Vector Calculus. 1 Line Integrals


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1 Vector lculus 1 Line Integrls Mss problem. Find the mss M of very thin wire whose liner density function (the mss per unit length) is known. We model the wire by smooth curve between two points P nd Q in 3spce. Given ny point (x, y, z) on, we let f(x, y, z) denote the corresponding vlue of the density function. 1. Divide into n smll sections. Let M k be the mss of the kth section, nd let s k be the length of the rc between P k 1 nd P k. 2. hoose P k (x k, y k, z k ) on the kth rc M k f(x k, y k, z k) s k 3. The mss M of the entire wire is M M k f(x k, yk, zk) s k 4. Tke mx s k 0, nd get M lim f(x k, yk, zk) s k mx s k 0 f(x, y, z) ds The lst term is the nottion for the limit of the Riemnn sum, nd it is clled the line integrl of f(x, y, z) with respect to s long. The sme definition is for f(x, y). 1
2 The mss M of the wire is M f(x, y, z) ds The length L of the wire is L ds If is curve in the xyplne nd f(x, y) is nonnegtive function on, then f(x, y) ds is equl to the re of the sheet tht is swept out by verticl line segment tht extends upwrd from (x, y) to height f(x, y) nd moves long A k f(x k, y k) s k A f(x, y) ds 2
3 Evluting line integrls Let be smoothly prmetrised r x(t) i + y(t) j, t b. Then s k t k t k 1 r (t) r (t k ) t k nd f(x, y) ds lim mx s k 0 lim mx t k 0 b b f(x k, yk) s k f(x(t k), y(t k)) r (t k) t k f(x(t), y(t)) r (t) (dx ) 2 f(x(t), y(t)) + ( ) 2 dy The line integrl does not depend on prmetristion of, in prticulr on n orienttion of. Exmple. Find (1 + x2 y)ds for 1. : 1 2 (t + t2 ) i (t + t2 ) j, 0 t 1 2. : (2 2t) i + (1 t) j, 0 t 1 3
4 Similrly, if is curve in 3spce smoothly prmetrised r x(t) i + y(t) j + z(t) k, then f(x, y, z) ds b b f(x(t), y(t), z(t)) r (t) (dx ) 2 f(x(t), y(t), z(t)) + ( ) 2 dy + ( ) 2 dz Exmple. Find (xy + z3 )ds for : cos t i + sin t j + t k, 0 t π Answer : 2π 4 /4 Line integrls with respect to x, y nd z Let s replce s k by x k ( or y k or z k ) in the definition of the line integrl. Then, we get the line integrl of f(x, y, z) with respect to x long f(x, y, z) dx lim mx s k 0 f(x k, yk, zk) x k f(x, y, z) dy f(x, y, z) dz lim mx s k 0 lim mx s k 0 f(x k, yk, zk) y k f(x k, yk, zk) z k 4
5 The sign of these line integrls depends on the orienttion of. Reversing the orienttion chnges the sign. Thus, one should find prmetric equtions for in which the orienttion of is in the direction of incresing t, nd then f(x, y, z) dx b Exmple. Find (1 + x2 y)dy for 1. : 1 2 (t + t2 ) i (t + t2 ) j, 0 t 1 2. : (2 2t) i + (1 t) j, 0 t 1 f(x(t), y(t), z(t)) x (t) Let be smooth oriented curve, nd let denote the oriented curve with opposite orienttion but the sme points s. Then f(x, y) dx f(x, y) dx, f(x, y) dy f(x, y) dy while f(x, y) ds + f(x, y) ds onvention f(x, y) dx + g(x, y) dy f(x, y) dx + g(x, y) dy We hve f(x, y) dx+g(x, y) dy b (f(x(t), y(t)) x (t) + g(x(t), y(t)) y (t)) 5
6 Integrting vector field long curve Definition. A vector field in plne is function tht ssocites with ech point P in the plne unique vector F (P ) prllel to the plne F (P ) F (x, y) f(x, y) i + g(x, y) j Similrly, vector field in 3spce is function tht ssocites with ech point P in the 3spce unique vector F (P ) in the 3spce F (P ) F (x, y, z) f(x, y, z) i + g(x, y, z) j + h(x, y, z) k One cn sy tht vector field is vectorvlued function with the number of components equl to the number of independent vribles (coordintes). Introduce d r dx i + dy j + dz k If F (x, y, z) f(x, y, z) i + g(x, y, z) j + h(x, y, z) k is continuous vector field, nd is smooth oriented curve, then the line integrl of F long is F d r (f i + g j + h k) (dx i + dy j + dz k) f(x, y, z) dx + g(x, y, z) dy + h(x, y, z) dz If r r(t) x(t) i + y(t) j + z(t) k, then b F d r F ( r(t)) r (t) 6
7 Exmple. The curve is line segment connecting the points ( π/2, π) nd (3π/2, 2π/3). Prmeterise, nd evlute F d r where Answer : F (x, y) ( 6xy + 3π 3 sin 3x) i (3x 2 + 2π 3 cos 3y 2 ) j F d r π Let t s, where s is n rc length prmeter. Then F d r b F ( r(s)) r (s) ds F T ds b where T r (s) is unit tngent vector long. F ( r(s)) T ds F T F cos θ F F T F 7
8 Line integrls long piecewise smooth curves If is curve formed from finitely mny smooth curves 1, 2,..., n joined end to end, then n Exmple. Let the curve between the points ( π/2, π) nd (3π/2, 2π/3) be curve formed from two line segments 1 nd 2, where 1 is joining ( π/2, π) nd (3π/2, π), nd 2 is joining (3π/2, π) nd (3π/2, 2π/3). Prmeterise 1 nd 2, nd evlute F d r where F (x, y) ( 6xy + 3π 3 sin 3x) i (3x 2 + 2π 3 cos 3y 2 ) j 8
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