Fields

A scalar determined at every point in a given domain, analytically or otherwise, constitutes a scalar field. Vectors similarly determined constitute a vector field. The defining analytical expressions of a three-dimensional field are commonly differentiable with respect to space; hence in a cartesian coordinate system they are amenable to partial differentiation with respect to x1, x2, and x3. In this context it is useful to define several differential operators. The operator ∇ is called the “del” or the “nabla” and is defined as follows: . . . ∇ ≡ i 𝜕/𝜕x1+j 𝜕/𝜕x2+k 𝜕/𝜕x3, (2.1) . . . or: . . . (∇)i ≡ 𝜕/𝜕xi. (2.2) . . . It can be seen that the del is a vector. By convention, however, it is not rendered in boldface. Before we define additional differential operators, we extend the subscript notation further and let a subscribed comma indicate partial differentiation. A comma preceding a letter subscript, say i, is taken to imply differentiation with respect to xi. Thus, if φ (xi) is a scalar function of position and thus defines a scalar field, its gradient, another differential operator, is defined by the equation: . . . (grad φ)i ≡ φ,i≡ 𝜕 φ/𝜕xi. (2.3) . . . Thus the gradient of a scalar is a vector.