Math 240 Calculus III

Transcription

1 Line Math 114 Calculus III Summer 2013, Session II Monday, July 1, 2013

2 Agenda Line 1.,, and 2. Line vector

3 Line Definition Let f : X R 3 R be a differentiable scalar function on a region of 3-dimensional space. The gradient of f is the vector field grad f = f = f i + f y j + f z k. f The direction of the gradient, f, is the direction in which f is increasing the fastest. The norm, f, is the rate of this increase. Eample If f(, y, z) = 2 + y 2 + z 2 then f = 2 i + 2y j + 2z k.

4 Line Definition Let F : X R 3 R 3 be a differentiable vector field with components F = F i + F y j + F z k. The divergence of F is the scalar function div F = F = F + F y y + F z z. The divergence of a vector field measures how much it is epanding at each point. Eamples 1. If F = i + y j then F = If F = y i + j then F = 0.

5 Line Definition Let F : X R 3 R 3 be a differentiable vector field with components F = F i + F y j + F z k. The curl of F is the vector field i j k curl F = F = y z F F y F z ( Fz y F ) ( y F i + z z F z = ) ( Fy j + F ) k. y The magnitude of the curl, F, measures how much F F rotates around a point. The direction of the curl, F, is the ais around which it rotates.

6 Line Eample If F = y i + j then F = 2 k.

7 Math 114 Line Theorem Let f : X R 3 R be a C 2 scalar function. Then ( f) = 0, that is, curl (grad f) = 0. Theorem Let F : X R 3 R 3 be a C 2 vector field. Then ( F) = 0, that is, div (curl F) = 0. To summarize, the composition of any two consecutive arrows in the diagram yields zero. scalar functions grad 0 vector curl vector 0 div scalar functions

8 Line Definition Let : [a, b] X R 3 be a C 1 path and f : X R 3 a continuous function. The scalar line integral of f along is b f ds = f((t)) (t) dt. a In two dimensions, a scalar line integral measures the area under a curve with base and height given by f.

9 Line Eample Let : [0, 2π] R 3 be the heli (t) = (cos t, sin t, t) and let f(, y, z) = y + z. Let s compute 2π f ds = f((t)) (t) dt. We find (t) = sin 2 t + cos 2 t + 1 = 2, 0 so now 2π 0 f((t)) (t) dt = 2π = 2 0 2π (cos t sin t + t) 2 dt 0 ( 1 2 sin 2t + t) dt = 2 2π 2.

10 Line Definition Let : [a, b] X R 3 be a C 1 path and F : X R 3 a continuous vector field. The vector line integral of F along is b F ds = F((t)) (t) dt. a If F has components F = F i + F y j + F z k, the vector line integral can also be written F ds = F d + F y dy + F z dz. Physically, a vector line integral measures the work done by the force field F on a particle moving along the path.

11 Line Eample Let : [0, 1] R 3 be the path (t) = (2t + 1, t, 3t 1) and let F = z i + j + y k. Let s compute F ds = zd + dy + ydz. First, we find (t) = (2, 1, 3), and now we can do 1 zd + dy + ydz = (3t 1)(2) + (2t + 1) + t(3)dt = t + 3dt = 5 2.

12 Line Changing orientation (b) (b) opp (a) opp (a) EXAMPLE EXAMPLE 7 If 7 : path path opp :[a, opp :[a, b] b] (See (See Figure Figure 6.8.) 6.8. Th (a) opp opp (b) (b) u(t) u(t) = a = + a b+ bt. Cl y of. of. Figure AA path pathand and its its Figure: opposite. and y have opposite orientations In addition In addition to ret the the speed. speed. This This follof f ds = f ds y F ds = F ds So the velocity vecto y u (t)) So the velocity v u of the velocity (t)) of the velo This can be achieved by negating t: Speed o Spe y(t) = ( t).

13 vector Line Definition A continuous vector field F is called a conservative vector field, or a gradient field, if F = f for some C 1 scalar function f. In this case we also say that f is a scalar potential of F. Theorem Suppose F is a continuous vector field defined on a connected, open region R R 3. Then F = f if and only if F has path independent line in R.

14 Path independence Line We say F : R R 3 R 3 has path independent line if any of the following hold: 1. F ds = F ds whenever and y are two simple C 1 y paths in R with the same initial and terminal points, 2. F ds = 0 for any simple, closed C 1 path lying in R (meaning the initial and terminal points of coincide), 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.

15 Physical interpretation Line To justify our terminology, if f is a scalar potential for the vector field F, it means that we can interpret f as measuring the potential energy associated with the force represented by F. In this setting, criterion 3 from the previous slide says that work = F ds = f(b) f(a) = change in potential energy, C meaning that the force represented by F obeys conservation of energy.

16 A test for conservative Line Theorem Suppose F is a C 1 vector field defined in a simply-connected region, R, (intuitively, R has no holes going all the way through). Then F = f for some C 2 scalar function if and only if F = 0 at all points in R. Eample Let ( F = 2 +y 2 +z 2 6 ) i + y 2 +y 2 +z 2 j + z 2 +y 2 +z 2 k. F is C 1 on R 3 {(0, 0, 0)}, which is a simply-connected domain. Check that F = 0 everywhere F is defined. Therefore, F is conservative.