cid 90 b a cid 90 cid 90 16 3 The Fundamental Theorem of Line Integrals Recall the Fundamental Theorem of Calculus for a single variable function f f cid 48 x dx f b f a It says that we may evaluate the integral of a derivative simply by knowing the values of the function at the endpoints of the interval of integration a b The Fundamental Theorem of Line Integrals is a precise analogue of this for multi variable functions The primary change is that gradient f takes the place of the derivative f cid 48 in the original theorem Theorem Fundamental Theorem of Line Integrals Suppose that C is a smooth curve from points A to B parameterized by r t for a t b Let f be a differentiable function whose domain includes C and whose gradient vector f is continuous on C Then The long caveats about differentiability and continuity are merely so that the original Fundamen C f dr f r b f r a f B f A tal Theorem of Calculus can be invoked in the proof Proof n 2 or 3 for the purposes of this course C f dr dx1 dxn cid 90 f x1 C dx1 dt f xn dxn dt fx1 cid 90 cid 18 f cid 90 b cid 90 b fxn C a x1 d dt a f r b f r a f xn dxn dx1 cid 19 cid 90 b dt d dt a where we applied FTC in the nal step f x1 t xn t dt f r t dt chain rule The Theorem can be alternatively stated if F is a conservative vector eld with potential function f then cid 90 F dr f end of C f start of C C We say that a line integral in a conservative vector eld is independent of path Examples 1 Let C be the curve parameterized by r t cid 90 C x2y3 dr x2y3 cid 12 cid 12 1 2 cid 17 cid 16 1 sin2 t 1 0 8 3 0 8 3 t sin t for 0 t 2 Then 1 cid 90 cid 90 cid 90 cid 16 t3 1 cid 17 cid 90 C x sin y dr x sin y cid 12 cid 12 26 8 for 2 t 3 Then 26 sin 3 7 3 2 t t 1 8 3 7 sin 3 2 2 Let C be the curve parameterized by r t sin y dx x cos y dy 3 Evaluate the line integrals cid 82 C y dx x dy where C1 is the straight line from 0 0 and 1 1 and C2 is the parabola y x2 between the same points For the rst curve we have r t t Ci t so y dx x dy For the second curve we have r t cid 0 t 2t dt 1 C1 0 cid 1 so t2 y dx x dy t2 dt 2t2 dt 1 cid 90 1 cid 90 1 0 cid 90 We expected the two solutions to be the same since y simply have applied the Fundamental Theorem x xy is conservative We could cid 19 cid 18 y cid 90 4 Evaluate cid 82 The integral is cid 82 cid 90 Ci x C2 C dr Ci xy dr xy 1 0 1 C y2z dx 2xyz dy xy2 dz along any curve joining 1 0 0 and 2 1 1 C F dr where F xy2z so the path is irrelevant and we obtain y2z dx 2xyz dy xy2 dz xy2z 2 0 0 cid 12 cid 12 cid 12 cid 12 1 1 cid 12 cid 12 cid 12 cid 12 2 1 1 1 0 0 Conservation of Energy The terminology conservative potential etc all comes from Physics There are two primary forms of energy potential stored and kinetic motion Suppose that a particle of mass m follows a curve C through a conservative force eld F f We parameterize the curve so that the particle is at position r t at time t Its velocity vector is then v t r cid 48 t The particle has kinetic energy K Now we evaluate the line integral cid 82 cid 90 cid 90 t1 1 Newton s second law F ma mv cid 48 says that1 d dt v cid 48 t v t dt m t0 is the change in kinetic energy over the path F dr m t0 C cid 90 t1 1By the product rule d dt v v v cid 48 v v v cid 48 2v cid 48 v 2 v 2 2 1 m v 2 and is said to have potential energy f 2 C F dr in two ways m v 2 cid 12 cid 12 cid 12 t1 t0 cid 52 K 1 2 v 2 dt 1 2 01y01xC1C2 2 Alternatively we may use the Fundamental Theorem cid 90 F dr C cid 90 C f dr f r t cid 12 cid 12 t1 t0 cid 52 f is negative the change in potential energy of the particle over the path Therefore cid 52 f cid 52 K 0 and so total energy is conserved Since Physicists always want energy to be conserved they typically choose potential functions to have a negative sign F f In math ematics we omit the negative Path Independence The Fundamental Theorem has the amazing interpretation that line integrals in conservative vector elds depend only on a curve s endpoints We want to turn this idea on its head Is it the case that a line integral or integrals being independent of path forces a vector eld to be conservative De nition A line integral cid 82 C F dr is independent of path if cid 82 C F dr cid 82 C F dr for any curve C with the same endpoints as C Before we can state the relevant theorems we need to understand the meaning of several terms De nition A region D is open if it contains no boundary points For example the inside of the unit disk D x y x2 y2 1 is open De nition A region D is path connected if every pair of points A B in D can be joined by a curve lying entirely in D A connected region with curve C joining A B A disconnected region cannot join A B with a curve lying in D 3 C CABOpenNotOpenDABCDABC The adjectives open and connected apply only to domains regions in this course The nal adjective applies only to curves De nition A curve is closed if it starts and nishes at the same point Independence of Path and Closed Curves The following important Theorem relates being inde pendent of path to line integrals round closed curves C F dr is independent of path if and only if C F dr independent of path then the line integrals over all curves must be independent of path This says that independence of path is really a property of the vector eld F rather than …