Question

(1 point) (a) Show that each of the vector fields F = 4yi + 4xj, G= x y zit vol y J, and ] = vertinant virtuaj are gradient vector fields on some domain (not necessarily the whole plane) by finding a potential function for each. For F, a potential function is f(x, y) = For G, a potential function is g(x, y) = For i, a potential function is h(x, y) = (b) Find the line integrals of F, G, H around the curve C given to be the unit circle in the xy-plane, centered at the origin, and traversed counterclockwise. SoF. dr = Scã. dr = Sců.dr = (c) For which of the three vector fields can Green's Theorem be used to calculate the line integral in part (b)? It may be used for ? (Be sure that you are able to explain why or why not.)

Answer 1

a(1)

Griven, F = 4yî +4x 9 - TEL is a gradient dient vector field of a function f(x,y). For 7, a potential function is f (xy) - 4x4 as grad. fa vf og O).say 4yê + 4 x 3 7 a i + ax 2

a(2)

a(3)

b(1) and part (c)

F 448 + 4 We can x & g a see that is a simple closed curre as it us a circle and the function F is a polynomial of a and therefore it is continuous fonton recher field having continuous derivatives in R baum Region bounded by a closed a Curve C. There it is qualified for Green The Green the applying Green Shorem. & r. dr . SS ra 4x 2 49 44) de dy C R 2 SS (4-4) dedy, R & F. dr 2 O Green theorem Therefore (c) There can be used to calculate the F is the only vector field where line integral

b(2)

Тc5 + 3 2 the in plane xy 2 Jy 5х x+y2 grad g where ga z dom" () G It is clear that grad g of any region outside a small cincle enclosing origin. however, nout defined at the origin The line integral round elosed any the origin. (0,0) are g & G Cunue I= & ordi C 6 542 522. dx î + dy 3 2 x² + y2 syds- Ex dy x +y 2. $-5 C 2 e diſkuu (5 -5 f d {bant a einele with radius I and centre at origin then putting We get, I = -5 par tant (4) Sant (land) t Now C is y = ahnt. X2 a Cost, ai dt. o 2 2 - 5x20 2-10T. G . dr = – 10K

b(3)