Slides7-01.Pdf

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 = ∇f = i + j + k. Vector line ∂x ∂y ∂z integrals Conservative fields ∇f The direction of the gradient, k∇fk , is the direction in which f is increasing the fastest. The norm, k∇fk, is the rate of this increase. Example If f(x, y, z) = x2 + y2 + z2 then ∇f = 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 = ∇ · 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 ∇ · F = 2. 2. If F = −y i + x j then ∇ · 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 = ∇ × 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, k∇ × 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 ∇ × 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 ∇ × (∇f) = 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 ∇ · (∇ × 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 √ 0 p x (t) = sin2 t + cos2 t + 1 = 2, so now Z 2π Z 2π √ 0 f(x(t)) x (t) dt = (cos t sin t + t) 2 dt 0 0 √ Z 2π √ 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. , → n 1 ⊆ If yTHEOREM:[c, d] → R 1.4n is anyLet reparametrizationx:[a b] R ofbex, a then piecewise C path and let f : X Rn → R be a continuous function whose domain X contains the image of x. , → n If y:[c d] R is any reparametrizationfds= fds. of x, then y x fds= fds. PROOF We will explicitly prove the resulty in thex case where x (and, therefore, y) is of class C1. (When x is only piecewise C1, we can always break up the integral 1 appropriately.)PROOF We In will the C explicitlycase, we prove have, the by Definitionresult in the 1.1 case and where the observationsx (and, therefore, in y) equationis of class (5), thatC1. (When x is only piecewise C1, we can always break up the integral 1 appropriately.)d In the C case, we have,d by Definition 1.1 and the observations in    equationfds (5),= thatf (y(t)) y (t) dt = f (x(u(t))) x (u(t)) u (t) dt. y c c d d      If y is orientation-preserving,fds= f (y(t then)) yu((tc))=dta=, u(d) =f (bx,( andu(t)))u (tx) (=u(tu))(t).u Thus,(t) dt. using substitutiony of variables,c c d d   If y is orientation-preserving, then u(c) = a, u(d) = b, and u (t) = u (t). Thus, f (x(u(t))) x(u(t)) u(t) dt = f (x(u(t))) x(u(t)) u (t) dt usingc substitution of variables, c d b d      f (x(u(t))) x (u(t)) u (t=) dt =f (x(u))f (xx((uu()t)))dux=(u(t))fdsu. (t) dt c a c x b = f (x(u)) x(u) du = fds. a x Math 114 Review Conservative vector fields Math 240

Grad, Div, Curl Gradient Divergence Curl How they’re related Definition Line integrals A continuous vector field F is called a conservative vector Scalar line 1 integrals field, or a gradient field, if F = ∇f for some C scalar Vector line integrals function f. In this case we also say that f is a scalar Conservative fields potential of F. Theorem Suppose F is a continuous vector field defined on a connected, 3 open region R ⊆ R . Then F = ∇f if and only if F has path independent line integrals in R. Math 114 Review Path independence Math 240

Grad, Div, Curl Gradient 3 3 Divergence We say F : R ⊆ R → R has path independent line Curl How they’re integrals if any of the following hold: related Z Z Line integrals 1 Scalar line 1. F · ds = F · ds whenever x and y are two simple C integrals x y Vector line integrals paths in R with the same initial and terminal points, Conservative fields I 2. F · ds = 0 for any simple, closed C1 path x lying in R x (meaning the initial and terminal points of x coincide), Z 3. F · ds = f(B) − f(A) for any differentiable curve C in C R running from point A to point B, and for any scalar potential f. Math 114 Review Physical interpretation Math 240

Grad, Div, Curl Gradient Divergence Curl How they’re To justify our terminology, if f is a scalar potential for the related Line integrals vector field F, it means that we can interpret f as measuring Scalar line integrals the potential energy associated with the force represented by F. Vector line integrals Conservative fields In this setting, criterion 3 from the previous slide says that Z work = F · ds = f(B) − f(A) = change in potential energy, C meaning that the force represented by F obeys conservation of energy. Math 114 Review A test for conservative fields Math 240

Grad, Div, Curl Theorem Gradient 1 Divergence Suppose F is a C vector field defined in a simply-connected Curl How they’re related region, R, (intuitively, R has no holes going all the way 2 Line integrals through). Then F = ∇f for some C scalar function if and Scalar line integrals only if ∇ × F = 0 at all points in R. Vector line integrals Conservative fields Example Let

 x  y z F = x2+y2+z2 − 6x i + x2+y2+z2 j + x2+y2+z2 k.

1 3 F is C on R − {(0, 0, 0)}, which is a simply-connected domain. Check that ∇ × F = 0 everywhere F is defined. Therefore, F is conservative.