Math 2a: Multivariable calculus Oliver Knill, ummer 208 hecklist III Partial Derivatives f x (x,y) = f(x,y) partial derivative x L(x,y) = f(x 0,y 0 )+f x (x 0,y 0 )(x x 0 )+f y (x 0,y 0 )(y y 0 ) linear approximation usel(x,y)toestimatef(x,y)nearf(x 0,y 0 ). Theresultisf(x 0,y 0 )+a(x x 0 )+b(y y 0 ) tangent line: ax+by = d with a = f x (x 0,y 0 ),b = f y (x 0,y 0 ), d = ax 0 +by 0 tangent plane: ax+by +cz = d with a = f x,b = f y,c = f z, d = ax 0 +by 0 +cz 0 estimate f(x,y,z) by L(x,y,z) near (x 0,y 0,z 0 ) f xy = f yx lairaut s theorem r u (u,v), r v (u,v) tangent to surface parameterized by r(u,v) Partial Differential Equations f t = f xx heat equation f tt f xx = 0 wave equation f x f t = 0 transport equation = advection equation f xx +f yy = 0 Laplace equation f t +ff x = f xx Burgers equation f t = f xf x x 2 f xx Black choles Gradient f(x,y) = [f x,f y ] T, f(x,y,z) = [f x,f y,f z ] T, gradient direction = vector of length = unit vector D v f = f v directional derivative, v is unit vector d f( r(t)) = f( r(t)) r (t) chain rule dt f(x 0,y 0 ) is orthogonal to the level curve f(x,y) = c containing (x 0,y 0 ) f(x 0,y 0,z 0 ) is orthogonal to the level surface f(x,y,z) = c containing (x 0,y 0,z 0 ) d f( x+t v) = D dt vf by chain rule (x x 0 )f x (x 0,y 0 )+(y y 0 )f y (x 0,y 0 ) = 0 tangent line (x x 0 )f x (x 0,y 0,z 0 )+(y y 0 )f y (x 0,y 0,z 0 )+(z z 0 )f z (x 0,y 0,z 0 ) = 0 tangent plane directionalderivativeat(x 0,y 0 )ismaximalinthe v = f(x 0,y 0 )/ f(x 0,y 0 ) direction f(x,y) increases in the f/ f direction at points which are not critical points partial derivatives are special directional derivatives: D i f = f x if D v f( x) = 0 for all v, then f( x) = 0 f(x,y,z) = c defines y = g(x,y), and g x (x,y) = f x (x,y,z)/f z (x,y,z) implicit diff angle between two surfaces: angle between the normal vectors
Extrema f(x,y) = [0,0] T, critical point or stationary point f(x,y,z) = [0,0,0] T, critical point of function of three variables D = f xx f yy f 2 xy discriminant, useful in second derivative test f(x 0,y 0 ) f(x,y) in a neighborhood of (x 0,y 0 ) local maximum f(x 0,y 0 ) f(x,y) in a neighborhood of (x 0,y 0 ) local minimum f(x,y) = λ g(x,y),g(x,y) = c, λ Lagrange equations f(x,y,z) = λ g(x,y,z),g(x,y,z) = c, λ Lagrange equations second derivative test: f = (0,0),D > 0,f xx < 0 local max, f = (0,0),D > 0,f xx > 0 local min, f = (0,0),D < 0 saddle point f(x 0,y 0 ) f(x,y) everywhere, global maximum f(x 0,y 0 ) f(x,y) everywhere, global minimum Double Integrals f(x,y) dydx double integral b d a c b d(x) a c(x) d b(y) c a(y) f(x,y) dydx integral over rectangle f(x,y) dydx type I region f(x,y) dxdy type II region f(r,θ) r drdθ polar coordinates r u r v dudv surface area b d f(x,y) dydx = d b a c c a dxdy area of region f(x,y) dxdy Fubini f(x,y) dxdy signed volume of solid bound by graph of f and xy-plane Triple Integrals f(x,y,z) dzdydx triple integral b d v a c u b g2 (x) a g (x) b a d c f(x,y,z) dzdydx integral over rectangular box h2 (x,y) h f(x,y) dzdydx type I region (x,y) f(r,θ,z) r dzdrdθ integral in cylindrical coordinates f(ρ,θ,z) ρ2 sin(φ) dzdrdθ integral in spherical coordinates v f(x,y,z) dzdydx = v f(x,y,z) dxdydz Fubini u u V = dzdydx volume of solid E E M = σ(x,y,z) dzdydz mass of solid E with density σ E d c b a Line Integrals F(x,y) = [P(x,y),Q(x,y)] T vector field in the plane F(x,y,z) = [P(x,y,z),Q(x,y,z),(x,y,z)] T vector field in space F dr = b a F( r(t)) r (t) dt line integral F(x,y) = f(x,y) gradient field = potential field = conservative field 2
Fundamental theorem of line integrals FTL: F(x,y) = f(x,y), b a F( r(t)) r (t)) dt = f( r(b)) f( r(a)) closed loop property F dr = 0, for all closed curves always equivalent: closed loop property, path independence and gradient field mixed derivative test curl( F) 0 assures F is not a gradient field Green s Theorem F(x,y) = [P,Q] T, curl in two dimensions: curl( F) = Q x P y Green s theorem: boundary of, then F dr = curl( F) dxdy area computation: Take F with curl( F) = Q x P y = like F = [ y,0] T or F = [0,x] T Green s theorem is useful to compute difficult line integrals or difficult 2D integrals Flux integrals F(x,y,z) vector field, = r() parametrized surface r u r v normal vector, n = ru rv r u r v unit normal vector r u r v dudv = d = n d normal surface element F d = F( r(u,v)) ( r u r v ) dudv flux integral tokes Theorem F(x,y,z) = [P,Q,] T, curl([p,q,] T ) = [ y Q z,p z x,q x P y ] T = F tokes s theorem: boundary of surface, then F dr = curl( F) d tokes theorem allows to compute difficult flux integrals or difficult line integrals Grad url Div = [ x, y, z ] T, F = f, curl( F) = F, div( F) = F div(curl( F)) = 0 and curl(grad( f)) = 0 div(grad(f)) = f Laplacian incompressible = divergence free field: div( F) = 0 everywhere. Implies F = curl( G) irrotational = curl( F) = 0 everywhere. Implies F = grad(f) Divergence Theorem div([p,q,] T ) = P x +Q y + z = F divergence theorem: solid E, boundary then F d = E div( F) dv the divergence theorem allows to compute difficult flux integrals or difficult 3D integrals 3
ome topology interior of a region D: points in D for which small neighborhood is still in D boundary of a curve: the end points of the curve if they exist boundary of a surface points on surface belonging to parameters (u,v) not in the interior of the parameter domain boundary of a solid D: the surfaces which bound the solid, points in the solid which are not in the interior of D closed surface: a surface without boundary like for example the sphere closed curve: a curve with no boundary like for example a knot ome surface parameterizations sphere of radius ρ: r(u,v) = [ρcos(u)sin(v),ρsin(u)sin(v),ρcos(v)] T graph of function f(x,y): = r(u,v) = [u,v,f(u,v)] T example: Paraboloid: r(u,v) = [u,v,u 2 +v 2 ] T. plane containing P and vectors u, v: r(s,t) = OP +s u+t v surface of revolution: distance g(z) of z axis : r(u,v) = [g(v)cos(u),g(v)sin(u),v] T example: ylinder: r(u,v) = [cos(u),sin(u),v] T example: one: r(u,v) = [vcos(u),vsin(u),v] T example: Paraboloid: r(u,v) = [ vcos(u), vsin(u),v] T Integration overview Line integral: b a F( r(t)) r (t) dt Flux integral: F( r(u,v)) ( r u r v ) dudv Theorems and Derivatives overview grad grad 2 curl grad 3 curl 3 div FT FTL 2 Green FTL 3 tokes 3 Gauss 4
Maths 2a: Multivariable calculus Oliver Knill, 208 Integral theorems overview The integral theorems are summarized by the fundamental theorem of multivariable alculus: df = F G where df is a derivative of F and δg is the boundary of G. δg FT FTL 2 Green FTL 3 tokes 3 Gauss Fundamental theorem of line integrals: if is an oriented curve with boundary {A,B} and f is a function, then f dr = f(b) f(a) emarks. ) For closed curves, f dr = 0. 2) Gradient fields are path independent: if F = f, then B F dr does not depend on the A path connecting A with B. 3) The theorem justifies the name conservative for gradient vector fields. 4) The term potential was coined by George Green who lived from 793-84. Let f(x,y,z) = x 2 +y 4 +z. Find the line integral of F(x,y,z) = f(x,y,z) along the path r(t) = [cos(5t),sin(2t),t 2 ] T from t = 0 to t = 2π. olution. r(0) = [,0,0] T and r(2π) = [,0,4π 2 ] T and f( r(0)) = and f( r(2π)) = +4π 2. The fundamental theorem of line integral gives f dr = f( r(2π)) f( r(0)) = 4π 2. Green s theorem. If is a region with oriented boundary and F is a vector field, then curl( F) dxdy = F dr. emarks. ) The curve is oriented in such a way that the region is to the left. 2) The boundary of the curve can consist of piecewise smooth pieces. 3)If : t r(t) = [x(t),y(t)] T,thelineintegralis b [P(x(t),y(t)),Q(x(t),y(t))]T [x (t),y (t)] T dt. a 4) Green s theorem was found by George Green (793-84) in 827 and by Mikhail Ostrogradski (80-862). 5) If curl( F) = 0 everywhere in the plane, then the field is a gradient field. 7) F(x,y) = [0,x] T we get an area formula. 8) Don t mix up Green with tokes. Green is a theorem in flatland, where F has two components. and curl( F) is a scalar.
2 Find the line integral of the vector field F(x,y) = [x 4 +sin(x)+y,x+y 3 ] T along the path r(t) = [cos(t),5sin(t)+log(+sin(t))] T with t [0,π]. olution. ince curl( F) = 0 we can take the simpler path r(t) = [ t,0] T, t, which has velocity r (t) = [,0] T. The line integral is [t4 sin(t), t] T [,0] T dt = t 5 /5 = 2/5. tokes theorem. If is a surface with boundary and F is a vector field, then curl( F) d = F dr. emarks. ) tokes theorem implies Green s theorem: if F = [P,Q,0] T is z-independent and the surface is contained in the xy-plane. 2) The orientation of is such that if you walk along with head in the direction of the normal vector r u r v, of, then to your left. 3) tokes theorem was also found by André Ampère (775-836) in 825. George tokes (89-903) gave it as a multivariable exam problem. 4) The flux of the curl of F through only depends on the boundary of. 5) The flux of the curl through a closed surface is zero. 3 Find the line integral of F(x,y,z) = [x 3 +xy,y,z] T along the polygonal path connecting (0,0,0),(2,0,0),(2,,0),(0,,0). olution. The path bounds a surface : r(u,v) = [u,v,0] T parameterized by = [0,2] [0,]. By tokes theorem, the line integral is equal to the flux of curl( F)(x,y,z) = [0,0, x] T through. The normal vector of is r u r v = [,0,0] T [0,,0] T = [0,0,] T so that curl( F) d = 2 0 0 [0,0, u]t [0,0,] T dudv = 2 0 0 u dudv = 2. Divergence theorem: If is the oriented boundary of a solid region E in space and F is a vector field, then div( F) dv = F d. B emarks. ) The surface is oriented so that the normal vector points away from E. 2) The divergence theorem is also called Gauss theorem. 3) It can be helpful to determine the flux of vector fields through surfaces. 4) The theorem was discovered in 764 by Joseph Louis Lagrange (736-83), later it was rediscovered by arl Friedrich Gauss (777-855) and by George Green. 5) For divergence-free vector fields F, the flux through a closed surface is zero. uch fields F are also called incompressible or source free. 4 ompute the flux of the vector field F(x,y,z) = [ x,y,z 2 ] T through the boundary of the rectangular box [0,3] [,2] [,2]. olution. By Gauss theorem, the flux is equal to the triple integral of div(f) = 2z over the box: 3 0 2 2 2z dxdydz = (3 0)(2 ( ))(4 ) = 27. 2
Maths 2a: Multivariable calculus Oliver Knill, ummer 208 Integral theorems overview INTEGATION. Line integral: Flux integral: Double integral: Triple integral: DIFFEENTIATION. b F( r(t)) r (t) dt a F( r(u,v)) r u r v dudv f(x,y) dxdy. f(x,y,z) dxdydz. E Area da = dxdy Arc length b a r (t) dt urface area r u r v dudv Volume B dxdydz IDENTITIE. div(curl( F)) = 0 curl(grad(f)) = 0 div(grad(f)) = f. Velocity: r (t) = d/dt r(t). Partial derivative: f x (x,y,z). Gradient: grad(f) = [f x,f y,f z ] T url in 2D: curl([p,q] T ) = Q x P y JAGON. url in 3D: curl[p,q,] T = [ y Q z,p z x,q x P y ] T Div: div( F) = div[p,q,] T = P x +Q y + z. ONEVATIVE FIELD: ) Gradient fields: F = grad(f). 2) losed curve property: F dr = 0 for any closed curve. 3) onservative:, 2 paths from A to B, then F dr = 2 F dr. 4) Mixed derivative test: curl(f) = 0 if F = f. 4) If F is irrotational, then F = f in simply connected. div( F) = 0 incompressible curl F) = 0 irrotational TOPOLOGY. ) Interior of region D: points which have a neighborhood contained in D. 2) Boundary of curve: endpoints. Boundary of surface: curves. Boundary of solid: surfaces. 3) imply connected: a closed curve in can be deformed inside to a point. 4) losed curve urve without boundary. 5) losed surface surface without boundary. LINE INTEGAL THEOEM. If : r(t) = [x(t),y(t),z(t)] T, t [a,b] is a curve and f(x,y,z) is a function. f dr = f( r(b)) f( r(a)) GEEN THEOEM. If is a region with boundary and F = [P,Q] T is a vector field, then curl(f) dxdy = F dr TOKE THEOEM. If is a surface with boundary and F is a vector field, then curl( F) d = F dr
DIVEGENE THEOEM. If is the boundary of a solid E in space with boundary surface and F is a vector field, then E div( F) dv = F d GENEAL TOKE. All theorems are of the form G df = where df is the derivative of F and δg is the oriented boundary of G. δg F George Gabriel tokes Mikhail Ostrogradsky arl Friedrich Gauss André-Marie Ampère Joseph Louis Lagrange George Green Last update: summer 208, Harvard University Math2a, Oliver Knill 2