Question

Find a potential function f for the field F. F = (inx + ser ?(6x + 6%)* (sec°(ex+6)= m n ) (yhtye Select the correct choice and fill in any answer boxes in your choice below. O A. f(x,y,z) = (Type an exact answer. Use C as the arbitrary constant.) OB. A potential function does not exist.

Answer 1

Vet Given sector Hield is Es Linx + secY6*+63)) î +(sec°(6x+69) + Â +(942) NOW PxP": fata + ) *F* NOW az |(272a + sexpaty) (se(62 1653- | Contato | = ê hetev) (sec%eze67) * "thr)+ 2 (Pony + se fez+) - )7+: Ř (Salse1647639+ man) - 33 (Preusementy X. 12 sec (6x769) tom (62764)–12 SecY6X769) tanC6x+6y)] = oitoai to 02 tot to AS so consenuat Peton VA F 20. F is a function B12,7,Z). there exist a potential such that Farba

( Then we have ale = lnx +scev(64+63) — 0 en het sec (64*69) + letter - @ en de - BG Integrating a partially with respect to a, keeping I and Z constant , we have to llora dx + f secv(6x+64) da or, t = Ina fabe - Jelgaciones | dra] da ti sec (5x+63) d (8416) ng, ki x Xnz - xxx dx + c om (6 +6 9) om, b = xmx-fda + s tom (62769) on, xlna-x+ tan(6x+6y) +,(9,) where 8,(9,) is an Am Poitrany function of y and z. Integrating a partially with respect to y, keeb x and ² constant, we have f = | sec (62 +69) dy + dy 00, * = tej sec"182469) 2(6x+69) + / do on. Batam (6x+64) + In (9V+2°) + 22(2,7) — where qalx,7) is an arbitrary Bunction of aandz. Integrating partially with siespect to z kech x and y constant, we have ta vyhradz to z, keeping

on, 1 (dlyW+z” o ] 9V+zV on, of all In (3424) + 4.3(3,9) --- ® where &3 (2,7) is an anlitrany Bunction of a andy. Comparison of 6 5 and ② shows that there will be a common value of f it we choose 9, (9, Z) = 1 n ly Y Zr +C c is the arbitrary . Constant 8. (X, 7) - alnx - x + go(X,Y)= alna-at no tan (6x+69) + C so that &(2,9, 2) a almx-x + 1 tan (68+69) + 1 lm (yhte