WI1402-LR Calculus II Delft University of Technology

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1 WI402-LR lculus II elft University of Technology Yer Michele Fcchinelli Version.0 Lst modified on Februry, 207

2 Prefce This summry ws written for the course WI402-LR lculus II, tught t the elft University of Technology. All the mteril treted is tken from [J. tewrt. lculus: Erly Trnscendentls. Brooks/ole, 7th edition, 202.] Throughout the summry, references to chpters nd sections cn be found. These re lbelled with the id of the symbol nd cn be found in the forementioned book, where exercises nd more explntions re given. In cse of ny comments bout the content of the summry, plese do not hesitte to contct me t m.fcchinelli@yhoo.it. Mthemtics is the music of reson. Jmes Joseph ylvester i

3 hngelog This is version.0. Below re listed the chnges pplied to ech version. Version te hnges.0 Februry, 207 First version ii

4 hpter hin Rule 4.6 df(g(t), h(t)) dt f(g(s, t), h(s, t)) t irectionl erivtive where û is unit vector. = f dx x dt + f dy y dt = f x x dt + f y y dt u f(x, y, z) = f(x, y, z) û for one vrible for two vribles (sme holds for s) (i) Another definition is u f(x, y) = f x (x, y) + f y (x.y) b, where û = 4.7 riticl Point t (, b) eterminnt of Hesse hpter 5 5., 5.2, 5.3 ouble Integrls if f x (, b) = 0 nd f y (, b) = 0 = f xx f yx f xy f yy if > 0 nd f xx < 0 if > 0 nd f xx > 0 if < 0 if = 0 V = R = f xxf yy fxy 2 f(x, y) da (i) The verge vlue of function is found by f vg = A(R) (ii) If R = {(x, y) x b, c y d}, then d c b f(x, y) dx dy; (iii) If R, then R g(x)h(y) da = b g(x) dx d c g(y) dy; (iv) If = {(x, y) x b, g (x) y g 2 (x)}, then (v) If, then A() = da. 5.4 Polr oordintes mximum minimum sddle point no informtion R [ b f(x, y) da; ]. R f(x, y) da = b R f(x, y) da = b r 2 = x 2 + y 2 x = r cos θ y = r sin θ d c f(x, y) dy dx = g2 (x) g (x) f(x, y) dy dx; (i) If R = {(x, y) 0 r b, α θ β}, then β b α f(r cos θ, r sin θ)r dr dθ; (ii) Note tht da = d(x, y) = r d(r, θ). R f(x, y) da =

5 5.5 Appliction of ouble Integrls mss m = ρ(x, y) da centroid x = m xρ(x, y) da, ȳ = m yρ(x, y) da moment of inerti I x = m y2 ρ(x, y) da, I y = m x2 ρ(x, y) da 5.7 Triple Integrls (i) If B = [, b] [c, d] [e, o], then b W = d c E f(x, y, z) dv o e f(x, y, z) dz dy dx; (ii) If E = {(x, y, z) x b, c y d, e z o}, then V (E) = E dv ; (iii) Mss: m = E ρ(x, y, z) dv ; (iv) entroid: x = m E xρ(x, y, z) dv, ȳ = m E yρ(x, y, z) dv, z = m 5.8 ylindricl oordintes r 2 = x 2 + y 2 x = r cos θ y = r sin θ z = z E zρ(x, y, z) dv. (i) If E = {(x, y, z) 0 r b, α θ β, c z d}, then β b d α c f(r cos θ, r sin θ, z)r dz dr dθ; (ii) Note tht dv = d(x, y, z) = r d(r, θ, z). 5.9 phericl oordintes E f(x, y, z) dv = ρ 2 = x 2 + y 2 + z 2 x = ρ sin φ cos θ y = ρ sin φ sin θ z = ρ cos φ (i) If E = {(x, y, z) 0 ρ b, α θ β, γ φ δ}, then β δ b α γ f(ρ sin φ cos θ, ρ sin φ sin θ, ρ cos φ)ρ2 sin φ dρ dφ dθ; E f(x, y, z) dv = (ii) Note tht dv = d(x, y, z) = ρ 2 sin φ d(ρ, φ, θ). 5.0 Jcobin (x, y) (u, v) = hnge of Vribles x u y u x v y v f(x, y) da = the sme holds for three dimentions. of the trnsformtion T given by (x, y) f(x(u, v), y(u, v)) d(u, v) (u, v) { x = g(u, v) y = h(u, v) 2

6 hpter 6 6. Vector Fields Grdient Field [ ] in R 2 P(x, y) F(x, y) = Q(x, y) P(x, y, z) in R 3 F(x, y, z) = Q(x, y, z) R(x, y, z) f(x, y) f(x, y) = [ f/ x f/ y ] F(x, y) then F is conservtive vector field nd f is potentil function for F. 6.2 Line Integrl long mooth urve ˆ f(x, y, z) ds = ˆ b f(r(t)) ṙ(t) dt (i) Mss nd entroid: ρ(x, y) ds, x = m xρ(x, y) ds, ȳ = m yρ(x, y) ds; (ii) pecil cses: f(x, y) dx = b f(x(t), y(t))ẋ(t) dt, f(x, y) dy = b f(x(t), y(t))ẏ(t) dt; (iii) ombintion P(x, y) dx + Q(x, y) dy = P(x, y) dx + Q(x, y) dy. Line Integrl of Vector Field ˆ F dr = ˆ b F(r(t)) ṙ(t) dt or F dr = P dx + Q dy + R dx, where F(x, y, z) = 6.3 Fundmentl Theorem of Line Integrls ˆ f dr = f(r(b)) f(r()) P(x, y, z) Q(x, y, z) R(x, y, z) (i) F dr is clled independent of pth if F dr = 2 F dr, where nd 2 hve sme initil nd strting points; (ii) F dr is independent of pth if nd only if F dr = 0. (iii) If F is continuous in n open nd connected regionnd F dr is independent of pth, then F is conservtive (hence F = f); [ ] P(x, y) (iv) Let F = on n open simply-connected region nd suppose Q(x, y) P y = Q x, then F is conservtive.. 3

7 6.4 Green s Theorem F dr = Are of Region { P(x, y) = 0 Q(x, y) = x { P(x, y) = y Q(x, y) = 0 { P(x, y) = 2 y Q(x, y) = 2 x 6.5 url P dx + Q dy = Q x P y = Q x P y = Q x P y = 2 i j k curl F = F = x y P Q R ( Q x P ) da y z x dy y dx x dy y dx (i) If F is defined in R 3 nd curl F = 0, then F is conservtive; (ii) Green s Theorem cn be rewritten s F T ds = F dr = ( F) k da in R2. ivergence div F = F = x y z P Q R = P x + Q y + R z (i) If F is defined in R 3, div (curl F) = 0; (ii) Green s Theorem cn be rewritten s F n ds = F dr = ( F) da in R urfce Are A() = r u r v da which is the re of the smooth prmetric surfce r(u, v) = with r u = x/ u y/ u z/ u nd r v = x/ v y/ v z/ v (i) Tngent Plne : r(u, v) = r 0 + ur u + vr v. 6.7 urfce Integrl. f(x, y, z) d = x(u, v) y(u, v) z(u, v) f(r(u, v)) r u r v da (i) Mss: m = ρ(x, y, z) d; (ii) entroid: x = m xρ(x, y, z) d, ȳ = m yρ(x, y, z) d, z = m Flux Integrl where n is the norml vector. F d = F n d = F (r u r v ) da in the domin, nd zρ(x, y, z) d. 4

8 6.8 tokes Theorem where =. 6.9 ivergence (or Guss) Theorem (i) If the survce is not closed: Generl Reltions F dr = ( F) d = ( F) (r u r v ) da lirut s Theorem If f hs continuous prtil derivtives, then F d = ( F) dv E F d = F d F d 2 f x y = 2 f y x Grdient Theorem If f represents sclr field, then f d = f dv E 5

9 This ws: WI402-LR lculus II Version.0 More summries re vilble for the following courses: AE0 Intro to AE I - Module E: Aerodynmics AE0 Intro to AE I - Module F: Flight Mechnics AE08-I Aerospce Mterils AE240-I Physics I AE240-II Physics II AE32-II Production of Aerospce ystems AE4447 Aircrft Performnce Optimiztion WI402-LR lculus II WI403-LR Liner Algebr

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