Vector Fields  and Scalar Fields

 1 In a Nut Shell:  Functions that assign vectors in the plane or space are termed vector fields.  You are familiar with vectors such as the unit vectors  i, j, k.   A vector field  in a plane takes the form   F(x,y)  =  P(x,y) i  +  Q(x,y) j   where  P(x,y) and Q(x,y) are scalar fields. (scalar function)   A vector field can also be defined by a vector valued function at each point (x,y,z) in space such as                             F(x, y, z)  =  P(x,y,z) i     +  Q(x,y,z) j     +  R(x,y,z) k         where  P(x,y,z), Q(x,y,z), and R(x,yz) are scalar fields. (scalar function)   There are several theorems dealing with vector fields that are important in your study of multi-variable calculus.  So it is important that you understand and can work with vector fields. 2 A scalar function (Field) in a plane is defined as   F  =  f(x,y).  Likewise a scalar function in space is defined as    F  =  f(x,y,z). 3 The Gradient of a scalar field is defined as follows:    in a plane      Grad F  =  ∂F/∂x i  +  ∂F/∂y j      in space          Grad F  =  ∂F/∂x i  +  ∂F/∂y j   +  ∂F/∂z k 4 Definition of the Divergence of a Vector Field,  F      in a plane            F(x, y, z)  =  P(x,y,z) i     +  Q(x,y,z)                         div F   =  ∂P/∂x i  +  ∂Q/∂y j      in space               F(x, y, z)  =  P(x,y,z) i     +  Q(x,y,z) j     +  R(x,y,z) k                                 div F   =  ∂P/∂x i  +  ∂Q/∂y j   +  ∂R/∂z k    Click here to continue discussion of vector fields.