Question

(2) This is problem 3 from the homework 9. Let F = 2.xyzi + x2zj + ryk. Show that F is conservative. Find f such that VS = F (do not guess, use formula) and evaluate F. ds where c is a piecewise smooth curve from the point (1,1,1) to the point (1, 2, 4). F

Answer 1

Question :- e- Let F = axyz i + x² zi + x²yk, drow that F is conservative. Find f auch that of F and evoluote /F.de where e il a piecewise &mooth ceave from the point (1, 1, 1) point (1, 2, 4). the Answer 6-ti) The formula for curl of Eis j K curl VXF = where ř = pi top + R dlax olay olaz Р R 20 Ох al ay :) ( )2(*. 3:) - & cur all of * Also we've result ilag IR 3 [It Fit vector field defined on whioge component functions have continuous partial derivatives and curl ² =0, VECTOR THEN IS А. CONSERVATIVE FIELD ] Now we've, 7 cm, y, z) (2xyz)i + (x²z) i (x?z)j + bi ?y) † . V = x?z, x²y WA heu u = 2xyz PAGE 01

Where Ou ou 2x dyz. Qaz axy 11 ou az 2 oy Qv ax 2xz 0 11 22 9 av dy az ow 2x x² = 0. axy Owo οι 9 - 9 aw oy wu all continuous partial derivatives Now we find cure P. A î k DXF blox olay olaz 2xyz x²z x²y DXF ê (x²z)) i a i h 2 ce’y) - ) - 2 (Pxyz) 3 tu) ду e (23) OZ дz + la (x²) by craya) >> DXF = (? - x2) (2xy + 2xy) į + (2x2 - 4x2) OK – QXE of o Hence = DXF Cure PAGE 02

Now from result of and using equation and equath , We can that rince curl and say all partial deratives continuous by result then the 9 we Can say that, F IS CONSERVATIVE .con PROVED Now we've to find Such that of = F (A) F = axy ) Here, + (x2-)j + Ge’y)k <fx , Fy, Fz> x²z F = 2xyz/ Fy x²z <axyz x²y7 9 9 F2 = x²y 9 Now, F ax y z W.8.6 we dx integrating get JFx dx = laxyz x²yz + holy, 2) [where he cy,z) is a function] any f worot Loe Now if we differentiate equation © ec get fy x²2 (y, z) we've Ify x²2 (from ②) PART 03

we Comparing both equations and equation © get, x²z+ 2e²2 Fy hy (42) x²z x²z + hy ly, z) = hy ly, z) worit y -) integrating above equation I hy ly, z) ay = Jody where gez) hly, z) is dome done gez) fn worot z. 1 putting this value in equation ③ we get g(z) + ) 2² y z 6 f oort 'z' Now differentiating @ Itz xezy g'(z) 7 + 8 also we've tz xzy (from ) and equation ® we get comparing equation tz x²y + g'(z) = g'(x) = 0 x²y PART 04

Now integrating 2 10t get I grader Jodi [where c is a Lons land Igez) - c in question Now putting valen lale get, Dedy? tc is the sequiruo funtion. Answel f od mooth cird We've To (2xyz)i + (22²2) h + 6x²y) & piece wise curive from the point (1,1,1) 70 (1,2,4). с Here F into can express x, y and z in terms going to parametrize the x, y, and a componenty of meto vas ti eble If we of t, y = y(t), z = 2(+) ast sb. Then JA (x, y, z) dx = F(x64), y), z(+)) x'(+) dt. x= x(+) , > a Une / cuerve that starts The vector representation of at oto and ends at it is :- 7 (4) (1-t) 1 t tt ost 1 (PAGE ON

Не.91 е welve 엥 61, 1, 1) and 91,= (1, 2, 4) (1-A) (1, 1, 17 + t (1, 2, 4) ( 1-4, 1-6, 1-4> + <t, at, 4t> 2 2 et < 1 1+4, 1+3+> is the vector supresentation of the ewive. 20 = 1, 1 tt, 1 +3t Jo 42@ dx = da = 3 9 dy 1 Now, fr F ds + Cox?z) dy C 1 0 (2 x y z ) dx + (x²y) dz putting values of J[(2.1.6i+4). (1+34)). (1++). (1+3+)).0 + (62)2.61+34)). 1 + (0232.(2+2). 3)] de (3+3+)] dt Ilo + CI+ 3) + (3+3+) for wat • [ufi on] 7 [4-0] + [ 3 -o] Ayower PAGE 06)