Question

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

Answer 1

f = tanx *i + y³*j

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