ENGI 4430 Line Integrals; Green s Theorem Page 8.01

Transcription

1 ENGI 4430 Line Integrals; Green s Theorem Page Line Integrals Two applications of line integrals are treated here: the evaluation of work done on a particle as it travels along a curve in the presence of a [vector field] force; and the evaluation of the location of the centre of mass of a wire. Work done: The work done by a force F in moving an elementary distance r along a curve is approximately the product of the component of the force in the direction of r and the distance r travelled: Integrating along the curve yields the total work done by the force F in moving along the curve : W F dr

2 ENGI 4430 Line Integrals; Green s Theorem Page 8.0 Example 8.01 Find the work done by F y x z T in moving once around the closed curve (defined in parametric form by x cos t, y sin t, z 0, 0 t ).

3 ENGI 4430 Line Integrals; Green s Theorem Page 8.03 Example 8.01 (continued) Example 8.0 Find the work done by x y z T F in moving around the curve (defined in parametric form by x cos t, y sin t, z 0, 0 t ).

4 ENGI 4430 Line Integrals; Green s Theorem Page 8.04 If the initial and terminal points of a curve are identical and the curve meets itself nowhere else, then the curve is said to be a simple closed curve. Notation: When is a simple closed curve, write F dr as F dr. F is a conservative vector field if and only if F dr 0 for all simple closed curves in the domain. Be careful of where the endpoints are and of the order in which they appear (the t1 t dr 0 dr orientation of the curve). The identity F dt F dt leads to the result t dt t dt 0 1 F dr F dr simple closed curves Another Application of Line Integrals: The Mass of a Wire Let be a segment t t t of wire of line density x, y, z 0 1. Then First moments about the coordinate planes: The location r of the centre of mass of the wire is r M, where the moment m t1 t1 t0 t0 ds ds ds dr dx dy dz M r dt, m dt and. dt dt dt dt dt dt dt

5 ENGI 4430 Line Integrals; Green s Theorem Page 8.05 Example 8.03 Find the mass and centre of mass of a wire (described in parametric form by x cos t, y sin t, z t, t ) of line density z.

6 ENGI 4430 Line Integrals; Green s Theorem Page 8.06 Example 8.03 (continued)

7 ENGI 4430 Line Integrals; Green s Theorem Page 8.07 Green s Theorem Some definitions: r t x t ˆ i y t ˆ, j a t b ) is A curve on (defined in parametric form by closed iff xa, ya x b, y b. The curve is simple iff t t r 1 r for all 1, t t such that a t1 t b ; (that is, the curve neither touches nor intersects itself, except possibly at the end points). Example 8.04 Two simple curves: open closed Two non-simple curves: open closed Orientation of closed curves: A closed curve has a positive orientation iff a point anticlockwise sense as the value of the parameter t increases. r t moves around in an

8 ENGI 4430 Line Integrals; Green s Theorem Page 8.08 Example 8.05 Positive orientation Negative orientation Let D be the finite region of bounded by. When a particle moves along a curve with positive orientation, D is always to the left of the particle. For a simple closed curve enclosing a finite region D of function T 1 and for any vector F f f that is differentiable everywhere on and everywhere in D, Green s theorem is valid: f f 1 F dr x y D da The region D is entirely in the xy-plane, so that the unit normal vector everywhere on D is ˆk. Let the differential vector da da k ˆ, then Green s theorem can also be written as ˆ F dr F k da curl F da D D Green s theorem is valid if there are no singularities in D. A [non-examinable] proof is provided at the end of this chapter.

9 ENGI 4430 Line Integrals; Green s Theorem Page 8.09 Example 8.06 F x r T 0 : Example 8.07 For x y F x y and as shown, evaluate. F dr

10 ENGI 4430 Line Integrals; Green s Theorem Page 8.10 Example 8.07 (continued) F x y x y

11 ENGI 4430 Line Integrals; Green s Theorem Page 8.11 Example 8.07 (continued) OR use Green s theorem!

12 ENGI 4430 Line Integrals; Green s Theorem Page 8.1 Example 8.08 Find the work done by the force ˆ ˆ F xy i y j in one circuit of the unit square.

13 ENGI 4430 Line Integrals; Green s Theorem Page 8.13 Path Independence Gradient Vector Fields: If F V, then F V x V y T Path Independence If F V or F V, then V is a potential function for F. Let the path travel from point P 0 to point P 1 :

14 ENGI 4430 Line Integrals; Green s Theorem Page 8.14 Domain A region of is a domain if and only if 1) For all points P 0 in, there exists a circle, centre P 0, all of whose interior points are inside ; and ) For all points P 0 and P 1 in, there exists a piecewise smooth curve, entirely in, from P 0 to P 1. Example 8.09 Are these domains? { (x, y) y > 0 } { (x, y) x 0 } If a domain is not specified, then, by default, it is assumed to be all of.

15 ENGI 4430 Line Integrals; Green s Theorem Page 8.15 When a vector field F is defined on a simply connected domain, these statements are all equivalent (that is, all of them are true or all of them are false): F V for some scalar field V that is differentiable everywhere in ; F is conservative; F dr is path-independent (has the same value no matter which path within is chosen between the two endpoints, for any two endpoints in ); Vend Vstart F dr (for any two endpoints in ); F dr 0 for all closed curves lying entirely in ; f x f y 1 everywhere in ; and F 0 everywhere in (so that the vector field F is irrotational). There must be no singularities anywhere in the domain in order for the above set of equivalencies to be valid. Example 8.10 Evaluate 3 0, 0 to x y dx x y dy where is any piecewise-smooth curve from 1,.

16 ENGI 4430 Line Integrals; Green s Theorem Page 8.16 Example 8.10 by direct evaluation of the line integral Let us pursue instead a particular path from (0, 0) to (1, ). The straight line path 1 is a segment of the line y x x 1 y. 1 3 I x y dx x y dy An alternative evaluation of I F dr is to use x as the parameter in both integrals (that is, to express y in terms of x throughout). Then I x y dx x y dy An alternative path involves going round the other two sides of the triangle, first from (0, 0) horizontally to (1, 0) then from there vertically to (1, ). On the first leg y 0 dy 0, so that the second part of the integral vanishes. On the second leg x 1 dx 0, so that the first part of the integral vanishes. Therefore 3 I x y dx x y dy

17 ENGI 4430 Line Integrals; Green s Theorem Page 8.17 Example 8.10 by direct evaluation of the line integral Yet another possibility is 3 an arc of the parabola y x. 3 3 I x y dx x y dy 1, Note that the above suggests that I F dr might be path-independent, because 0, 0 evaluations along three different paths have all produced the same answer. But this is not a proof of path independence. For a proof, one must establish that F is conservative, either by finding the potential function, or by showing that curl F 0.

18 ENGI 4430 Line Integrals; Green s Theorem Page 8.18 Outline of a Proof of Green s Theorem [not examinable] Let Px, y ˆ Qx, y F i ˆj. onsider a convex region D as shown. Left and right boundaries can be identified. Then D Q da x d q y c p y Q dx dy x d c Q x, y x q y x p y dy d d c,,,, Qq y y Q p y y dy Qq y y dy Q p y y dy c c d x q y from y c to y d followed by the path along x p y But the path along from y D d back to y c constitutes one complete circuit around the closed path. Q da x But the path along D Q dy Lower and upper boundaries for the region can also be identified. b hx P P da dy dx y a gx y D y g x from x a to x b b a b a P x, y a b y h x y g x dx,, P x h x P x g x dx,, P x h x dx P x g x dx followed by the path along y hx from x b back to x a constitutes one complete circuit around the closed path. P Q P da P dx y da P dx Q dy x y D b a

19 ENGI 4430 Line Integrals; Green s Theorem Page 8.19 Green s Theorem (continued) But F dr P dx P dx Q dy Q dy Therefore D Q P da F dr x y This proof can be extended to non-convex regions. Simply divide them up into convex sub-regions and apply Green s theorem to each sub-region. The line integrals along common interior boundaries cancel out because they are travelled in opposite directions along the same line. The boundary of each convex sub-region D is a simple closed curve theorem is valid:, for which Green s i i Q P F dr da x y i Di Q P F dr da x y i i i Di Therefore Green s theorem is also valid for any simply-connected region. [End of hapter 8]

20 ENGI 4430 Line Integrals; Green s Theorem Page 8.0 [Space for Additional Notes]