For each of the following vector fields f, decide whether it is conservative or not by computing curl f. type in a potential function f (that is, ∇f=f). assume the potential function has a value of zero at the origin. if the vector field is not conservative, type n.1. f(x, y)=(-6x+5y)i+(5x+10y)j
2. f(x, y,z)=-3xi-2yj+k
3. f(x, y)=(-siny)i+(10y-3xcosy)j
4. f(x, y,z)=-3x^2i+5y^2j+5z^2k

1,2 and 4 are conservatives

3 is not conservative

Step-by-step explanation:

We calculate  the Curl F

Remember that:

        Curl F = <>

1. Curl F = <0,0,5-5> = <0,0,0>

  The potential function f so that  ∇f=F

  f(x,y,z) =

  Then F is conservative

2. Curl F = < 0, 0 ,0>

  The potential function f so that  ∇f=F

  f(x,y,z) =

  Then F is conservative

3. Curl F = <0 ,0, 10+3xsin(y) - (-cos(y))>

              = <0 ,0 , 10 +3xsin(y) + cos(y)<

 How the field's divergence is not zero the vector field is not conservative

4. Curl F = <0, 0, 0>  

  The potential function f so that  ∇f=F

  f(x,y,z) =    

   Then F is conservative

heyyy

step-by-step explanation:

x =±sqrt(1/3)

step-by-step explanation:

x+4/6x=1/x

multiply each side by 6x to get rid of the fractions

6x*(x+4/6x)=1/x*6x

distribute

6x^2 +4 = 6

subtract 4 from each side

6x^2 +4-4 = 6-4

6x^2 =2

divide by 6

6x^2/6 = 2/6

x^2 = 1/3

take the square root of each side

sqrt(x^2) = ±sqrt(1/3)

x =±sqrt(1/3)