Question

(1 point) For each of the following vector fields F, decide whether it is conservative or not by computing the appropriate first order partial derivatives. Type in a potential function f (that is, V f = F) with f(0,0) = 0. If it is not conservative, type N. A. F (x, y) = (-140 – 4y) i + (-4x + 12y)j f (x, y) = B. F (x, y) = -7yi - 6xj f(x,y) = C. F (2, y) = (-7 sin y)i + (-8y - 7x cos y)j f (x, y) = Note: Your answers should be either expressions of x and y (e.g. "3xy + 2y"), or the letter "N"

Answer 1

A. ㅅ f(n,y) = (- 14x – 4y) î + (-42 +12y) (1) A FT +fz 心, (sa) Where fi = - 14% -47 and F2 = - An +12 y The first step is to check if f is conservative or not - ... = du a Fz an afi ау (-42 +12y) = -4 o ay (-kn-ay) and = 0 so af afi we conclude that an ay the scalar curl of f is zero and consequently . f is conservative tience, there exists some potential function f so that sf = f. .'. af af dy - an (fin fz) comparing, we get af =f2 af ay at af 27 on -4n+izy → (3) 14a-4y 1) → (3) ou

take can with condition (3) . we let's start af zxc (n,y) ~ 144may, and then integrating constant and treat y , we get Where (4) f(x,y) = -14 Sadre – ayf dr + $(y) '$'in an arbitrorry function = - 72 - Any + $(4) Now, differentiating partially with respect to 'y We get from (4) af ad dy → (5) - 42 t oy = • We get lry cis Comparing (s) with (2) af oy :: $(y) = Gyk + c Where constant tence from (4) We get f(0,42 = -7% - Any +by te (6) f(0,0) = =0, then from 9 (6) given that get We c :. f(wy) - fur- 4ny toyr

B -> (1) F(x, y) = -7y î -suh A (say) fir th and fi = -ty Where -62 Fz= ir fin conservatice & first we cheek whether not . and =-6 o', OF2 an 2 a (-6x) & fi ay (-7y) =-7. af we conclude that then of to ay an conservative F is NOT Z f(a, y) = >(1) c. f(n,y) = (-7 siny) î + (-By-7x cosy) fi î tf₂ A (say) Where fi - 7 sim y and f = -8y - In cos y whether f i conservative or First check we not. 7x cos :, ola or (- 7 smy) and afa - fcos y cosy) .8y. an -7cos y . afi 2 ay ay afz ah co hence F i conservative on dy there exists some potential function f , Hence

so that ... of = f. () = (1,6) (-7 sin y siny, -8y- In cosy cosy) af = -7 siny (2) on at sy-7ncosy (3) dy with condition (2) condition (2). We can Let's stort take the equation =-7 siny an and the integrating, constant and treat y We get f = -7 sing som + Ø (4) where ¢ is function of y (4) :. f(n,y) = te sinly) +0(4) Now, differentiating (1) partially with respect to We get ry', ad ay 5 7x cos (4) + af Jy Now, comparing (5) with (3) , we get ad =-8y ay Integrating, we get $(y) = - 4yhte putting the value f(x,y) = 7x sin (4) -> (6) in (4), we get 4 yr te → (7) Hence

Gwen that f(0,0) = 0 then from (7), we get c=0 f(,y) = - 7e sin (y) - 4yr

Please give a thumbs up if you are benefited.