Math 17 - Review. Review for Chapter 12

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1 Mth 17 - eview Ying Wu eview for hpter Given prmetric plnr curve x = f(t), y = g(t), where t b, how to eliminte the prmeter? (Use substitutions, or use trigonometry identities, etc). How to prmeterize curve f(x, y) = 0? (For polr form r = f(θ), one cn set x(t) = f(t) cos t nd y(t) = f(t) sin t; for the specil form y = f(x), one cn set x(t) = t nd y(t) = f(t).) 2. Differentition of prmetric curves nd the slope. For polr coordintes, dy dx = dy/dt dx/dt, d 2 y dx 2 = dy /dt dx/dt. dy dx = r sin θ + r cos θ r cos θ r sin θ, cot φ = 1 dr r dθ, where φ denotes the ngle between the tngent line t P nd the rdius OP from the origin. 3. eview of integrtion in hpter 6 nd how to pply the formuls to prmetric curves. Are under curve: A = ydx Vol. of rev. round x-xis: V x = πy 2 dx Vol. of rev. round y-xis: V y = πx 2 dy Arc length: s = ds = (dx) 2 + (dy) 2 Are of surf. of rev. round x-xis: A x = 2π y ds Are of surf. of rev. round y-xis: A y = 2π x ds 4. Opertion of vectors. he ddition nd the multipliction of vectors behve pretty much the sme wy s the opertions of numbers with some exceptions. he following re some reminders: (i) he dot product of two vectors outputs number. (ii) he sclr product nd the vector product output vectors. (iii) he vector product of two vectors is not commuttive. 5. Some importnt fcts: (i) If 0, then u = / is the unit vector tht hs the sme direction of ; 1

2 (ii) Perpendiculr test: b = 0 iff nd b re perpendiculr; (iii) Prllel test: b = 0 iff nd b re prllel. 6. Vector functions: he opertions on vector functions cn be done componentwise. (Appliction to motions) With position vector r(t), velocity v(t) nd ccelertion (t), one hs r = v, v =. eview for hpter he distnce formul nd its reltion to dot products. If r hs its hed in (x 1, y 1, z 1 ) nd til t (x 2, y 2, z 2 ), then r =< x 1 x 2, y 1 y 2, z 1 z 2 >, nd r 2 = r r = (x 1 x 2 ) 2 + (y 1 y 2 ) 2 + (z 1 z 2 ) Importnt fcts: (the ngle θ below is the ngle between the two vectors nd b) (i) b = b cos θ, nd so nd b re perpendiculr if nd only if b = 0. (ii) b 2 = 2 b 2 sin 2 θ, nd so nd b re prllel to ech other if nd only if b = 0, the zero vector. (iii) b is perpendiculr to both nd b. 3. Direction ngles nd numbers. Let r =< x, y, z >. Let the ngles between r nd the x, y nd z xes be α, β nd γ, respectively. hen these ngles re the direction ngles of r nd the direction numbers re 4. he component of long b is cos α = r i r, cos β = r j r, cos γ = r k. r omp b = b. b 5. Let L be line in spce tht is prllel to n =<, b, c > nd psses through (x 0, y 0, z 0 ). hen the prmetric eqution of L is nd the symmetric equtions of L is r(t) =< t + x 0, bt + y 0, ct + z 0 >, x x 0 = y y 0 b = z z 0. c 2

3 6. he plne psses (x 0, y 0, z 0 ) with norml vector n =<, b, c > hs eqution <, b, c > < x x 0, y y 0, z z 0 >= 0. he ngle between two plnes is the ngle between the two norml vectors. 7. Vector functions nd their limits, derivtives, nd ntiderivtives. One key thing to remember: do it componentwise. Note tht the product rules for derivtives re very similr to the product rule for ordinry functions, with exception in its vector product form. 8. Given r(t), how do you find the velocity, the speed, the ccelertion, the unit tngent vector, the principl norml vector, the tngent component nd the norml component of the ccelertion, nd the curvture. 9. Given the velocity nd the ccelertion together with some initil conditions, how do you find the position vector r(t)? 10. Given the eqution of surfce nd the cutting plnes, how do you describe the trces? 11. Given the eqution of plne curve nd n xis L, how do you find n eqution of the surfce generted by revolving bout L? 12. he ylindricl nd the sphericl coordintes. he formule bout cylindricl coordintes: x = r cos θ y = r sin θ z = z nd x 2 + y 2 = r 2, tn θ = y x. he formule bout sphericl coordintes: x = ρ cos θ sin φ y = ρ sin θ sin φ z = ρ cos φ nd x 2 + y 2 + z 2 = ρ 2. eview for hpter Limits of multivrible functions. All the limit lws we lernt in MAH 15 remin vlid nd the techniques re similr (substitution for continuous functions, cnceling zerofctors in both the numertor nd the denomintor, mking use of known limit, etc). 3

4 2. How to show nonexistence of some limits? (Use different pths to pproch the limit.) 3. How to find prtil derivtives? (egrd it s derivtive nd use everything vlid for derivtives.) 4. How to find tngent objects? (For surfce with eqution z = f(x, y), use norml vector n(x, y) =< f x (x, y), f y (x, y), 1 >. For surfce with eqution F (x, y, z) = 0, then the grdient is norml. Importnt: You ANNO use the vector functions n(x, y) or F (x, y, z) s normls. You HAVE O EVALUAE them t point in the tngent object. 5. How to find extrem of function with bounded domin region? (Step 1: Find criticl points INSIDE the region. Step 2: Find extrem on ech curve of the boundry of the region by Mth 15 techniques. Step 3: By comprison to pick up the bsolute extrem). 6. How to clssify the criticl points (if they re locl extrem nd wht kind)? (Use (x, y) = f xx f yy (f xy ) 2.) 7. Differentils nd its pplictions. df(x, y, z) = f(x, y, z) < dx, dy, dz >. One cn use df s n pproximtion to f(x + dx, y + dy, z + dz) f(x, y, z). 8. hin rules: (Go check them on pges 736 nd 739). 9. Implicit prtil differentition. (his time the eqution defines one vrible (z, sy) s function of the other vribles (x, y, sy). Hence when you prtilly differentite both sides of the eqution with respect to x or y, you cnnot regrd z s constnt.) 10. Appliction of implicit prtil differentition: when surfce is given by n eqution, one cn use implicit prtil differentition to find the tngent plne by using n =< z x, z y, 1 > s norml. 11. he grdient nd the directionl derivtives: f(x, y, z) =< f x, f y, f z > nd D n f(x, y, z) = f(x, y, z) n, if n = 1. eview for hpter 15 4

5 1. Evlution of double integrls: (x, y coordintes) b g2 (x) g 1 (x) d h2 (y) c h 1 (y) f(x, y)dydx, if is x b, g 1 (x) y g 2 (x). f(x, y)dydx, if is c y d, h 1 (y) x h 2 (y). When you cn evlute the integrl by either wy, you my wnt to choose simpler wy. 2. Evlution of double integrls: (Polr coordintes) b g2 (r) g 1 (r) b h2 (θ) h 1 (θ) A useful fct: da = rdrdθ = dxdy. Some pplictions of double integrls. f(r cos θ, r sin θ)rdθdr, if is r b, g 1 (r) θ g 2 (r). f(r cos θ, r sin θ)rdrdθ, if is θ b, h 1 (θ) r h 2 (θ). 2. Are of region is da. 2b. he volume between z = f(x, y) nd z = g(x, y) when (x, y) re in is (f(x, y) g(x, y))da. 2c. (Applictions in Physics) Let ρ(x, y) be the density of lmin whose region is. hen the mss nd the centroid of the lmin (x, y) is mss m = ρ(x, y)da 1 x = m xρ(x, y)da 1 y = m yρ(x, y)da 3. Evlution of triple integrls: (rectngulr coordintes) he min ide is the sme s the cross section ide. he following gives wy to reduce triple integrl to double integrl. f(x, y, z)dv = h1 (x,y) ( f(x, y, z)dz)da if is h 1 (x, y) z h 2 (x, y), (x, y) in. h 2 (x,y) 4. Evlution of triple integrls: (cylindricl coordintes nd sphericl coordintes) he min reltionship mong rectngulr, cylindricl nd sphericl coordintes is 5. Some pplictions of triple integrls. dv = dxdydz = rdrdθdz = ρ 2 sin φdφdθdρ. 5. he volume of the solid is dv. 5

6 5b. he mss of solid with density ρ(x, y, z) is ρ(x, y, z)dv. 5c. he centroid of with density ρ(x, y, z) is (x, y, z), where x = xρ(x, y, z)dv, y = yρ(x, y, z)dv, z = zρ(x, y, z)dv 6. Use double integrl to find the surfce re. If surfce is given by r(u, v) =< x(u, v), y(u, v), z(u, v) >, where (u, v) is in, then the re of the surfce is r u r v da. eview for hpter A vector field is vector function F =< P (x, y, z), Q(x, y, z), (x, y, z) > or F = < P (x, y), Q(x, y) >. 2. he grdient opertor =< x, y, z 3. div F = F. 4. curl F = F. > is liner opertor. 5. Line integrls: If is curve with prmetric equtions < x(t), y(t), z(t) > with t b, then b f(x, y, z)dx = f(x(t), y(t), z(t))x (t)dt. f(x, y, z)ds = f(x, y, z)dy = f(x, y, z)dz = b b b f(x(t), y(t), z(t))y (t)dt. f(x(t), y(t), z(t))z (t)dt. f(x(t), y(t), z(t)) (x (t)) 2 + (y (t)) 2 + (z (t)) 2 dt. 6. onservtive fields: A vector field F = < P, Q > is conservtive if nd only if P y = Q x. In this cse, one cn find its potentil function f such tht f = F by the following steps: (1) Find f(x, y) = P dx + φ(y). (2) Set Q = f y to get differentil eqution of φ (y). (3) Solve the eqution to find φ(y) nd thereby getting f. 7. Importnt fct: If F = < P, Q, > is conservtive field, nd if is the unit tngent vector of the curve, then F ds is independent of pth. (herefore, you cn mke 6

7 use of it to simplify the integrl). 8. Green s heorem: Let be the closed curve bounding the region nd suppose tht P, Q re both continuous nd hve continuous first order of derivtives, then P dx + Qdy = ( Q x P y )da. 7