Determine whether the vector field F is conservative or not. If it is, find its potential function (i.e., find a real-valued function V such that F = -grad V). F ( x , y , z ) = x i + y 2 j + z k

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Determine whether the vector field F is conservative or not. If it is, find its potential function (i.e., find a real-valued function V such that F = -grad V).  F  (  x  ,  y  ,  z  )  =  x  i  +      y    2    j  +  z  k

grocbyntza · Accepted Answer

Given   F  (  x  ,  y  ,  z  )  =  x  i  +      y    2    j  +  z  kVector field F is conservative if curl F=0curl F=   ∇  ×  F  =      |                            i                          j                          k                                                  ∂                          ∂              x                                                            ∂                          ∂              y                                                            ∂                          ∂              z                                                            x                                      y            2                                    z                      |      =  i  (      ∂          ∂      y        (  z  )  −      ∂          ∂      z        (      y    2    )  )  −  j  (      ∂          ∂      x        (  z  )  −      ∂          ∂      z        (  x  )  )  +  k  (      ∂          ∂      x        (      y    2    )  −      ∂          ∂      y        (  x  )  )    =  i  (  0  )  −  j  (  0  )  +  k  (  0  )    =  0Hence the vector field F is conservativeNow find the potential functionThat is find a real valued function V such that F=-grad V  F  =  −  ∇  V    here   F  (  x  ,  y  ,  z  )  =  x  i  +      y    2    j  +  z  k    so,   x  i  +      y    2    j  +  z  k  =  −  ∇  V        V    x    (  x  ,  y  ,  z  )  =  −  x  ,      V    y    (  x  ,  y  ,  z  )  =  −      y    2    ,      V    z    (  x  ,  y  ,  z  )  =  −  z    V  (  x  ,  y  ,  z  )  =  ∫      V    x    (  x  ,  y  ,  z  )  d  x    =  ∫  −  x  d  x    =  −            x      2        2    +  g  (  y  ,  z  )    and     V  (  x  ,  y  ,  z  )  =  ∫      V    y    (  x  ,  y  ,  z  )  d  x    =  ∫  −      y    2    d  y    =  −            y      3        3    +  h  (  x  ,  z  )    and     V  (  x  ,  y  ,  z  )  =  ∫      V    z    (  x  ,  y  ,  z  )  d  x    =  ∫  −  z  d  y    =  −            z      2        2    +  k  (  x  ,  z  )    on combining the above     V  (  x  ,  y  ,  z  )  =  −            x      2        2    −            y      3        3    −            z      2        2      Hence the potential function is   V  (  x  ,  y  ,  z  )  =  −            x      2        2    −            y      3        3    −            z      2        2

Determine whether the vector field F is conservative or not. If it is, find its potential function F(x,y,z)=xi+y^2 j+zk

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