Math 114 Review Math 240 Grad, Div, Curl Gradient Divergence Curl How they're related Math 114 Review Line integrals Scalar line integrals Vector line integrals Conservative Math 240 | Calculus III fields Summer 2013, Session II Monday, July 1, 2013 Math 114 Review Agenda Math 240 Grad, Div, Curl Gradient Divergence Curl How they're 1. Gradient, Divergence, and Curl related Gradient Line integrals Scalar line Divergence integrals Vector line integrals Curl Conservative fields How they're related 2. Line integrals Scalar line integrals Vector line integrals Conservative vector fields Math 114 Review Gradient Math 240 Grad, Div, Curl Definition Gradient 3 Divergence Let f : X ⊆ R ! R be a differentiable scalar function on a Curl How they're region of 3-dimensional space. The gradient of f is the vector related Line integrals field Scalar line @f @f @f integrals grad f = rf = i + j + k: Vector line @x @y @z integrals Conservative fields rf The direction of the gradient, krfk , is the direction in which f is increasing the fastest. The norm, krfk, is the rate of this increase. Example If f(x; y; z) = x2 + y2 + z2 then rf = 2x i + 2y j + 2z k: Math 114 Review Divergence Math 240 Grad, Div, Curl Gradient Definition Divergence 3 3 Curl Let F : X ⊆ R ! R be a differentiable vector field with How they're related components F = Fxi + Fyj + Fzk. The divergence of F is the Line integrals scalar function Scalar line integrals Vector line @F @F @F integrals x y z Conservative div F = r · F = + + : fields @x @y @z The divergence of a vector field measures how much it is \expanding" at each point. Examples 1. If F = x i + y j then r · F = 2. 2. If F = −y i + x j then r · F = 0. Math 114 Review Curl Math 240 Grad, Div, Curl Gradient Definition Divergence 3 3 Curl Let F : X ⊆ ! be a differentiable vector field with How they're R R related components F = Fxi + Fyj + Fzk. The curl of F is the vector Line integrals Scalar line field integrals Vector line integrals i j k Conservative fields @ @ @ curl F = r × F = @x @y @z Fx Fy Fz @F @F @F @F @F @F = z − y i + x − z j + y − x k: @y @z @z @x @x @y The magnitude of the curl, kr × Fk, measures how much F ∇×F rotates around a point. The direction of the curl, k∇×Fk , is the axis around which it rotates. Math 114 Review Curl Math 240 Grad, Div, Curl Gradient Example Divergence Curl If F = −y i + x j then r × F = 2 k. How they're related Line integrals Scalar line integrals Vector line integrals Conservative fields Math 114 Review How they're related Math 240 Grad, Div, Curl Gradient Theorem Divergence 3 2 Curl Let f : X ⊆ R ! R be a C scalar function. Then How they're related r × (rf) = 0, that is, curl (grad f) = 0: Line integrals Scalar line integrals Theorem Vector line 3 3 2 integrals Let F : X ⊆ R ! R be a C vector field. Then Conservative fields r · (r × F) = 0, that is, div (curl F) = 0: To summarize, the composition of any two consecutive arrows in the diagram yields zero. 0 scalar grad vector curl $ vector div scalar functions / fields / fields / functions 9 0 Math 114 Review Scalar line integrals Math 240 Grad, Div, Curl Gradient Divergence Definition Curl 3 1 3 How they're Let x :[a; b] ! X ⊆ R be a C path and f : X ! R a related continuous function. The scalar line integral of f along x is Line integrals Scalar line integrals Z Z b Vector line 0 integrals f ds = f(x(t)) x (t) dt: Conservative x a fields In two dimensions, a scalar line integral measures the area under a curve with base x and height given by f. Math 114 Review Scalar line integrals Math 240 Grad, Div, Curl Example Gradient 3 Divergence Let x : [0; 2π] ! be the helix x(t) = (cos t; sin t; t) and let Curl R How they're related f(x; y; z) = xy + z. Let's compute Line integrals Z Z 2π Scalar line 0 integrals f ds = f(x(t)) x (t) dt: Vector line integrals x 0 Conservative fields We find p 0 p x (t) = sin2 t + cos2 t + 1 = 2; so now Z 2π Z 2π p 0 f(x(t)) x (t) dt = (cos t sin t + t) 2 dt 0 0 p Z 2π p 1 2 = 2 2 sin 2t + t dt = 2 2π : 0 Math 114 Review Vector line integrals Math 240 Grad, Div, Curl Gradient Definition Divergence Curl Let x :[a; b] ! X ⊆ 3 be a C1 path and F : X ! 3 a How they're R R related continuous vector field. The vector line integral of F along x Line integrals Scalar line is integrals Z Z b Vector line 0 integrals F · ds = F(x(t)) · x (t) dt: Conservative fields x a If F has components F = Fxi + Fyj + Fzk, the vector line integral can also be written Z Z F · ds = Fxdx + Fydy + Fzdz: x x Physically, a vector line integral measures the work done by the force field F on a particle moving along the path x. Math 114 Review Vector line integrals Math 240 Grad, Div, Curl Gradient Divergence Example Curl How they're 3 related Let x : [0; 1] ! R be the path x(t) = (2t + 1; t; 3t − 1) and Line integrals let F = −z i + x j + y k: Let's compute Scalar line integrals Z Z Vector line integrals F · ds = −zdx + xdy + ydz: Conservative fields x x First, we find x0(t) = (2; 1; 3), and now we can do Z Z 1 −zdx + xdy + ydz = −(3t − 1)(2) + (2t + 1) + t(3)dt x 0 Z 1 5 = −t + 3dt = 2 : 0 P1: OSO coll50424˙ch06P1: OSO PEAR591-Colley July 26, 2011 13:31 coll50424˙ch06 PEAR591-Colley July 26, 2011 13:31 Math 114 6.1 Scalar and Vector Line Integrals 417 Review Changing orientation 6.1 Scalar and Vector Line Integrals 417 Math 240 x b x a n 1 x b( ) x oppa ( ) EXAMPLE 7 If x:[a, b] →n R is any1 C path, then we may define the opposite ( ) opp( ) EXAMPLE 7 If x:[a, b] → R is any C path, then we may define the opposite Grad, Div, path x :[a, b] →n Rn by Curl path xopp:[oppa, b] → R by Gradient = + − . Divergence x (tx)opp=(xt()a +xb(a− t).b t) Curl opp How they're related = , → , (See(See Figure Figure 6.8.) 6.8.) That That is, xopp is,(tx)opp=(xt)(u(t)),x(u where(t)), whereu:[a, bu]:[→a [ba], b]isgivenby[a b]isgivenby Line integrals = + − xx aa x x b b u(tu) (t)a= ab+ bt.− Clearly,t. Clearly, then, then,xopp isx anopp orientation-reversingis an orientation-reversing reparametrization reparametrization Scalar line (( )) oppopp( () ) integrals x y of xof. x. ◆ ◆ Vector line integrals FigureFigure 6.8 6.8AA path path and and its its Conservative Figure: x and y have opposite orientations fields opposite.opposite. In additionIn addition to reversing to reversing orientation, orientation, a reparametrization a reparametrization of a path of can a path change can change = ◦ Z Z thethe speed. speed. This This follows follows readily readily from from the chain the chainrule: If rule:y Ifx yu=, thenx ◦ u, then f ds = f ds d y x y (t) = (x(ud(t))) = x (u(t))u (t). (4) y (tdt) = (x(u(t))) = x (u(t)) u (t). (4) Z Z dt F · ds = − F · ds So the velocity vector of the reparametrization y is just a scalar multiple (namely, y x So the velocity vector of the reparametrization y is just a scalar multiple (namely, u (t)) of the velocity vector of x. In particular, we have This can be achieved by negating t: u (t)) of the velocity vector of x. In particular, we have Speed of y = y(t) = u(t)x(u(t)) = = y(t) = x(−t): Speed of y y (t) u (t) x (u(t)) = u(t) x(u (t)) = u(t)· (speed of x). (5) = u(t) x(u(t)) = u(t) · (speed of x). (5) Since u is one-one, it follows that either u(t) ≥ 0 for all t ∈ [a, b]oru(t) ≤ 0 forSince all t ∈u[ais, b one-one,]. The first it case follows occurs that precisely either u when(t) ≥y is0 orientation-preservingfor all t ∈ [a, b]oru(t) ≤ 0 andfor the all secondt ∈ [a when, b].y Theis orientation-reversing. first case occurs precisely when y is orientation-preserving andHow the does second the line when integraly is orientation-reversing. of a function or a vector field along a path differ from theHow line integral does the (of line the integral same function of a function or vector or field) a vector along field a reparametriza- along a path differ tionfrom of a the path? line Not integral much (of at all. the The same precise function results or vector are stated field) in along Theorems a reparametriza- 1.4 andtion 1.5. of a path? Not much at all. The precise results are stated in Theorems 1.4 and 1.5. THEOREM 1.4 Let x:[a, b] → Rn be a piecewise C1 path and let f : X ⊆ Rn → R be a continuous function whose domain X contains the image of x.