MATH 21-259 ummry of hpter 13 1. Vector Fields re vector functions of two or three vribles. Typiclly, vector field is denoted by F(x, y, z) = P (x, y, z)i+q(x, y, z)j+r(x, y, z)k where P, Q, R re clled sclr fields. The grdient of sclr function of two nd three vribles is relly vector field in plne or in spce nd is clled grdient vector field. A vector field is clled conservtive if it is the grdient of some sclr function, tht is, F = f for some f. This function f is clled potentil function for F. 2. Line Integrls () Line Integrl of sclr function: If f is function of two vribles nd is curve in plne prmetrized by r(t) = x(t)i + y(t)j, t b, then the line integrl of f over is defined by f(x, y) ds = f(x(t), y(t)) r (t) dt. If f is function of three vribles nd is curve in spce prmetrized by r(t) = x(t)i + y(t)j + z(t)k, t b, then the line integrl of f over is defined by f(x, y, z) ds = f(x(t), y(t), z(t)) r (t) dt. (b) Line Integrl of Vector Function: If F = P i + Qj is vector field of two vribles nd is curve in plne prmetrized by r(t) = x(t)i+y(t)j, t b, then the line integrl of F over is defined by F.dr = F.ˆT ds = P dx + Qdy = F (x(t), y(t)).r (t) dt. If F = P i + Qj + Rk is vector field of three vribles nd is curve in spce prmetrized by r(t) = x(t)i + y(t)j + z(t)k, t b, then the line integrl of F over is defined by F.dr = F.ˆT ds = P dx + Qdy + Rdz = F (x(t), y(t), z(t)).r (t) dt. If F represents the force field then the line integrl of F long curve represents the work done by the force field F in moving prticle from the strting point of the curve to the finl point of the curve. 3. Fundmentl Theorem of Line Integrl sttes tht if f is differentible of function of two or three vribles nd is the smooth curve given by the vector function r(t), t b, then f.dr = f(r(b)) f(r()). 1
onsequences/remrks: () The integrl of the grdient vector field long smooth closed curve is lwys zero. In mthemticl nottion, f.dr = 0. (b) If F is vector field defined everywhere then F = f F.dr is pth independent F.dr = 0 for every closed curve. (c) If F = P i + Qj is vector field defined everywhere then F = f Q x = P y. To find f, we solve the equtions f x = P nd f y = Q simultneously. (d) If F = P i + Qj + Rk is vector field defined everywhere then F = f Q x = P y, Q z = R y, nd P z = R x curlf = 0 where curl F is defined below. To find f, we solve the equtions f x = P, f y = Q, nd f z = R simultneously. 4. How to find f so tht F(x, y, z) = f(x, y, z)? () tep I. To find f we need to solve the following three equtions: f x = P (1) f y = Q (2) f z = R (3) (b) tep II. hoose one of the eqution from bove nd undo the opertion of prtil derivtive by doing prtil integrtion. Idelly, we choose the one tht is esy to integrte. If for instnce we choose (1) then remember the constnt of integrtion would depend on y nd z. Thus, we get f(x, y, z) = P dx + g(y, z). (c) tep III. We my now trget on using other two equtions to find the unknown function g(y, z). We my strt by equting the prtil derivtive of f with respect to y tht we get from tep II nd eqution(2). This step would gurntee tht we find the vlue of g(y, z) in terms of known function of y nd n unknown function of z which we my we cll s h(z). After plugging this informtion bck into the originl function, we see tht only unknown we hve now is h(z). (d) tep IV. Lst step will focus on finding h(z) by equting the prtil derivtive of f tht we obtin from tep III nd eqution 3. Finlly, plug the vlue of h into the function tht we obtined to tep III to give the nswer. (4) 2
5. Green s theorem sttes tht if is piecewise smooth closed curve oriented counterclockwise enclosing region nd if F = P i + Qj where P nd Q hve continuous prtil derivtive on n open region tht contins, then F.dr = Appliction of Green s theorem () To compute the line integrls: P dx + Qdy = (Q x P y ) da. Wrning: Only true for closed curves! (b) To compute the re of the regions: Note tht Are() = 1 da = F.dr where is the boundry of the domin nd F cn be chosen s ny of the following: < 0, x > or < y, 0 > or < y/2, x/2 >. It is generlly dvised to use F =< y/2, x/2 > when deling with circulr curves nd other two for other types of curves. 6. How to compute line integrl? () tep I. First of ll, check if the given vector field F is conservtive or not. (b) tep II. If the nswer to the bove turns out to be YE (Feeling Lucky) then you my use fundmentl theorem to compute the line integrl provided finding potentil function is not tht difficult. But if the nswer is NO then you should sk nother question, tht is, Is the given curve closed? (c) tep III. If the curve turns out to be closed then it is generlly good ide to use Green s theorem nd compute the double integrl insted of the line integrl. But if the given curve is not closed then you my hve to continue with the direct computtion of the line integrl. All of the bove steps cn be summrized in the form of the following chrt: Is F = f? Yes No, Is closed curve? Use Fundmentl Theorem of line integrl Yes No Use Green s theorem irect computtion 7. url nd ivergence: Let F =< P, Q, R > be vector function then i j k curlf = / x / y / z P Q R = (R y Q z )i (R x P z )j+(q x P y )k nd divf = P x +Q y +R z. 3
8. urfce Are: Given prmetrized surfce : r(u, v) = x(u, v)i+y(u, v)j+z(u, v)k where (u, v) belongs to domin, then the surfce re of this surfce is given by A() = r u r v da, where r u nd r v re prtil derivtives of r with respect to u nd v respectively. 9. urfce Integrl () urfce Integrl of sclr functions: Given sclr field f(x, y, z) nd surfce : r(u, v) = x(u, v)i + y(u, v)j + z(u, v)k where (u, v) belongs to domin, then the surfce integrl of f is defined by f(x, y, z) d = f(r(u, v)) r u r v da. In prticulr, surfce z = g(x, y), (x, y) in cn be prmetrized s follows: r(x, y) = xi + yj + g(x, y)k, (x, y) in. Note tht norml vector to the tngent plne to this surfce is given by r x r y = g x i g y j + k. Thus the surfce re of such surfce is given by (g x ) 2 + (g y ) 2 + 1 da. (b) urfce Integrl of vector functions: Given sclr field F(x, y, z) =< P, Q, R > nd surfce : r(u, v) = x(u, v)i + y(u, v)j + z(u, v)k where (u, v) belongs to domin, then the surfce integrl of F is defined by F.d = F(r(u, v)).ˆn d = F(r(u, v)).r u r v da. In prticulr, if z = g(x, y), (x, y) in is the given surfce then F.d = P g x Qg y + Rg z da. Remrks: 1. Note tht vector r x r y bove points in the upwrd direction s indicted by the position component of k. 2. We sy tht the surfce is positively oriented if the norml vector is pointing upwrds. 3. As convention, for closed surfce we tke the positive orienttion to be the one when the norml vector point outwrds. 4. losed surfces hve no boundry, e.g., sphere hs no boundry. 4
10. tokes Theorem sttes tht if is positively oriented surfce tht is bounded by closed curve with positive orienttion nd F =< P, Q, R > where P, Q, nd R hve continuous first order prtil derivtives. Then F.dr = curl(f).d. 11. ivergence Theorem sttes tht if E is simple solid region which hs s positively oriented boundry surfce nd F =< P, Q, R > where P, Q, nd R hve continuous first order prtil derivtives. Then F.d = div(f) dv. E 5