Mathematical Methods for Physics and Engineering - Matematica.NET

10.6 SCALAR AND VECTOR FIELDSA normal n to the surface at this point is then given by∣ ∣∣∣∣∣∣n = ∂r∂θ × ∂ri j k∂φ = a cos θ cos φ acos θ sin φ −a sin θ−a sin θ sin φ asin θ cos φ 0∣= a 2 sin θ(sin θ cos φ i +sinθ sin φ j +cosθ k),which has a magnitude of a 2 sin θ. Therefore, the element of area at P is, from (10.23),dS = a 2 sin θdθdφ,and the total surface area of the sphere is given byA =∫ πdθ∫ 2π0 0dφ a 2 sin θ =4πa 2 .This familiar result can, of course, be proved by much simpler methods! ◭10.6 Scalar and vector fieldsWe now turn to the case where a particular scalar or vector quantity is definednot just at a point in space but continuously as a field throughout some regionof space R (which is often the whole space). Although the concept of a field isvalid for spaces with an arbitrary number of dimensions, in the remainder of thischapter we will restrict our attention to the familiar three-dimensional case. Ascalar field φ(x, y, z) associates a scalar with each point in R, while a vector fielda(x, y, z) associates a vector with each point. In what follows, we will assume thatthe variation in the scalar or vector field from point to point is both continuousand differentiable in R.Simple examples of scalar fields include the pressure at each point in a fluidand the electrostatic potential at each point in space in the presence of an electriccharge. Vector fields relating to the same physical systems are the velocity vectorin a fluid (giving the local speed and direction of the flow) and the electric field.With the study of continuously varying scalar and vector fields there arises theneed to consider their derivatives and also the integration of field quantities alonglines, over surfaces and throughout volumes in the field. We defer the discussionof line, surface and volume integrals until the next chapter, and in the remainderof this chapter we concentrate on the definition of vector differential operatorsand their properties.10.7 Vector operatorsCertain differential operations may be performed on scalar and vector fieldsand have wide-ranging applications in the physical sciences. The most importantoperations are those of finding the gradient of a scalar field and the divergenceand curl of a vector field. It is usual to define these operators from a strictly347