f(~r(a))Cr(b)r(a)Green’s Theorem:ZZD∂Q∂x−∂P∂ydA =ZCP (x, y) dx + Q(x, y) dyDCStokes’ Theorem:ZZScurl~F · d~S =ZC~F · d~rnSCDivergence Theorem:ZZZEdiv~F dV =ZZS~F ·