Question

The vector field F(x,y)= is conservative. Find such that F= ∇f

Answer 1

Answer:

f = tanx *i   + y³*j

Step-by-step explanation:

F(x,y) = ??

Fx,y) = gradf

Then

f(x,y)  = ∫ sec²x dx  + ∫ 3y²dy

∫ sec²x dx    =   ∫  1/ cos²x  dx

By change of variables

if  z  =  tan x      then   z  =  sinx/cosx

dz/dx = cosx*cosx - (-sinx*sinx) / cos²x

dz/dx = cos²x  +  sin²x /cos²x

dz/dx=  1 /cos²x

dz = dx/ cos²x

∫  1/ cos²x  dx    =   ∫dz = z = tanx

And  ∫3*y²   = y³

Then  f = tanx *i   + y³*j