Question

Although it is not defined on all of space R', the field associated with the line integral below is simply connected, and the component test can be used to show it is conservative. Find a potential function for the field and evaluate the integral. (1,2,9) 27x? dx + 2 dy + 8z In y dz (1,1,5) Find the potential function. f(x,y,z) = 0 Evaluate the line integral. (1,2,9) 27° dx + 4zł dy +8ziny dz = 0 y (1,1,5) (Type an exact answer.)

Answer 1

- - (1,2,9) (272²dn+ 42 dy + 82 kny dz (2,1,5) . Find the potential function: Solenn oft 2772—11) oz tiis dx = 8zeny this Integration of equ? (i) worita keeping 9 ) constant 662,4,7) = 27.72 + +19,2) $19,9,2)j gm? + A 14,z) -ill then partial differentiation af equmois wirity and compare with equil ot. Ally, z) = 4 2 1 म JA Ly, z) = 192 dy A(Y,Z) = 4zAny & A(z) then t(1,9, 2) = 94² +42² hytate) www. in

- Oz then partial differentiation of equirêu) writ zand compare with equncillo ot - 834nyt A'(z) = 8zenty) A'CZ) =D [ A(Z) = constant I they *(my, z) = 973 +42’lny + constant , (3,2,9). then fin y z) as a HB-414) >[9(28+ 413nu)-( 913ft als” ents)] »[9 +32460))-( 9+1000)) 99-9 41,1,5)