Question

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

Answer 1

Answer:

1,2 and 4 are conservatives

3 is not conservative

Step-by-step explanation:

We calculate  the Curl F

Remember that:

        Curl F = <$\frac{dFz}{dy} - \frac{dFy}{dz}, \frac{dFz}{dx} - \frac{dFx}{dz}, \frac{dFy}{dx} - \frac{dFx}{dy}$>

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

  The potential function f so that  ∇f=F

  f(x,y,z) = $-3x^{2} +5xy + 5y^{2}$

  Then F is conservative

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

  The potential function f so that  ∇f=F

  f(x,y,z) = $-3/2x^{2} -y^{2}+z$

  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) = $x^{3}+(5/3)y^{3}+(5/3)z^{3}$    

   Then F is conservative