MathHelp

Vector Fields and Scalar Fields

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.

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).

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

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

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