From Fundamentals of Electromagnetic Fields


The Del operator ( ?) is defined as:

This vector operator possesses properties analogous to those of ordinary vectors. It is useful in defining a quantitie s gradient, curl, and divergence by operating ? on a scalar field, and on a vector field by cross and dot product respectively.

The Gradient

Let ?(x,y,z) be a scalar function defined and differentiable at each point (x,y,z) in a certain region in space., i.e., ?(x,y,z) is a differentiable scalar field. Then, the gradient of ?(x,y,z) is defined as:

Physical Interpretation of Gradient

Let us have a function of three variables, say, T(x,y,z) that represents temperature in a room at point (x,y,z). Then, its derivative is supposed to tell us how fast the temperature (function) varies with position coordinates. But, it is not as simple as it seems to be since it is important to know the direction in which one moves. Here, one has to consider the rate of change of temperature T along different directions i.e., a x, a y, and a z. A known theorem on partial derivatives simply represents rate of change of T(x,y,z) as:

The above equation tells us how fast T changes when we alter all three variables by infinitesimal small amounts dx, dy, and dz. Writing the above equation in a more convenient way as following where the relation is reminiscent...

Copyright Infinity Science Press LLC 2007 under license agreement with Books24x7

Products & Services
Color Meters and Appearance Instruments
Color meters and appearance instruments are used to measure the properties of paints and coatings including color, gloss, haze and transparency.
Gradient Index Lenses
Gradient index (GRIN) lenses focus light through a precisely controlled radial variation of the lens material's index of refraction from the optical axis to the edge of the lens.
Industrial Valves
Industrial valves are classified in many different ways. They can be distinguished by material of construction, media handled, and/or application.
CNC Routers
CNC routers are computer-controlled machines that are capable of cutting complex two-dimensional (2D) and three-dimensional (3D) shapes with router bits that spin at high speeds.

Topics of Interest

1.8 DIFFERENTIAL GEOMETRY IN CURVILINEAR COORDINATES Consider a generalized curvilinear coordinate system in which a point will be located by the construction of three mutually perpendicular...

2.11 The Laplacian Operator All of the differential operations discussed so far pertain to first-order differential operators. One second-order differential operator that occurs frequently in the...

Chapter List Chapter 4: Kinematics of Continuum Chapter 5: Kinetics of Continuum Chapter 6: Thermodynamics of Continuum Chapter 7: Nonlinear Elasticity 4.1 Preliminaries Let ( x) be a scalar...

2.8 The Gradient of a Scalar Function Let f(x, y, z) be a real-valued differentiable function of x, y, and z, as shown in Figure 2.28. The differential change in f from point P to Q , from equation...

In this chapter, we present the Stokes equations and their relation to the general linear elasticity equations. The present adaptive-refinement algorithm uses modified multigrid to solve linear...