Scalar Fields

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Section 1.18 Scalar Fields

The electrostatic potentials, \(V\text{,}\) and the gravitational potential, \(\Phi\text{,}\) are examples of scalar fields. A scalar field is any scalar-valued physical quantity (i.e. a number with units attached) at every point in space. It may be useful to think of the temperature in a room, \(T\text{,}\) as your iconic example of a scalar field.

The symbol \(\rr\) represents the position vector which points from an arbitrary fixed origin (that you get to pick once and for all at the beginning of any problem and must use consistently thereafter) to a given point in space. We often write the symbol that represents a scalar field as \(V(\rr)\) where the position vector \((\rr)\) not only reminds us that the scalar field may vary from point to point in space, but also give us a coordinate independent symbol to describe the point at which we are evaluating the field. Similarly, we write \(V(\rr)\text{,}\) \(\Phi(\rr)\text{,}\) or \(T(\rr)\) for the electrostatic potential, the gravitational potential, or the temperature, respectively. Even though the symbol \(\rr\) contains a vector sign, the name of the scalar field (e.g. \(V\text{,}\) \(\Phi\text{,}\) or \(T\)) does not, since the value of the scalar field at each point is a number, not a vector. Several alternative notations are commonly used to denote the point at which the scalar field is being evaluated, for example \(V(P)\) (where \(P\) is intended to represent the “point” at which the field is evaluated) or \(V(x,y,z)\) (where a coordinate system has been chosen).