# STUDY GUIDE, CALCULUS III, 2017 SPRING

Size: px
Start display at page:

Transcription

1 TUY GUIE, ALULU III, 2017 PING ontents hpter 13. Functions of severl vribles Plnes nd surfces Grphs nd level curves Limit of function of two vribles Prtil derivtives hin rule irectionl derivtive nd the grdient Tngent plnes nd liner pproximtion Optimiztion problems 4 hpter 14. Multiple integrtion ouble integrls over rectngulr regions ouble integrls over generl regions ouble integrls in polr coordintes Triple integrls Triple integrls in cylindricl nd sphericl coordintes hnge of vribles 6 hpter 15. Vector clculus Vector fields Line integrls onservtive vector fields Green s Theorem ivergence nd curl urfce integrls tokes Theorem ivergence Theorem 11 Abstrct. The following list includes the key topics nd importnt homework questions. Going through them helps your preprtion of the Finl. It is, however, by no mens exhustive. You re entitled to rewrd of 2 points towrd Test if you re the first person to report bon-fide mthemticl mistke (i.e. not including lnguge typos nd grmmticl errors.) emrk. There re no choice questions in the Tests nd Finl. o when you review the choice questions in the online homework, understnd the correct nswer nd mke sure you cn nswer the question without ll the choices being given. hpter 13. Functions of severl vribles (i). ompute the derivtives, including pplictions of chin rules nd implicit differentition. (ii). Grph the six qudric surfces in 3 ; drw the level curves nd contour mp on 2 ; derive the eqution of the tngent plnes to these surfces using prtil derivtives. (iii). ompute the limit of function t point if the point is in the domin of the function (nd therefore the limit exists); prove the limit does not exist by choosing two pths. 1

2 2 TUY GUIE, ALULU III, 2017 PING (iv). ompute the directionl derivtive u f, where u is unit directionl vector nd u f is rte of chnge; compute the grdient vector f nd use it to find the mximl nd miniml rtes of chnge nd the corresponding directions u mx nd u min. (v). Use first-order prtil derivtive to find the liner pproximtion of function. (vi). Use first-order prtil derivtive to find the criticl points nd use second-order prtil derivtives to clssify them s locl mximum, locl minimum, sddle points, or inconclusive Plnes nd surfces. Equtions of plnes in 3. Grphs of qudric surfces through intercepts nd trces: ellipsoid, elliptic prboloid, hyperboloid of one sheet, hyperboloid of two sheets, elliptic cone, nd hyperbolic prboloid. Importnt online questions: 2, 3, 9, 10, 12, 14, 17, Grphs nd level curves. omin nd rnge of function f(x, y) with two independent vribles x nd y. Level curves nd contour mp of function. The contour mps consisting of the level curves re on the xy-plne, not in the xyzspce! Importnt online questions: 1, 4, 5, Limit of function of two vribles. If (, b) is in the domin of the function f nd f is continuous t (, b), then lim f(x, y) = f(, b). (x,y) (,b) The polynomil, rtionl, exponentil, rdicl, trigonometric functions re ll continuous in their nturl domins. This mens tht, if the point is in the domin of such function, then the limit of the function pproching the point is simply the function vlue t this point. To prove lim f(x, y) does not exist, (x,y) (,b) one needs to construct two pths of (x, y) (, b) such tht there re two distinct limits long the two pths. Importnt online questions: 1, 4, 5, 7, Prtil derivtives. ompute the prtil derivtives. lirut Theorem: f xy = f yx. Importnt online questions: 1, 5, 6, 10, hin rule. hin rule with one independent vrible: z = f(x, y), x = x(t), nd y = y(t). Then dz dt = z dx x dt + z dy y dt. hin rule with two independent vribles: z = f(x, y), x = x(s, t), nd y = y(s, t). Then z s = z x x s + z y nd z y s t = z x x t + z y y t. hin rule cn be generlized to more independent vribles.

3 TUY GUIE, ALULU III, 2017 PING 3 Implicit differentition: If F (x, y, z) = 0, then z x = F x nd z F z y = F y. F z emember the minus sign in the bove formuls! Importnt online questions: 3, 4, 7, 8, 13, irectionl derivtive nd the grdient. Let f = f(x, y) be differentible t (, b) nd u = u 1, u 2 be unit vector in 2. Then the directionl derivtive of f t (, b) in the direction of u is u f(, b) = u 1 f x (, b) + u 2 f y (, b) = f x (, b), f y (, b) u 1, u 2 = f u. The directionl derivtive is sclr! The grdient of f t (, b) is f(, b) = f x (, b), f y (, b) = f x (, b) i + f y (, b) j. The grdient is vector nd it is lwys orthogonl to the level curves of the function. Let f = f(x, y, z) be differentible t (, b, c) nd u = u 1, u 2, u 3 be unit vector in 3. Then the directionl derivtive of f t (, b, c) in the direction of u is in which u f(, b, c) = f x (, b, c), f y (, b, c), f z (, b, c) u 1, u 2, u 3 = f u, f(, b, c) = f x (, b, c), f y (, b, c), f z (, b, c) = f x (, b, c) i + f y (, b, c) j + f z (, b, c) k. At (, b), the mximl directionl derivtive is chieved when the directionl vector f(, b) u mx = f(, b), nd umx f(, b) = f(, b), while the miniml directionl derivtive is chieved when the directionl vector f(, b) u min = f(, b), nd umin f(, b) = f(, b). Keep in mind tht u mx nd u min re two (unit) directionl vectors while umx f(, b) nd umin f(, b) re two sclrs! If u is prllel to f, then u f = 0 nd there is no chnge of the function in the direction of u. Importnt online questions: 3, 5, 9, 10, 12, 15, 18, 21, Tngent plnes nd liner pproximtion. The tngent plne to F (x, y, z) = 0 t (, b, c) hs the eqution F x (, b, c)(x ) + F y (, b, c)(y b) + F z (, b, c)(z c) = 0. The tngent plne to z = f(x, y) t (, b, f(, b)) hs the eqution z = f x (, b)(x ) + f y (, b)(y b) + f(, b).

4 4 TUY GUIE, ALULU III, 2017 PING Around the bse point (, b), the function f(x, y) cn be pproximted by the liner pproximtion f(x, y) L(x, y) = f x (, b)(x ) + f y (, b)(y b) + f(, b). Keep in mind tht the prtil differentition f x nd f y here re evluted t the bse point (, b) nd remember the term f(, b). The totl differentil dz = f x (, b)dx + f y (, b)dy. Importnt online questions: 1, 2, 3, 4, 5, 8, 9, Optimiztion problems. The criticl points (, b) of f(x, y) stisfies f x (, b) = 0 nd f y (, b) = 0. econd derivtive test: Let (x, y) = f xx f yy f 2 xy. uppose tht (, b) is criticl point of f. If (, b) > 0 nd f xx (, b) < 0, then f hs locl mximum vlue t (, b); If (, b) > 0 nd f xx (, b) > 0, then f hs locl minimum vlue t (, b); If (, b) < 0, then f hs sddle point t (, b); If (, b) = 0, then the test is inconclusive. Find the bsolute mximl nd miniml vlues of function f on bounded domin : (1) etermine the vlues of f t ll criticl points in ; (2) Find the mximum nd miniml vlues of f on the boundry of ; (3) The bsolute mximl vlue is the gretest vlue in the bove two steps, nd the bsolute miniml vlue is the lest vlue in the bove two steps. Importnt online questions: 1, 3, 4, 9, 12, 18, 20. (i). Evlute double integrls hpter 14. Multiple integrtion f(x, y) da, where is rectngulr, y-simple, x-simple, polr rectngulr, r-simple in polr coordintes. In prticulr, use polr if the function f contins x 2 + y 2 or the the domin is piece of disc. (ii). The re of region in 2 is 1 da. (iii). Evlute triple integrls f(x, y, z) dv, where is cubic, z-simple, y-simple, x-simple, in cylindricl coordintes (r, θ, z), in sphericl coordintes (ρ, ϕ, θ). In prticulr, use cylindricl if the function f contins x 2 + y 2 ; use sphericl is the function f contins x 2 + y 2 + z 2 or the domin is piece of bll. (iv). The volume of solid in 3 is 1 dv. (v). The extr fctor in polr nd cylindricl is r; while the extr fctor in sphericl is ρ 2 sin ϕ. (vi). Use chnge of vribles to evlute double integrl nd remember the bsolute vlue of the Jcobin.

5 TUY GUIE, ALULU III, 2017 PING ouble integrls over rectngulr regions. If is rectngulr, i.e. = [, b] [c, d] = {(x, y) x b, c y d}, then d b b d f(x, y) da = f(x, y) dxdy = f(x, y) dydx. c If the region is rectngulr, then one cn chnge the order of integrtion freely s long s one keeps the bounds of x nd y correspondingly. Importnt online questions: 4, 7, 8, ouble integrls over generl regions. If is y-simple, i.e. = {(x, y) x b, g 1 (x) y g 2 (x)}, then b g2 (x) f(x, y) da = f(x, y) dydx. g 1 (x) If the region is y-simple, then do y-integrtion first! If is x-simple, i.e. = {(x, y) c y d, h 1 (y) x h 2 (y)}, then d h2 (y) f(x, y) da = f(x, y) dxdy. c h 1 (y) If the region is x-simple, then do x-integrtion first! One cn chnge the order of integrtion between y-simple nd x-simple. Alwys sketch the integrting region nd find ll the required informtion. The re of region in 2 is Are () = 1 da. The volume of solid in 3 bove region in 2 nd is bounded bove by z = h 1 (x, y) nd bounded below by z = h 2 (x, y) is [h 1 (x, y) h 2 (x, y)] da. Importnt online questions: 9, 13, 15, 17, 18, ouble integrls in polr coordintes. If is polr rectngulr, i.e. = {(r, θ) α θ β, 0 r b}, then β b f(x, y) da = f(r cos θ, r sin θ)r drdθ. α emember the extr fctor r; use polr coordintes if the integrting function f contins x 2 + y 2 or the integrting region is piece of disc. If is r-simple in polr coordintes, i.e. = {(r, θ) α θ β, 0 g 1 (θ) r g 2 (θ)}, then β g2 (θ) f(x, y) da = f(r cos θ, r sin θ)r drdθ. α g 1 (θ) Importnt online questions: 1, 2, 3, 5, 8, 12, 14. c

6 6 TUY GUIE, ALULU III, 2017 PING Triple integrls. If is cubic, i.e. = [, b] [c, d] [p, q] = {(x, y, z) x b, c y d, p z q}, then q d b f(x, y, z) dv = f(x, y, z) dxdydz =. p c If the region is cubic, then one cn chnge within the six orders of integrtion freely s long s one keeps the bounds of x, y, nd z correspondingly. If is z-simple, i.e. = {(x, y) (x, y), h 1 (x, y) z h 2 (x, y)}, then h2 (x,y) f(x, y, z) dv = f(x, y, z) dzda. h 1 (x,y) If the region is z-simple, then do z-integrtion first! There re lso y-simple nd x-simple regions. The volume of solid is 1dV. Importnt online questions: 4, 6, 9, Triple integrls in cylindricl nd sphericl coordintes. If = {(r, θ, z) α θ β, 0 r b, h 1 (r, θ) z h 2 (r, θ)}, then β b h2 (r,θ) f(x, y, z) dv = f(r cos θ, r sin θ, z) dzr drdθ. α h 1 (r,θ) emember the extr fctor r; use cylindricl coordintes if the integrting function f contins x 2 + y 2 nd cylindricl is simply polr plus the z-vrible. In sphericl coordintes, point P in 3 cn be represented s P = (ρ, ϕ, θ). ρ is the distnce from the origin to P, hence ρ 0; ϕ is the ngle between the positive z-xis nd the ry OP, hence, 0 ϕ π; θ is the ngle tht mesures the rottion with respect to the z-xis, hence, 0 θ 2π. If is sphericl rectngle, i.e. = {(ρ, ϕ, θ) α θ β, p ϕ q, ρ b}, then β q b f(x, y, z) dv = f(ρ sin ϕ cos θ, ρ sin ϕ sin θ, ρ cos ϕ)ρ 2 sin ϕ dρdϕdθ. α p emember the extr fctor ρ 2 sin ϕ; use sphericl coordintes if the integrting function f contins x 2 + y 2 + z 2 or the integrtion region is piece of bll. If is ρ-simple in sphericl coordintes, i.e. = {(ρ, ϕ, θ) α θ β, p ϕ q, h 1 (ϕ, θ) ρ h 2 (ϕ, θ)}, then β q h2 (ϕ,θ) f(x, y, z) dv = f(ρ sin ϕ cos θ, ρ sin ϕ sin θ, ρ cos ϕ)ρ 2 sin ϕ dρdϕdθ. α p h 1 (ϕ,θ) Importnt online questions: 3, 4, 5, 10, hnge of vribles. If x = x(u, v) nd y = y(u, v), then f(x, y) da = f(x(u, v), y(u, v)) (x, y) (u, v) da,

7 TUY GUIE, ALULU III, 2017 PING 7 in which (x, y) (u, v) = x x u v y y u v is the Jcobin determinnt. Keep in mind the extr fctor is the bsolute vlue of the Jcobin determinnt, nd you need to find the correct integrting domin in (u, v). If you hve u nd v s functions of x nd y, then you need to solve for x nd y in order to compute the Jcobin! Importnt online questions: 3, 6, 7, 12, hpter 15. Vector clculus Prmetric eqution of curve is vector function r(t). (i). : r(t) = x(t), y(t), where t b, defines curve in 2 with n orienttion. (ii). : r(t) = x(t), y(t), z(t), where t b, defines curve in 3 with n orienttion. The rclength of is b r (t) dt. Importnt curves: stright line, circle, ellipse, helix, etc. Two types of vector field: F = f(x, y), g(x, y) in 2 nd F = f(x, y, z), g(x, y, z), h(x, y, z) in 3. We cn discuss two topics for both of them: (iii). Is F conservtive? There re severl equivlent criteri. If it is, then find its potentil function ϕ such tht ϕ = F. (iv). Evlute the line integrls of F in the circultion form long curve : F T ds = F r ds = F d r. Then we discuss the fundmentl theory of clculus in 2 nd in 3, seprtely: (vi). Fundmentl theory of clculus for vector fields in 2 : A closed curve enclosing region in 2, Green s Theorem connects double integrl in the interior nd line integrl on the boundry. ircultion form: F d r = f dx + g dy = (g x f y ) da. In the specil cse, when F is conservtive, g x = f y so F d r = 0. Flux form: F n ds = f dy g dx = (f x + g y ) da. (vii). Prmetric eqution of surfce in 3 : The re of is r(u, v) = x(u, v), y(u, v), z(u, v), where (u, v). 1 d = r u r v da. Importnt surfces: plne, cylinder, cone, sphere, prboloid, etc.

8 8 TUY GUIE, ALULU III, 2017 PING (viii). Fundmentl theory of clculus for vector fields in 3. I: A closed curve s the boundry of surfce in 3, tokes Theorem connects flux surfce integrl of the curl in the interior nd circultion line integrl of the vector field on the boundry: F d r = ( F ) n d. (ix). Fundmentl theory of clculus for vector fields in 3. II: A closed surfce enclosing solid in 3, ivergence Theorem connects triple integrl of the divergence in the interior nd outwrd flux surfce integrl of the vector field on the boundry: F n d = F dv Vector fields. A vector field in 2 : F (x, y) = f(x, y), g(x, y) = f(x, y) i + g(x, y) j. A vector field in 3 : F (x, y, z) = f(x, y, z), g(x, y, z), h(x, y, z) = f(x, y, z) i + g(x, y, z) j + h(x, y, z) k. ecll tht the grdient vector field of ϕ(x, y) is defined s ϕ = ϕ x, ϕ y = ϕ x i + ϕ y j, nd the grdient vector field of ϕ(x, y, z) is defined s ϕ = ϕ x, ϕ y, ϕ z = ϕ x i + ϕ y j + ϕ z k. Importnt online questions: 4, 5, 6, 7, Line integrls. A curve in 2 is vector function r(t) = x(t), y(t) = x(t) i + y(t) j where t b. The bove prmetric form of the curve defines n orienttion long the curve s the direction when t increses from to b. Prmetrize stright line, circle, ellipse, etc. The line integrl of sclr function f(x, y) long curve is b f ds = f(x(t), y(t)) r (t) dt, in which The rclength of is r (t) = x (t), y (t) nd r (t) = (x (t)) 2 + (y (t)) 2. b r (t) dt. The line integrl of vector field F = f(x, y), g(x, y) long curve hs two forms:

9 ircultion form: F T ds = = b Flux form: TUY GUIE, ALULU III, 2017 PING 9 F r ds = f(t), g(t) x (t), y (t) dt = = F n ds = b F d r = b f dx + g dy f(t)x (t) + g(t)y (t) dt. f dy g dx f(t)y (t) g(t)x (t) dt. Importnt online questions: 1, 4, 5, 8, 14, onservtive vector fields. The following sttements re equivlent. A vector field F = f, g is conservtive. F hs potentil function, i.e. F = ϕ for some sclr function ϕ. F d r = ϕ(b) ϕ(a) if is curve from initil point A to terminl point B. Tht is, the line integrl from A to B of conservtive vector field is pth independent. F d r = 0 if is closed curve. f y = g x, i.e. curl F = 0 nd F is irrottionl. Find the potentil function if vector field is conservtive. Importnt online questions: 1, 2, 3, 6, Green s Theorem. Green s Theorem hs two forms. Let be closed curve enclosing region in 2. ircultion form: F d r = f dx + g dy = (g x f y ) da. In the specil cse, when F is conservtive, g x = f y so F d r = 0. Flux form: F n ds = f dy g dx = (f x + g y ) da. The key observtion of Green s Theorem is tht the double integrl in the interior equls the line integrl on the boundry. An ppliction of Green s Theorem is to use circultion to compute the re of : Are () = 1 da = x dy = y dx = 1 (x dy y dx). 2 Importnt online questions: 10, 11, 13, 14, 15.

10 10 TUY GUIE, ALULU III, 2017 PING ivergence nd curl. Let F be vector field in 3 : F (x, y, z) = f(x, y, z), g(x, y, z), h(x, y, z) = f(x, y, z) i + g(x, y, z) j + h(x, y, z) k. Then the divergence of F is div F = f x + g y + h z = x, y, z f, g, h = F, nd the curl of F is curl F i j k = x y z f g h = ( h y g ) ( f i + z z h ) ( g i + x x f ) i = F y. The divergence is sclr nd the curl is vector. If div F = 0, then F is incompressible (i.e. source-free); if curl F = 0, then F is irrottionl. The following sttements re equivlent. A vector field F = f, g, h is conservtive. F hs potentil function, i.e. F = ϕ for some sclr function ϕ. F d r = ϕ(b) ϕ(a) if is curve from initil point A to terminl point B. Tht is, the line integrl from A to B of conservtive vector field is pth independent. F d r = 0 if is closed curve. f y = g x, f z = h x, nd h y = g z, i.e. curl F = 0 nd F is irrottionl. Find the potentil function of vector field if it is conservtive. how tht the curl of grdient vector field is zero, i.e. the vector field conservtive: curl ( ϕ) = ( ϕ) = 0. how tht the divergence of curl is zero: Importnt online questions: 4, 5, 10, urfce integrls. A surfce in 3 is vector function div (curl F ) = ( F ) = 0. r(u, v) = x(u, v), y(u, v), z(u, v) = x(u, v) i + y(u, v) j + z(u, v) k, where (u, v). Prmetrize plne, cylinder, cone, sphere, prboloid, etc. The surfce integrl of sclr function f(x, y, z) on surfce is f d = f(x(u, v), y(u, v), z(u, v)) r u r v da, in which The re of is r u (t) = x u, y u, z u nd r v(t) = x v, y v, z v. Are () = r u r v da.

11 TUY GUIE, ALULU III, 2017 PING 11 The surfce integrl of vector field F = x(u, v), y(u, v), z(u, v) on surfce is F d = F n d = F (x(u, v), y(u, v), z(u, v)) ( r u r v) da. Notice tht one hs to choose between r u r v nd r u r v so tht it is consistent with the norml vector n. Importnt online questions: 1, 2, 3, 6, 7, 9, 10, 15, tokes Theorem. Let be surfce in 3 with boundry closed curve nd their orienttions re consistent by the right-hnd rule. Then ( ) F d r = ( F ) n d = ( F ) d. The key observtion of tokes Theorem is tht the flux surfce integrl of the curl in the interior equls circultion line integrl of the vector field on the boundry. Also keep in mind tht the curve c : r(t) = x(t), y(t), z(t) here is in 3. A specil cse when F is conservtive, i.e. F = 0, so the surfce integrl is lwys zero nd hence the circultion line integrl of conservtive vector field long ny closed curve is zero. Importnt online questions: 1, 2, 3, 4, 7, 8, ivergence Theorem. Let be closed surfce enclosing solid region in 3. Then ( ) F d = F n d = F dv. The key observtion of ivergence Theorem is tht the triple integrl of the divergence in the interior equls outwrd flux surfce integrl of the vector field on the boundry. A specil cse when F is incompressible, i.e. F = 0, so the triple integrl is lwys zero nd hence the flux surfce integrl of n incompressible vector field on ny closed surfce is zero. Importnt online questions: 1, 3, 5, 8, 9, 10, 12, 14.

### b = and their properties: b 1 b 2 b 3 a b is perpendicular to both a and 1 b = x = x 0 + at y = y 0 + bt z = z 0 + ct ; y = y 0 )

***************** Disclimer ***************** This represents very brief outline of most of the topics covered MA261 *************************************************** I. Vectors, Lines nd Plnes 1. Vector

### Study Guide # Vectors in R 2 and R 3. (a) v = a, b, c = a i + b j + c k; vector addition and subtraction geometrically using parallelograms

Study Guide # 1 MA 26100 - Fll 2018 1. Vectors in R 2 nd R 3 () v =, b, c = i + b j + c k; vector ddition nd subtrction geometriclly using prllelogrms spnned by u nd v; length or mgnitude of v =, b, c,

### Chapter 12 Vectors and the Geometry of Space 12.1 Three-dimensional Coordinate systems

hpter 12 Vectors nd the Geometry of Spce 12.1 Three-dimensionl oordinte systems A. Three dimensionl Rectngulr oordinte Sydstem: The rtesin product where (x, y, z) isclled ordered triple. B. istnce: R 3

### WI1402-LR Calculus II Delft University of Technology

WI402-LR lculus II elft University of Technology Yer 203 204 Michele Fcchinelli Version.0 Lst modified on Februry, 207 Prefce This summry ws written for the course WI402-LR lculus II, tught t the elft

### Lecture 20. Intro to line integrals. Dan Nichols MATH 233, Spring 2018 University of Massachusetts.

Lecture 2 Intro to line integrls Dn Nichols nichols@mth.umss.edu MATH 233, Spring 218 University of Msschusetts April 12, 218 (2) onservtive vector fields We wnt to determine if F P (x, y), Q(x, y) is

### Example. Check that the Jacobian of the transformation to spherical coordinates is

lss, given on Feb 3, 2, for Mth 3, Winter 2 Recll tht the fctor which ppers in chnge of vrible formul when integrting is the Jcobin, which is the determinnt of mtrix of first order prtil derivtives. Exmple.

### 13.1 Double Integral over Rectangle. f(x ij,y ij ) i j I <ɛ. f(x, y)da.

CHAPTE 3, MULTIPLE INTEGALS Definition. 3. Double Integrl over ectngle A function f(x, y) is integrble on rectngle [, b] [c, d] if there is number I such tht for ny given ɛ>0thereisδ>0 such tht, fir ny

### Section 17.2: Line Integrals. 1 Objectives. 2 Assignments. 3 Maple Commands. 1. Compute line integrals in IR 2 and IR Read Section 17.

Section 7.: Line Integrls Objectives. ompute line integrls in IR nd IR 3. Assignments. Red Section 7.. Problems:,5,9,,3,7,,4 3. hllenge: 6,3,37 4. Red Section 7.3 3 Mple ommnds Mple cn ctully evlute line

### Vector Calculus. 1 Line Integrals

Vector lculus 1 Line Integrls Mss problem. Find the mss M of very thin wire whose liner density function (the mss per unit length) is known. We model the wire by smooth curve between two points P nd Q

### Math 116 Calculus II

Mth 6 Clculus II Contents 7 Additionl topics in Integrtion 7. Integrtion by prts..................................... 7.4 Numericl Integrtion.................................... 7 7.5 Improper Integrl......................................

### Polar Coordinates. July 30, 2014

Polr Coordintes July 3, 4 Sometimes it is more helpful to look t point in the xy-plne not in terms of how fr it is horizontlly nd verticlly (this would men looking t the Crtesin, or rectngulr, coordintes

### VectorPlot[{y^2-2x*y,3x*y-6*x^2},{x,-5,5},{y,-5,5}]

hapter 16 16.1. 6. Notice that F(x, y) has length 1 and that it is perpendicular to the position vector (x, y) for all x and y (except at the origin). Think about drawing the vectors based on concentric

### 10.4 AREAS AND LENGTHS IN POLAR COORDINATES

65 CHAPTER PARAMETRIC EQUATINS AND PLAR CRDINATES.4 AREAS AND LENGTHS IN PLAR CRDINATES In this section we develop the formul for the re of region whose oundry is given y polr eqution. We need to use the

### 9.4. ; 65. A family of curves has polar equations. ; 66. The astronomer Giovanni Cassini ( ) studied the family of curves with polar equations

54 CHAPTER 9 PARAMETRIC EQUATINS AND PLAR CRDINATES 49. r, 5. r sin 3, 5 54 Find the points on the given curve where the tngent line is horizontl or verticl. 5. r 3 cos 5. r e 53. r cos 54. r sin 55. Show

### Lecture 16. Double integrals. Dan Nichols MATH 233, Spring 2018 University of Massachusetts.

Leture 16 Double integrls Dn Nihols nihols@mth.umss.edu MATH 233, Spring 218 University of Msshusetts Mrh 27, 218 (2) iemnn sums for funtions of one vrible Let f(x) on [, b]. We n estimte the re under

### Section 16.3 Double Integrals over General Regions

Section 6.3 Double Integrls over Generl egions Not ever region is rectngle In the lst two sections we considered the problem of integrting function of two vribles over rectngle. This sitution however is

### Geometric quantities for polar curves

Roerto s Notes on Integrl Clculus Chpter 5: Bsic pplictions of integrtion Section 10 Geometric quntities for polr curves Wht you need to know lredy: How to use integrls to compute res nd lengths of regions

### Test Yourself. 11. The angle in degrees between u and w. 12. A vector parallel to v, but of length 2.

Test Yourself These are problems you might see in a vector calculus course. They are general questions and are meant for practice. The key follows, but only with the answers. an you fill in the blanks

### Triangles and parallelograms of equal area in an ellipse

1 Tringles nd prllelogrms of equl re in n ellipse Roert Buonpstore nd Thoms J Osler Mthemtics Deprtment RownUniversity Glssoro, NJ 0808 USA uonp0@studentsrownedu osler@rownedu Introduction In the pper

### Double Integrals over Rectangles

Jim Lmbers MAT 8 Spring Semester 9- Leture Notes These notes orrespond to Setion. in Stewrt nd Setion 5. in Mrsden nd Tromb. Double Integrls over etngles In single-vrible lulus, the definite integrl of

### FP2 POLAR COORDINATES: PAST QUESTIONS

FP POLAR COORDINATES: PAST QUESTIONS. The curve C hs polr eqution r = cosθ, () Sketch the curve C. () (b) Find the polr coordintes of the points where tngents to C re prllel to the initil line. (6) (c)

### 2.1 Partial Derivatives

.1 Partial Derivatives.1.1 Functions of several variables Up until now, we have only met functions of single variables. From now on we will meet functions such as z = f(x, y) and w = f(x, y, z), which

### FUNCTIONS OF SEVERAL VARIABLES AND PARTIAL DIFFERENTIATION

FUNCTIONS OF SEVERAL VARIABLES AND PARTIAL DIFFERENTIATION 1. Functions of Several Variables A function of two variables is a rule that assigns a real number f(x, y) to each ordered pair of real numbers

### Polar coordinates 5C. 1 a. a 4. π = 0 (0) is a circle centre, 0. and radius. The area of the semicircle is π =. π a

Polr coordintes 5C r cos Are cos d (cos + ) sin + () + 8 cos cos r cos is circle centre, nd rdius. The re of the semicircle is. 8 Person Eduction Ltd 8. Copying permitted for purchsing institution only.

### Definitions and claims functions of several variables

Definitions and claims functions of several variables In the Euclidian space I n of all real n-dimensional vectors x = (x 1, x,..., x n ) the following are defined: x + y = (x 1 + y 1, x + y,..., x n +

LECTURE 9: QUADRATIC RESIDUES AND THE LAW OF QUADRATIC RECIPROCITY 1. Bsic roerties of qudrtic residues We now investigte residues with secil roerties of lgebric tye. Definition 1.1. (i) When (, m) 1 nd

### INTRODUCTION TO TRIGONOMETRY AND ITS APPLICATIONS

CHAPTER 8 INTRODUCTION TO TRIGONOMETRY AND ITS APPLICATIONS (A) Min Concepts nd Results Trigonometric Rtios of the ngle A in tringle ABC right ngled t B re defined s: sine of A = sin A = side opposite

### Differentiable functions (Sec. 14.4)

Math 20C Multivariable Calculus Lecture 3 Differentiable functions (Sec. 4.4) Review: Partial derivatives. Slide Partial derivatives and continuity. Equation of the tangent plane. Differentiable functions.

### Review guide for midterm 2 in Math 233 March 30, 2009

Review guide for midterm 2 in Math 2 March, 29 Midterm 2 covers material that begins approximately with the definition of partial derivatives in Chapter 4. and ends approximately with methods for calculating

### Exam 2 Review Sheet. r(t) = x(t), y(t), z(t)

Exam 2 Review Sheet Joseph Breen Particle Motion Recall that a parametric curve given by: r(t) = x(t), y(t), z(t) can be interpreted as the position of a particle. Then the derivative represents the particle

### Calculus IV Math 2443 Review for Exam 2 on Mon Oct 24, 2016 Exam 2 will cover This is only a sample. Try all the homework problems.

Calculus IV Math 443 eview for xam on Mon Oct 4, 6 xam will cover 5. 5.. This is only a sample. Try all the homework problems. () o not evaluated the integral. Write as iterated integrals: (x + y )dv,

### Vocabulary Check. Section 10.8 Graphs of Polar Equations not collinear The points are collinear.

Section.8 Grphs of Polr Equtions 98 9. Points:,,,,.,... The points re colliner. 9. Points:.,,.,,.,... not colliner. Section.8 Grphs of Polr Equtions When grphing polr equtions:. Test for symmetry. () )

### Independent of path Green s Theorem Surface Integrals. MATH203 Calculus. Dr. Bandar Al-Mohsin. School of Mathematics, KSU 20/4/14

School of Mathematics, KSU 20/4/14 Independent of path Theorem 1 If F (x, y) = M(x, y)i + N(x, y)j is continuous on an open connected region D, then the line integral F dr is independent of path if and

### Exam 2 Summary. 1. The domain of a function is the set of all possible inputes of the function and the range is the set of all outputs.

Exam 2 Summary Disclaimer: The exam 2 covers lectures 9-15, inclusive. This is mostly about limits, continuity and differentiation of functions of 2 and 3 variables, and some applications. The complete

### REVIEW, pages

REVIEW, pges 510 515 6.1 1. Point P(10, 4) is on the terminl rm of n ngle u in stndrd position. ) Determine the distnce of P from the origin. The distnce of P from the origin is r. r x 2 y 2 Substitute:

### METHOD OF LOCATION USING SIGNALS OF UNKNOWN ORIGIN. Inventor: Brian L. Baskin

METHOD OF LOCATION USING SIGNALS OF UNKNOWN ORIGIN Inventor: Brin L. Bskin 1 ABSTRACT The present invention encompsses method of loction comprising: using plurlity of signl trnsceivers to receive one or

### Multivariable integration. Multivariable integration. Iterated integration

Multivrible integrtion Multivrible integrtion Integrtion is ment to nswer the question how muh, depending on the problem nd how we set up the integrl we n be finding how muh volume, how muh surfe re, how

### WESI 205 Workbook. 1 Review. 2 Graphing in 3D

1 Review 1. (a) Use a right triangle to compute the distance between (x 1, y 1 ) and (x 2, y 2 ) in R 2. (b) Use this formula to compute the equation of a circle centered at (a, b) with radius r. (c) Extend

### Kirchhoff s Rules. Kirchhoff s Laws. Kirchhoff s Rules. Kirchhoff s Laws. Practice. Understanding SPH4UW. Kirchhoff s Voltage Rule (KVR):

SPH4UW Kirchhoff s ules Kirchhoff s oltge ule (K): Sum of voltge drops round loop is zero. Kirchhoff s Lws Kirchhoff s Current ule (KC): Current going in equls current coming out. Kirchhoff s ules etween

### 47. Conservative Vector Fields

47. onservative Vector Fields Given a function z = φ(x, y), its gradient is φ = φ x, φ y. Thus, φ is a gradient (or conservative) vector field, and the function φ is called a potential function. Suppose

### Exercise 1-1. The Sine Wave EXERCISE OBJECTIVE DISCUSSION OUTLINE. Relationship between a rotating phasor and a sine wave DISCUSSION

Exercise 1-1 The Sine Wve EXERCISE OBJECTIVE When you hve completed this exercise, you will be fmilir with the notion of sine wve nd how it cn be expressed s phsor rotting round the center of circle. You

### MATH 118 PROBLEM SET 6

MATH 118 PROBLEM SET 6 WASEEM LUTFI, GABRIEL MATSON, AND AMY PIRCHER Section 1 #16: Show tht if is qudrtic residue modulo m, nd b 1 (mod m, then b is lso qudrtic residue Then rove tht the roduct of the

### Unit 1: Chapter 4 Roots & Powers

Unit 1: Chpter 4 Roots & Powers Big Ides Any number tht cn be written s the frction mm, nn 0, where m nd n re integers, is nn rtionl. Eponents cn be used to represent roots nd reciprocls of rtionl numbers.

### VECTOR CALCULUS Julian.O 2016

VETO ALULUS Julian.O 2016 Vector alculus Lecture 3: Double Integrals Green s Theorem Divergence of a Vector Field Double Integrals: Double integrals are used to integrate two-variable functions f(x, y)

### Practice problems from old exams for math 233

Practice problems from old exams for math 233 William H. Meeks III October 26, 2012 Disclaimer: Your instructor covers far more materials that we can possibly fit into a four/five questions exams. These

### NONCLASSICAL CONSTRUCTIONS II

NONLSSIL ONSTRUTIONS II hristopher Ohrt UL Mthcircle - Nov. 22, 2015 Now we will try ourselves on oncelet-steiner constructions. You cn only use n (unmrked) stright-edge but you cn ssume tht somewhere

### Section 10.2 Graphing Polar Equations

Section 10.2 Grphing Polr Equtions OBJECTIVE 1: Sketching Equtions of the Form rcos, rsin, r cos r sin c nd Grphs of Polr Equtions of the Form rcos, rsin, r cos r sin c, nd where,, nd c re constnts. The

### 1. Vector Fields. f 1 (x, y, z)i + f 2 (x, y, z)j + f 3 (x, y, z)k.

HAPTER 14 Vector alculus 1. Vector Fields Definition. A vector field in the plane is a function F(x, y) from R into V, We write F(x, y) = hf 1 (x, y), f (x, y)i = f 1 (x, y)i + f (x, y)j. A vector field

### Translate and Classify Conic Sections

TEKS 9.6 A.5.A, A.5.B, A.5.D, A.5.E Trnslte nd Clssif Conic Sections Before You grphed nd wrote equtions of conic sections. Now You will trnslte conic sections. Wh? So ou cn model motion, s in E. 49. Ke

### MAXIMUM FLOWS IN FUZZY NETWORKS WITH FUNNEL-SHAPED NODES

MAXIMUM FLOWS IN FUZZY NETWORKS WITH FUNNEL-SHAPED NODES Romn V. Tyshchuk Informtion Systems Deprtment, AMI corportion, Donetsk, Ukrine E-mil: rt_science@hotmil.com 1 INTRODUCTION During the considertion

### (1) Non-linear system

Liner vs. non-liner systems in impednce mesurements I INTRODUCTION Electrochemicl Impednce Spectroscopy (EIS) is n interesting tool devoted to the study of liner systems. However, electrochemicl systems

### CS 135: Computer Architecture I. Boolean Algebra. Basic Logic Gates

Bsic Logic Gtes : Computer Architecture I Boolen Algebr Instructor: Prof. Bhgi Nrhri Dept. of Computer Science Course URL: www.ses.gwu.edu/~bhgiweb/cs35/ Digitl Logic Circuits We sw how we cn build the

### Math 148 Exam III Practice Problems

Math 48 Exam III Practice Problems This review should not be used as your sole source for preparation for the exam. You should also re-work all examples given in lecture, all homework problems, all lab

### Domination and Independence on Square Chessboard

Engineering nd Technology Journl Vol. 5, Prt, No. 1, 017 A.A. Omrn Deprtment of Mthemtics, College of Eduction for Pure Science, University of bylon, bylon, Irq pure.hmed.omrn@uobby lon.edu.iq Domintion

### Lecture 19. Vector fields. Dan Nichols MATH 233, Spring 2018 University of Massachusetts. April 10, 2018.

Lecture 19 Vector fields Dan Nichols nichols@math.umass.edu MATH 233, Spring 218 University of Massachusetts April 1, 218 (2) Chapter 16 Chapter 12: Vectors and 3D geometry Chapter 13: Curves and vector

### Synchronous Machine Parameter Measurement

Synchronous Mchine Prmeter Mesurement 1 Synchronous Mchine Prmeter Mesurement Introduction Wound field synchronous mchines re mostly used for power genertion but lso re well suited for motor pplictions

### Spiral Tilings with C-curves

Spirl Tilings with -curves Using ombintorics to Augment Trdition hris K. Plmer 19 North Albny Avenue hicgo, Illinois, 0 chris@shdowfolds.com www.shdowfolds.com Abstrct Spirl tilings used by rtisns through

### Math Final Exam - 6/11/2015

Math 200 - Final Exam - 6/11/2015 Name: Section: Section Class/Times Instructor Section Class/Times Instructor 1 9:00%AM ( 9:50%AM Papadopoulos,%Dimitrios 11 1:00%PM ( 1:50%PM Swartz,%Kenneth 2 11:00%AM

### (CATALYST GROUP) B"sic Electric"l Engineering

(CATALYST GROUP) B"sic Electric"l Engineering 1. Kirchhoff s current l"w st"tes th"t (") net current flow "t the junction is positive (b) Hebr"ic sum of the currents meeting "t the junction is zero (c)

### University of California, Berkeley Department of Mathematics 5 th November, 2012, 12:10-12:55 pm MATH 53 - Test #2

University of California, Berkeley epartment of Mathematics 5 th November, 212, 12:1-12:55 pm MATH 53 - Test #2 Last Name: First Name: Student Number: iscussion Section: Name of GSI: Record your answers

### Design and Modeling of Substrate Integrated Waveguide based Antenna to Study the Effect of Different Dielectric Materials

Design nd Modeling of Substrte Integrted Wveguide bsed Antenn to Study the Effect of Different Dielectric Mterils Jgmeet Kour 1, Gurpdm Singh 1, Sndeep Ary 2 1Deprtment of Electronics nd Communiction Engineering,

### [f(t)] 2 + [g(t)] 2 + [h(t)] 2 dt. [f(u)] 2 + [g(u)] 2 + [h(u)] 2 du. The Fundamental Theorem of Calculus implies that s(t) is differentiable and

Midterm 2 review Math 265 Fall 2007 13.3. Arc Length and Curvature. Assume that the curve C is described by the vector-valued function r(r) = f(t), g(t), h(t), and that C is traversed exactly once as t

### Calculus 3 Exam 2 31 October 2017

Calculus 3 Exam 2 31 October 2017 Name: Instructions: Be sure to read each problem s directions. Write clearly during the exam and fully erase or mark out anything you do not want graded. You may use your

### i + u 2 j be the unit vector that has its initial point at (a, b) and points in the desired direction. It determines a line in the xy-plane:

1 Directional Derivatives and Gradients Suppose we need to compute the rate of change of f(x, y) with respect to the distance from a point (a, b) in some direction. Let u = u 1 i + u 2 j be the unit vector

### TIME: 1 hour 30 minutes

UNIVERSITY OF AKRON DEPARTMENT OF ELECTRICAL AND COMPUTER ENGINEERING 4400: 34 INTRODUCTION TO COMMUNICATION SYSTEMS - Spring 07 SAMPLE FINAL EXAM TIME: hour 30 minutes INSTRUCTIONS: () Write your nme

### Module 9. DC Machines. Version 2 EE IIT, Kharagpur

Module 9 DC Mchines Version EE IIT, Khrgpur esson 40 osses, Efficiency nd Testing of D.C. Mchines Version EE IIT, Khrgpur Contents 40 osses, efficiency nd testing of D.C. mchines (esson-40) 4 40.1 Gols

### MATH 259 FINAL EXAM. Friday, May 8, Alexandra Oleksii Reshma Stephen William Klimova Mostovyi Ramadurai Russel Boney A C D G H B F E

MATH 259 FINAL EXAM 1 Friday, May 8, 2009. NAME: Alexandra Oleksii Reshma Stephen William Klimova Mostovyi Ramadurai Russel Boney A C D G H B F E Instructions: 1. Do not separate the pages of the exam.

### Notes on Spherical Triangles

Notes on Spheril Tringles In order to undertke lultions on the elestil sphere, whether for the purposes of stronomy, nvigtion or designing sundils, some understnding of spheril tringles is essentil. The

### Fubini for continuous functions over intervals

Fuini for ontinuous funtions over intervls We first prove the following theorem for ontinuous funtions. Theorem. Let f(x) e ontinuous on ompt intervl =[, [,. Then [, [, [ [ f(x, y)(x, y) = f(x, y)y x =

### EET 438a Automatic Control Systems Technology Laboratory 5 Control of a Separately Excited DC Machine

EE 438 Automtic Control Systems echnology bortory 5 Control of Seprtely Excited DC Mchine Objective: Apply proportionl controller to n electromechnicl system nd observe the effects tht feedbck control

### Understanding Basic Analog Ideal Op Amps

Appliction Report SLAA068A - April 2000 Understnding Bsic Anlog Idel Op Amps Ron Mncini Mixed Signl Products ABSTRACT This ppliction report develops the equtions for the idel opertionl mplifier (op mp).

### The Chain Rule, Higher Partial Derivatives & Opti- mization

The Chain Rule, Higher Partial Derivatives & Opti- Unit #21 : mization Goals: We will study the chain rule for functions of several variables. We will compute and study the meaning of higher partial derivatives.

### Synchronous Machine Parameter Measurement

Synchronous Mchine Prmeter Mesurement 1 Synchronous Mchine Prmeter Mesurement Introduction Wound field synchronous mchines re mostly used for power genertion but lso re well suited for motor pplictions

### Math 5BI: Problem Set 1 Linearizing functions of several variables

Math 5BI: Problem Set Linearizing functions of several variables March 9, A. Dot and cross products There are two special operations for vectors in R that are extremely useful, the dot and cross products.

### First Round Solutions Grades 4, 5, and 6

First Round Solutions Grdes 4, 5, nd 1) There re four bsic rectngles not mde up of smller ones There re three more rectngles mde up of two smller ones ech, two rectngles mde up of three smller ones ech,

### 1.6. QUADRIC SURFACES 53. Figure 1.18: Parabola y = 2x 2. Figure 1.19: Parabola x = 2y 2

1.6. QUADRIC SURFACES 53 Figure 1.18: Parabola y = 2 1.6 Quadric Surfaces Figure 1.19: Parabola x = 2y 2 1.6.1 Brief review of Conic Sections You may need to review conic sections for this to make more

### Review Problems. Calculus IIIA: page 1 of??

Review Problems The final is comprehensive exam (although the material from the last third of the course will be emphasized). You are encouraged to work carefully through this review package, and to revisit

### Independence of Path and Conservative Vector Fields

Independence of Path and onservative Vector Fields MATH 311, alculus III J. Robert Buchanan Department of Mathematics Summer 2011 Goal We would like to know conditions on a vector field function F(x, y)

### 10.1 Curves defined by parametric equations

Outline Section 1: Parametric Equations and Polar Coordinates 1.1 Curves defined by parametric equations 1.2 Calculus with Parametric Curves 1.3 Polar Coordinates 1.4 Areas and Lengths in Polar Coordinates

### Experiment 3: The research of Thevenin theorem

Experiment 3: The reserch of Thevenin theorem 1. Purpose ) Vlidte Thevenin theorem; ) Mster the methods to mesure the equivlent prmeters of liner twoterminl ctive. c) Study the conditions of the mximum

### MATH Review Exam II 03/06/11

MATH 21-259 Review Exam II 03/06/11 1. Find f(t) given that f (t) = sin t i + 3t 2 j and f(0) = i k. 2. Find lim t 0 3(t 2 1) i + cos t j + t t k. 3. Find the points on the curve r(t) at which r(t) and

### Name: ID: Section: Math 233 Exam 2. Page 1. This exam has 17 questions:

Page Name: ID: Section: This exam has 7 questions: 5 multiple choice questions worth 5 points each. 2 hand graded questions worth 25 points total. Important: No graphing calculators! Any non scientific

### CHAPTER 3 EDGE DETECTION USING CLASICAL EDGE DETECTORS

CHAPTER 3 EDE DETECTION USIN CLASICAL EDE DETECTORS Edge detection is one o te most importnt opertions in imge nlsis. An edge is set o connected piels tt lie on te boundr between two regions. Te clssiiction

### Practice problems from old exams for math 233

Practice problems from old exams for math 233 William H. Meeks III January 14, 2010 Disclaimer: Your instructor covers far more materials that we can possibly fit into a four/five questions exams. These

### ABB STOTZ-KONTAKT. ABB i-bus EIB Current Module SM/S Intelligent Installation Systems. User Manual SM/S In = 16 A AC Un = 230 V AC

User Mnul ntelligent nstlltion Systems A B 1 2 3 4 5 6 7 8 30 ma 30 ma n = AC Un = 230 V AC 30 ma 9 10 11 12 C ABB STOTZ-KONTAKT Appliction Softwre Current Vlue Threshold/1 Contents Pge 1 Device Chrcteristics...

### The Discussion of this exercise covers the following points:

Exercise 4 Bttery Chrging Methods EXERCISE OBJECTIVE When you hve completed this exercise, you will be fmilir with the different chrging methods nd chrge-control techniques commonly used when chrging Ni-MI

### For each question, X indicates a correct choice. ANSWER SHEET - BLUE. Question a b c d e Do not write in this column 1 X 2 X 3 X 4 X 5 X 6 X 7 X 8 X

For each question, X indicates a correct choice. ANSWER SHEET - BLUE X ANSWER SHEET - GREEN X ANSWER SHEET - WHITE X ANSWER SHEET - YELLOW For each question, place an X in the box of your choice. X QUESTION

### c The scaffold pole EL is 8 m long. How far does it extend beyond the line JK?

3 7. 7.2 Trigonometry in three dimensions Questions re trgeted t the grdes indicted The digrm shows the ck of truck used to crry scffold poles. L K G m J F C 0.8 m H E 3 m D 6.5 m Use Pythgors Theorem

### Experiment 3: Non-Ideal Operational Amplifiers

Experiment 3: Non-Idel Opertionl Amplifiers 9/11/06 Equivlent Circuits The bsic ssumptions for n idel opertionl mplifier re n infinite differentil gin ( d ), n infinite input resistnce (R i ), zero output

### Experiment 3: Non-Ideal Operational Amplifiers

Experiment 3: Non-Idel Opertionl Amplifiers Fll 2009 Equivlent Circuits The bsic ssumptions for n idel opertionl mplifier re n infinite differentil gin ( d ), n infinite input resistnce (R i ), zero output

### Energy Harvesting Two-Way Channels With Decoding and Processing Costs

IEEE TRANSACTIONS ON GREEN COMMUNICATIONS AND NETWORKING, VOL., NO., MARCH 07 3 Energy Hrvesting Two-Wy Chnnels With Decoding nd Processing Costs Ahmed Arf, Student Member, IEEE, Abdulrhmn Bknin, Student

### SOLVING TRIANGLES USING THE SINE AND COSINE RULES

Mthemtics Revision Guides - Solving Generl Tringles - Sine nd Cosine Rules Pge 1 of 17 M.K. HOME TUITION Mthemtics Revision Guides Level: GCSE Higher Tier SOLVING TRIANGLES USING THE SINE AND COSINE RULES

### University of North Carolina-Charlotte Department of Electrical and Computer Engineering ECGR 4143/5195 Electrical Machinery Fall 2009

Problem 1: Using DC Mchine University o North Crolin-Chrlotte Deprtment o Electricl nd Computer Engineering ECGR 4143/5195 Electricl Mchinery Fll 2009 Problem Set 4 Due: Thursdy October 8 Suggested Reding:

### Review Sheet for Math 230, Midterm exam 2. Fall 2006

Review Sheet for Math 230, Midterm exam 2. Fall 2006 October 31, 2006 The second midterm exam will take place: Monday, November 13, from 8:15 to 9:30 pm. It will cover chapter 15 and sections 16.1 16.4,

### Joanna Towler, Roading Engineer, Professional Services, NZTA National Office Dave Bates, Operations Manager, NZTA National Office

. TECHNICA MEMOANDM To Cc repred By Endorsed By NZTA Network Mngement Consultnts nd Contrctors NZTA egionl Opertions Mngers nd Are Mngers Dve Btes, Opertions Mnger, NZTA Ntionl Office Jonn Towler, oding

### Solutions to the problems from Written assignment 2 Math 222 Winter 2015

Solutions to the problems from Written assignment 2 Math 222 Winter 2015 1. Determine if the following limits exist, and if a limit exists, find its value. x2 y (a) The limit of f(x, y) = x 4 as (x, y)

### Ch13 INTRODUCTION TO NUMERICAL TECHNIQUES FOR NONLINEAR SUPERSONIC FLOW

Ch13 INTRODUCTION TO NUMERICAL TECHNIQUES FOR NONLINEAR SUPERSONIC FLOW Goerning Eqtions for Unste Iniscid Compressible Flow Eler's eqtion Stte eqtions finite-difference nmericl techniqes Goerning Eqtions

### CHAPTER 11 PARTIAL DERIVATIVES

CHAPTER 11 PARTIAL DERIVATIVES 1. FUNCTIONS OF SEVERAL VARIABLES A) Definition: A function of two variables is a rule that assigns to each ordered pair of real numbers (x,y) in a set D a unique real number