Lecture 5 - Fundamental Theorem for Line Integrals and Green s Theorem

Lecture 5 - Fundamental Theorem for Line Integrals and Green s Theorem Math 392, section C September 14, 2016 392, section C Lect 5 September 14, 2016 1 / 22

Last Time: Fundamental Theorem for Line Integrals: 392, section C Lect 5 September 14, 2016 2 / 22

Last Time: Fundamental Theorem for Line Integrals: Theorem Let C be a smooth curve, parametrized by r(t), t [a, b]. Let f be a smooth function. Then f d r = f ( r(b)) f ( r(a)). C 392, section C Lect 5 September 14, 2016 2 / 22

So, if a vector field is conservative, calculating line integrals is very simple. 392, section C Lect 5 September 14, 2016 3 / 22

So, if a vector field is conservative, calculating line integrals is very simple. Question: When is F conservative? 392, section C Lect 5 September 14, 2016 3 / 22

So, if a vector field is conservative, calculating line integrals is very simple. Question: When is F conservative? Theorem Let F be a continuous vector field on an open, connected region D. If F d r is independent of path, then F is conservative. C 392, section C Lect 5 September 14, 2016 3 / 22

Definition We say that C F d r is independent of path whenever the value of this line integral is the same for any two paths with the same endpoints. 392, section C Lect 5 September 14, 2016 4 / 22

Definition We say that C F d r is independent of path whenever the value of this line integral is the same for any two paths with the same endpoints. 392, section C Lect 5 September 14, 2016 4 / 22

Criterion to decide whether F is conservative If we knew that F = (P, Q) = f for some twice differentiable function f, then P y = f xy = f yx = Q x 392, section C Lect 5 September 14, 2016 5 / 22

Criterion to decide whether F is conservative If we knew that F = (P, Q) = f for some twice differentiable function f, then P y = f xy = f yx = Q x So, this is a necessary condition for a vector field to be conservative. 392, section C Lect 5 September 14, 2016 5 / 22

Criterion to decide whether F is conservative Theorem If F(x, y) = (P(x, y), Q(x, y)) is defined on an open, simply-connected region D, and P y = Q x on D, then F is conservative. 392, section C Lect 5 September 14, 2016 6 / 22

Exercise 11, Section 3.3: Find the work done by F = (x 3 y 4, x 4 y 3 ) to move a particle along the curve C, parametrized by r(t) = ( t, 1 + t 3 ), t [0, 1]. 392, section C Lect 5 September 14, 2016 7 / 22

392, section C Lect 5 September 14, 2016 8 / 22

Exercise 20, Section 3.3: Find the work done by F = (e y, xe y ) to move a particle from (0, 1) to (2, 0). 392, section C Lect 5 September 14, 2016 9 / 22

392, section C Lect 5 September 14, 2016 10 / 22

Exercise 31, Section 3.3: Let F = ( y, x) x 2 + y 2 Show that F satisfies P y = Q x, and that the values of the line integrals of F along the upper and lower hemispheres, joining the points (1, 0) to ( 1, 0), are different. 392, section C Lect 5 September 14, 2016 11 / 22

392, section C Lect 5 September 14, 2016 12 / 22

392, section C Lect 5 September 14, 2016 13 / 22

Green s Theorem Let C be a simple, closed curve on the plane, that bounds a region D. Assume also that C is oriented counterclockwise 1. If P and Q have continuous partial derivatives on an open region containing D, then C Pdx + Qdy = D (Q x P y ) da 1 Counterclockwise orientation will be called the positive orientation 392, section C Lect 5 September 14, 2016 14 / 22

Green s Theorem Let C be a simple, closed curve on the plane, that bounds a region D. Assume also that C is oriented counterclockwise 1. If P and Q have continuous partial derivatives on an open region containing D, then C Pdx + Qdy = D (Q x P y ) da We will prove this theorem next class. Right now, we will work on how to use the theorem. 1 Counterclockwise orientation will be called the positive orientation 392, section C Lect 5 September 14, 2016 14 / 22

Example: Exercise 3, Section 3.4 Evaluate xydx + x 2 y 3 dy, where C is the triangle with vertices C (0, 0), (1, 0) and (1, 2), oriented counterclockwise. 392, section C Lect 5 September 14, 2016 15 / 22

392, section C Lect 5 September 14, 2016 16 / 22

Example: Exercise 9, Section 3.4 Evaluate C (y + e x )dx + (2x + cos(y 2 ))dy, where C is the boundary of the region enclosed by y = x 2 and x = y 2, positively oriented. 392, section C Lect 5 September 14, 2016 17 / 22

392, section C Lect 5 September 14, 2016 18 / 22

Example: Exercise 17, Section 3.4 Find the work done by F = (x 2 + xy, xy 2 ) to move a particle from the origin along the x-axis to (1, 0), then along a straight line to (0, 1), and then back to the origin, vertically. 392, section C Lect 5 September 14, 2016 19 / 22

392, section C Lect 5 September 14, 2016 20 / 22

Example/Application If F is such that Q x P y = 1 (for example, if P = y and Q = 0, or if Q = x and P = 0, or even if P = 1 2 y and Q = 1 2x, then Area (D) = 1dA D

Example/Application If F is such that Q x P y = 1 (for example, if P = y and Q = 0, or if Q = x and P = 0, or even if P = 1 2 y and Q = 1 2x, then Area (D) = 1dA = xdy D C

Example/Application If F is such that Q x P y = 1 (for example, if P = y and Q = 0, or if Q = x and P = 0, or even if P = 1 2 y and Q = 1 2x, then Area (D) = 1dA = xdy= ydx D C C

Example/Application If F is such that Q x P y = 1 (for example, if P = y and Q = 0, or if Q = x and P = 0, or even if P = 1 2 y and Q = 1 2x, then Area (D) = 1dA = xdy= ydx= 1 2 xdy ydx D C C C 392, section C Lect 5 September 14, 2016 21 / 22

Example: Find the area of the ellipse (bx) 2 + (ay) 2 = (ab) 2 392, section C Lect 5 September 14, 2016 22 / 22