MAT 241- Calculus 3- Prof. Santilli Toughloves Chapter 16

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1 MAT 41- alulus 3- Prof. antilli Toughloves hapter 16 1.) Vetor Fields: funtions that assign vetors to points in spae..) tandard form of a vetor field: F (x, y) = M(x,y)ˆ i + N(x, y) ˆ j over a plane F (x, y,z) = M(x, y,z)ˆ i + N (x, y,z) ˆ j + P(x,y, z) k ˆ over spae 3.) entral Vetor Fieldsa.) always points towards the origin b.) magnitudes of the vetors are the same at all points equidistant from the origin 4.) Inverse quare Fieldsa.) Has the form of F (x, y,z) 1 r r ˆ 5.) Gradient is a vetor field 6.) onservative Vetor field a.) Vetors of the field are normal to the level urves from whih they emanate. b.) The field is the gradient of some funtion (potential funtion). If F = f then F is a onservative vetor field and f is the potential funtion. 7.) onservative Vetor field- onservation of energy exist 8.) Test for onservative field in a plane: a.) If F (x, y) = M(x,y)ˆ i + N(x, y) ˆ j is onservative then M y = N x 9.) Differential Operator (3-D)- Dell: = i x ˆ + y ˆ j + k z ˆ =,, x y z 10.) url of Vetor Field: i ˆ j ˆ k ˆ urlf = F = measures the field s ability to rotate (url around an axis) x y z M N P 11.) Irrotational Field: url F = F = 0 1.) Test for onservative field in spae: a.) If F (x, y,z) = M(x, y,z)ˆ i + N (x, y,z) ˆ j + P(x,y, z) k ˆ is onservative then M y = N x, M z = P x, P y = N z or url F = F = 0 13.) Irrotational Field: url F = F = 0, therefore onservative fields are irrotational. 14.) Laplae Operator (Laplaian): = = x + y + z

2 15.) Laplaian of a vetor field: F = Mˆ i + Nˆ j + Pk ˆ 16.) Divergene of a vetor field: divf = F = M x + N y + P whih measures the z tendeny for the vetor field to spread out (diverge) or ome together (onverge) at any point in the field. 17.) If div F = F = 0 then F is divergene free, inompressible or solenoidal. 18.) div( urlf )= ( F )= 0 Divergene of url of a vetor field (triple salar produt) div f = f = f = + + Divergene of the gradient of a funtion 19.) ( ) 0.) url ( f ) = f = 0 url of the gradient of a funtion x f y f z 1.) url( urlf ) ( F ) = ( F ) F f = url of url of a vetor field (triple produt).) url ( ff ) = ( ff ) = f F + f ( F ) 3.) div ( ff ) = ( ff ) = F f + f ( F ) 4.) div ( F G ) = ( F G ) = ( F ) G F ( G ) 5.) [ f ( g)] = f ( g) f 6.) Ar Length (review): dr Given the vetor funtion: r (t) = x(t)ˆ i + y(t) ˆ j + z(t) k ˆ a.) dr (t) = dx(t)ˆ i + dy(t) ˆ j + dz(t) k ˆ b.) dr (t) = r (t)dt.) dr (t) = T dt d.) dr (t) = TT ˆ dt e.) dr (t) = r (t) T ˆ dt f.) dr (t) = dst ˆ ds Given the vetor funtion: r (t) = x(t)ˆ i + y(t) ˆ j + z(t) k ˆ a.) ds = dr b.) ds = r (t)dt or ds dt = r (t).) ds ds = r (s) = 1

3 d.) ds = Tdt e.) ds = dx + dy + dz f.) ds = g.) ds = dx + dy + dz dt 1 + dy dx = 1 + dx dy = r + dr dθ dx dy dθ 7.) Line Integrals: Arlength Form: fds = f r (t) dt = f dx + dy + dz dt Vetor Form: F dr = F Tds ˆ = F r ( t) dt Differential Form: F dr = Mdx + Ndy + Pdz 8.) For Vetor and Differential Form: Value of the line integral depends on the diretion of the path along the urve. 9.) For arlength Form: Value of the line integral DOE NOT depend on the diretion of the path along the urve. 30.) Line integrals (integration along urves) Possible appliationsa.) Lateral surfae area of ylinders b.) mass of urved wires 31.) Fundamental theorem of line integrals: f dr = defined as r (t) in the interval a t b. f ( r (b)) f ( r (a)) where the urve is 3.) For onservative fields: F dr = f dr = f ( r (b)) f ( r (a))= potential funtion at b- potential funtion at a a.) Line integral of a onservative field is independent of path 33.) Line integral of onservative field along a losed path is zero, i.e., F onservative dr = 0.

4 34.) losed paths have + orientation if the region enlosed by the path is to the LEFT of the path diretion. Otherwise, the orientation is negative. 35.) Path Types imple, not losed Not simple, not losed imple, losed Not simple, losed 36.) egion Types imply- onneted Non simply- onneted 37.) Green s Theorem For a simply losed path around the boundary of the simply onneted region: N Mdx + Ndy = x M da y a.) Green s theorem for finding Area- edue to a line integrals to evaluate area: 1 1.) xdy ydx = Area of egion.) xdy = Area of egion 3.) ydx = Area of egion 38.) For non simply onneted regions, turn the regions into multiple simply onneted regions then apply Green s Theorem to eah new region. 39.) First Vetor Form of Green s Theorem (tokes Theorem in -D) For a simply losed path around the boundary of the simply onneted region: F dr = F T ˆ ds N = x M da = ( F y ) k ˆ da = ( urlf ) k ˆ da

5 40.) eond Vetor Form of Green s Theorem (Divergene (Gauss) Theorem in -D) For a simply losed path around the boundary of the simply onneted region: M N F Nds ˆ = + da = ( F ) da = ( divf ) da x y 41.) tandard form of a parametri surfae: r (u,v) = x(u,v)ˆ i + y(u,v) ˆ j + z(u,v) k ˆ where x = x(u,v), y = y(u,v), and z = z(u,v) are the parametri equations for the surfae. 4.) Normal vetor to a parametri surfae: N = r u r v, and = r u r v where r u r v r u = x u (u,v)ˆ i + y u (u,v)ˆ j + z u (u,v) k ˆ and r v = x v (u,v)ˆ i + y v (u,v) ˆ j + z v (u,v) k ˆ 43.) Orientation of a surfae: Outside surfae = + orientation (normal points outwards), inside surfae = orientation (normal points inwards) 44.) Parametri surfae and normal for a sphere: r = asinϕ osθˆ i + a sinϕ sinθˆ j + aosϕk ˆ where ˆ n = 1 a r (note that ˆ n is in the same diretion as r ) 45.) Parametri surfae of evolution, θ is the angle of revolution: otate y = f (x) about the x-axis x = x y = f (x)osθ z = f (x)sinθ otate x = f (y) about the y-axis x = f (y)osθ y = y z = f (y)sinθ otate x = f (z) about the z-axis x = f (z)osθ y = f (z)sinθ z = z 46.) urfae Area d Given the parametri surfae: r (u,v) = x(u,v)ˆ i + y(u,v) ˆ j + z(u,v) k ˆ a.) d = r u r v da Parametri form b.) d = N da beause N = r u r v whih is the normal to the surfae.) d = G da beause G(x,y, z) = z f (x, y). Normal to level surfaes is the gradient, i.e., N = G d.) d = 1 + f x + fy da beause G(x,y, z) = z f (x, y)

6 47.) urfae Area d Given the parametri surfae: r (u,v) = x(u,v)ˆ i + y(u,v) ˆ j + z(u,v) k ˆ a.) d = d b.) d = G G d beause = G G where G(x,y, z) = z f (x, y).) d = GdA beause d = G da. d.) d = m f x,m f y, ±1 da beause G(x,y, z) = z f (x, y). +1 k ˆ, oriented up 1k ˆ, oriented down e.) d = ( r u r v )da beause d = d = r u r v da = N N r u r v da = r u r v r r u r u r v da. v 48.) urfae Integrals (Know the differene between d and alar Form: 1.) urfae area: = d.) urfae integral fd a.) mass = b.) x = xρd ρd ρd, y = yρd ρd urfae, z = zρd ρd da) egion.) I z = (x + y )ρd, I x = (y + z )ρd, I y = (x + z )ρd

7 Vetor Form: 1.) Flux: Φ = F d = volume rossing normal to the surfae/ unit time a.) Mass Flux: Φ mass = ρf d a.) Heat Flux: Φ heat = H d = = mass passing unit time k T d, where H is the heat flow, k is the thermal ondutivity of the material and T is the temperature distribution throughout the surfae. 49.) tokes Theorem boundary F dr = boundary F ˆ T ds = ( urlf ) d = enlosed surfae ( F ) d enlosed surfae irulation of vetor field around a losed path or work done by the field on a partile traveling along a losed path in the field = flux of the url of the vetor field. 50.) Divergene Theorem (Gauss s Theorem) F d = surfae enlosed volume ( divf )dv = enlosed volume ( F )dv For: divf > 0 oure (expanding fluid) divf < 0 ink (ompressing fluid) divf = 0 Inompressible Flux of the vetor field = dv/dt 51.) OVEVIEW- examples of mass A B V b mass = ρdx mass = ρds mass = ρda mass = a urve region ρd mass = solid surfae ρdv

8 5.) Fundamental Theorem of alulus Integral of a derivative over a region = Original funtion evaluated on the boundary of the region b a f dr = f ( r ( b) ) f ( r ( a) ) F (x)dx = F(b) F(a) N x M da = Mdx + Ndy y F d = Q F dr F dv = F d a r ( a) Q b r ( b)