Line Integrals Independence of Path -(9.9) f x

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1 . Differentials and Eact Differentials: The differentials of f, and f,,z are: Line Integrals Independence of Path -(9.9) d d, and d d dz. A differential epression P,,zd Q,,zd R,,zdz is said to be an eact differential if there eists a function f,,z such that P,,z d Q,,z d R,,z dz that is f,,z P,,z, f,,z Q,,z, f z,,z R,,z. (Remember eact differential equations in differential equations?!) Eample Let f,,z e z ompute the differential. e z z e z d e z d 3/ z e z dz Eample Determine if the differential d d is eact. Two was to determine if a differential is eact: a. If it is eact, then there eists a function f, such that f, d 4, Such a does not eist since the right side of the equation is a function of and. This contradicts to the assumption of the eistence of f,. Therefore, the differential is not eact. b. If and for some f,, then since both and are polnomials. heck 4, f. Since, f, does eists and the differential is not eact.

2 Note that for f,,z we need to check three equations:.. Path Independence: Let A and B be two points in a space (a plane). A line integral F dr is said to be independent of the path if the value of the integral is the same for ever curve connecting A and B. Eample ompute F dr where F, i, and : from, to, b a straight line; : from, to, b the path: ; 3 : from, to, b the path: rt t 3/, t t t 3/, t t 5 : r t t,t, t ; : r t t, t, t ; 3 : r 3 t t 3/,t 5, t F dr t,t, t t F dr t,t, t t t 3 3 t3/ 3 F dr t 5, t 3/ 5t 4, 3 t / 5t 9 3 t t t3 Observe that F dr are the same for i,,3. Though we cannot conclude from these 3 eamples i that this line integral is independent of the path, we see it is possible that the line integral is independent of the path. A Fundamental Theorem for Line Integrals: Suppose that

3 P,,z d Q,,z d R,,z dz (the differential P,,z d Q,,z d R,,z dz is eact) and is a curve from A to B. Then a. the line integral P,,z d Q,,z d R,,z dz is independent of the path ; and b. P,,zd Q,,zd R,,z dz fb fa Proof: Let rt t, t, zt, a t b be a parametric representation of where A a,a,za and B b,b,zb. Then we know d d dz and P,,z d Q,,z d Now we can rewrite the line integral as: P,,zd Q,,zd R,,zdz P,,z d a b a b R,,z dz. d Q,,z d d dz R,,z dz ft,t,zt a b fb fa Notes: LetF,,z P,,z, Q,,z, R,,z for,,z in a region D which contains. Observe that F dr P,,z d Q,,z d R,,z dz. If the differential P,,z d Q,,z d R,,z dz is eact, then the integral F dr is independent of the path. If F dr is independent of the path, then F dr. If P,,zd Q,,zd R,,zdz is eact, then we know there eists a function f,,z such that P,,z, Q,,z, R,,z, that is f,,z F,,z. So, for a given F,,z, if there eists a scalar function f,,z such that f,,z F,,z, then F dr is path independent and F dr fb fa onservative vector fields: A vector field F,,z is said to be conservative if there is a scalar function f,,z such that f F. f is called a potential function of F and F is called a gradient field. 3

4 Note that if a force field F is conservative then the work done b F from the point A to the point B in space is the same for an path from A to B. If F dr depends on the path, then F is not conservative. A frictional force such as air resistance is not conservative. Two Was to Evaluate a Path Independent Line Integral: Let F dr be independent of path. Find f such that f F and then F dr fb fa. hose a curve from A to B and evaluate F dr. Eample Let F,,z 3,z, andbeacurvefroma,, to B,,7.ompute F dr in two was. a. Fin such that f F. 3, f,,z 3 d 3,z. heck z.,z zd z z, f,,z 3 z z. heck z z, z. f,,z 3 z F dr fb fa f,,7 f,, 7 6 b. Let rt,, t,,5 t, t, 5t.AtB, t, at A, t F dr 3t, t 5t, t,,5 3t 4 t 5t 5 t t 3 4 t 5 t 3 t3 5 t 3 t Test for Path Independence: Let F,,z P,,z, Q,,z, R,,z for,,z in an open simpl connected region D which contains where P, Q and R have continuous first partial derivatives. Then F dr is independent of the path if and onl if curl F,,z. Note that curl F,,z Q, Q, P P, Q, Q P P. implies F dr is independent of the path implies there eists a potential function f,,z such f F, that is 4

5 P, Q, R. Since P, Q and R have continuous first partial derivatives, f has continuous second partial derivatives. So, That is P Q ;,, P ;. Q. Eample Let F z, z zcosz, z cosz. Determine if F is conservative and,, evaluate F dr.,, Determine first if F is a gradient field of f. z, f,,z z d,z z,z z,z z zcosz,,z zcosz,,z zcoszd,,z sinz z, f,,z z sinz z z cosz z z cosz, z, z f,,z z sinz. So, F is conservative and its potential function is f,,z z sinz,, F dr f,, f,,,, 5