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22 DIFFERENTIAL AND INTEGRAL OPERATIONS
223 Surfacehtegrals
Surface integrals involve the integral operator,
l d*, (2.14)
where do is a vector that represents a differential area. This vector has a magnitude
equal to a differential area of S, and a direction perpendicular to the surface. If we
write the differential area as d u and the unit normal ii, the differential area vector
becomes d o = i 3 du. Because a surface has two sides, there is ambiguity in how
to define b. For a simple, closed surface such as the one in Figure 2.3(a), we define
ci to always point in the "outward" direction. If the surface is not closed, i.e., does not enclose a volume, the direction of b is typically determined by the closed path
C that defines the boundaries of the surface, and the right-hand rule, as shown in
Figure 2.3(b). Frequently, the surface integral operator acts on a vector field quantity by means of the dot product
/dT.V. (2.15)
S
h Cartesian coordinates.
where dui is positive or negative depending on the sign of ii. &i, as discussed in the
previous paragraph. This surface integral becomes
There are less common surface integrals of the form
2 Y '
Figure 2.3 Surface Integrals
DIFFERENTIAL OPERATIONS 23
which is an operation on a scalar which produces a vector quantity, and
P
which also generates a vector.
2.2.4 Volume Integrals
The volume integral is the simplest integral operator because the variables of inte- gration are scalars. It is written
where dT is a differential volume, and V represents the total volume of integration. The most common volume integral acts on a scalar field quantity and, as a result, produces a scalar
d r @.
In Cartesian coordinates, this is written
dxl dx2 dx3 @.
Volume integrals of vector quantities are also possible:
2. 3 DIFFERENTIAL OPERATIONS
By their definition, field quantities are functions of position. Analogous to how the change in a function of a single variable is described by its derivative, the position dependence of a scalar field can be described by its gradient, and the position dependence of a vector field by its curl and divergence. The Del operator v is used to describe all three of these fundamental operations. The operator v is written in coordinate-independent vector notation. It can be expressed in subscript/summation notation in a Cartesian coordinate system as
Keep in mind, this expression is only valid in Cartesian systems. It will need to be modified when we discuss non-Cartesian coordinate systems in Chapter 3.
