Josh Engwer (TTU) Line Integrals 11 November / 25

Transkript

1 Line Integrals alculus III Josh Engwer TTU 11 November 2014 Josh Engwer (TTU) Line Integrals 11 November / 25

2 PART I PART I: LINE INTEGRALS OF SALAR FIELDS Josh Engwer (TTU) Line Integrals 11 November / 25

3 Line Integral of a Scalar Field f in R 2 (Derivation) Josh Engwer (TTU) Line Integrals 11 November / 25

4 Line Integral of a Scalar Field f in R 2 (Derivation) Josh Engwer (TTU) Line Integrals 11 November / 25

5 Line Integral of a Scalar Field f in R 2 (Derivation) Josh Engwer (TTU) Line Integrals 11 November / 25

6 Line Integral of a Scalar Field f in R 2 (Derivation) Josh Engwer (TTU) Line Integrals 11 November / 25

7 Line Integral of a Scalar Field f in R 2 (Derivation) Josh Engwer (TTU) Line Integrals 11 November / 25

8 Line Integral of a Scalar Field f in R 2 (Derivation) Josh Engwer (TTU) Line Integrals 11 November / 25

9 Line Integral of a Scalar Field f in R 2 (Derivation) s k = ( x k ) 2 + ( y k ) 2 (dx) ) = ds = + (dy) 2 ( After taking limits on both sides Now, the parameterization of Γ : = dx dt = x (t) = dx = x (t) dt = dy dt = y (t) = dy = y (t) dt ds = [x (t)] 2 + [y (t)] 2 dt f (x, y) ds = Γ }{{} Line Integral b x = x(t) y = y(t) t [a, b] [ ] f x(t), y(t) [x (t)] 2 + [y (t)] 2 dt a } {{ } Ordinary Single Integral Josh Engwer (TTU) Line Integrals 11 November / 25

10 Line Integral of a Scalar Field in R 2 (Definition) Definition Given smooth curve Γ : x = x(t) y = y(t) t [a, b] and scalar field f (x, y) (Γ) Then the line integral of f over curve Γ is defined to be Γ f (x, y) ds := b a [ ] f x(t), y(t) [x (t)] 2 + [y (t)] 2 dt REMARK: Very often, the curve is denoted by : b [ ] f (x, y) ds := f x(t), y(t) [x (t)] 2 + [y (t)] 2 dt a Josh Engwer (TTU) Line Integrals 11 November / 25

11 Line Integral of a Scalar Field in R 3 (Definition) Definition Given smooth curve Γ : x = x(t) y = y(t) z = z(t) t [a, b] and scalar field f (x, y, z) (Γ). Then the line integral of f over curve Γ is defined to be Γ f (x, y, z) ds := b a [ ] f x(t), y(t), z(t) [x (t)] 2 + [y (t)] 2 + [z (t)] 2 dt REMARK: Very often, the curve is denoted by : b [ ] f (x, y, z) ds := f x(t), y(t), z(t) [x (t)] 2 + [y (t)] 2 + [z (t)] 2 dt a Josh Engwer (TTU) Line Integrals 11 November / 25

12 Line Integral over a Piecewise Smooth urve = = f ds = f ds + 1 f ds + 2 f ds 3 Josh Engwer (TTU) Line Integrals 11 November / 25

13 Parameterizing urves TASK: Parameterize a curve starting at t = a and ending at t = b. URVE y = f (x) x = g(y) x 2 + y 2 = r 2 x 2 A 2 + y2 B 2 = 1 x 2 A 2 y2 B 2 = 1 REMARK: PARAMETERIZATION x = t y = f (t) t [a, b] x = g(t) y = t t [a, b] x = r cos t y = r sin t t [a, b] x = A cos t y = B sin t t [a, b] x = A sec t y = B tan t t [a, b] Parameterizations are NOT unique. Josh Engwer (TTU) Line Integrals 11 November / 25

14 Line Integral (urve Orientation) Josh Engwer (TTU) Line Integrals 11 November / 25

15 Line Integral (urve Orientation) Josh Engwer (TTU) Line Integrals 11 November / 25

16 Parameterizing Line Segments Josh Engwer (TTU) Line Integrals 11 November / 25

17 Line Integral of a Scalar Field f in R 2 w.r.t. x, y Proposition x = x(t) Given smooth curve : y = y(t) t [a, b] Then: f (x, y) dx = f (x, y) dy = b a b a and continuous scalar field f (x, y). [ ] f x(t), y(t) x (t) dt [ ] f x(t), y(t) y (t) dt Josh Engwer (TTU) Line Integrals 11 November / 25

18 Line Integral of a Scalar Field f in R 3 w.r.t. x, y, z Proposition x = x(t) y = y(t) Given smooth curve : z = z(t) t [a, b] Then: f (x, y, z) dx = f (x, y, z) dy = f (x, y, z) dz = b a b a b a and continuous scalar field f (x, y, z). [ ] f x(t), y(t), z(t) x (t) dt [ ] f x(t), y(t), z(t) y (t) dt [ ] f x(t), y(t), z(t) z (t) dt Josh Engwer (TTU) Line Integrals 11 November / 25

19 Line Integrals (Thin Wires) Proposition (enter of Mass of a Thin Wire) Given smooth curve : {x = x(t), y = y(t), z = z(t), t [a, b]} Let a thin wire W have the shape of curve. Suppose the wire has mass-density ρ(x, y, z) at each point (x, y, z) on the wire. Then, the center of mass of W is ( x, ȳ, z), where: Mass(W) = ρ(x, y, z) ds 1 x = xρ(x, y, z) ds Mass(W) 1 ȳ = yρ(x, y, z) ds Mass(W) 1 z = zρ(x, y, z) ds Mass(W) REMARK: The center of mass may not lie on the wire itself! Josh Engwer (TTU) Line Integrals 11 November / 25

20 PART II PART II: LINE INTEGRALS OF VETOR FIELDS Josh Engwer (TTU) Line Integrals 11 November / 25

21 Line Integral of a Vector Field F in R 2 Definition Let vector function R(t) = x(t), y(t) trace out a smooth curve in R 2. Let vector field F(x, y) = M(x, y), N(x, y) be continuous on. Then the line integral of F along is defined by [ ] F d R := M(x, y) dx + N(x, y) dy where d R = dx, dy = x (t), y (t) dt Alternatively, F d R = b a ] F[ R(t) R (t) dt = b a [ M ( x(t), y(t) ) x (t) + N ( x(t), y(t) ) ] y (t) dt ] where vector function F[ R(t) = M ( x(t), y(t) ), N ( x(t), y(t) ) Josh Engwer (TTU) Line Integrals 11 November / 25

22 Line Integral of a Vector Field F in R 3 Definition Let vector function R(t) = x(t), y(t), z(t) trace out a smooth curve in R 3. Let vector field F(x, y, z) = M(x, y, z), N(x, y, z), P(x, y, z) be continuous on. Then the line integral of F along is defined by [ ] F d R := M(x, y, z) dx + N(x, y, z) dy + P(x, y, z) dz where d R = dx, dy, dz = x (t), y (t), z (t) dt Josh Engwer (TTU) Line Integrals 11 November / 25

23 Line Integrals in R 2 (Work) Recall from SST 9.3 that dot products allowed us to compute the work done by a constant force acting on an object along a line: Work = F PQ Line integrals allows us to compute the work done by a force field (i.e. variable force) acting on an object along a curve: Proposition Let vector function R(t) = x(t), y(t) trace out a smooth curve in R 2. Let force field F(x, y) be continuous on. Then the work done by F on an object moving it along the curve is Work = F d R Josh Engwer (TTU) Line Integrals 11 November / 25

24 Line Integrals in R 3 (Work) Recall from SST 9.3 that dot products allowed us to compute the work done by a constant force acting on an object along a line: Work = F PQ Line integrals allows us to compute the work done by a force field (i.e. variable force) acting on an object along a curve: Proposition Let vector function R(t) = x(t), y(t), z(t) trace out a smooth curve in R 3. Let force field F(x, y, z) be continuous on. Then the work done by F on an object moving it along the curve is Work = F d R Josh Engwer (TTU) Line Integrals 11 November / 25

25 Fin Fin. Josh Engwer (TTU) Line Integrals 11 November / 25