(Continued from previous post on the del operator,...

Thursday, July 21, 2011

(Continued from previous post on the del operator, here)

When ∇ is applied to a vector-valued function, it takes the form of what is called the divergence of the vector function, V(x,y,z), when V is defined by V(x,y,z) = P(x,y,z) i + Q(x,y,z) j + R(x,y,z) k, where P, Q, and R, are scalar functions. The divergence is the dot-product of the del operator and V:

The divergence of V may consequently be thought of as a scalar field. The physical significance of the divergence of a vector field is that it is the amount of density “flowing” into, or out of, a given region of space (visualizing the vector field as a quasi fluid-like entity — consequences in electromagnetism and fluid dynamics follow immediately). Alternatively, it may be visualized as how a particular point behaves as a “source” from a point (in which density is decreasing) or a “sink” toward a point (in which density is increasing).

Another way in which ∇ is applied to vector-valued functions is the scenario in which, instead of the dot-product, the cross-product (or vector-product) of ∇ and the vector function V is taken — creating what is known as the curl of V:

(Note: the matrix determinant does not directly correlate to the definition of the operation, but it is merely a method of better remembering how to carry out the curl calculation.) The curl represents the infinitesimal rotation of a 3-dimensional vector field. This rotation is represented by a vector, whose length is proportional to the magnitude of rotation, and whose direction is that of said rotation about V. The intuitive interpretation of this operation may be represented as this: if, in a vector field describing the motion of a viscous fluid; there is a ball with an imperfect surface, then the ball will rotate due to the motion of the fluid about it.

The Laplace operator (also known as the Laplacian) is a differential operator which is defined to be the divergence of the gradient of a scalar field (in Euclidean space):

The Laplace operator has analogs in many coordinate systems, and for that reason its intuitive interpretations have many forms. In physical systems, the Laplace operator may be interpreted as the flux density (flux per unit area) of the gradient flow of a scalar field. Common applications lie in the analysis of gravitational potential fields, electric potential fields, and heat flow. This operator gives rise to the d’Alembert operator, which is its analog in 4-dimensional space-time. albanhouse delves into this particular operator’s connection to electromagnetism and tensor analysis in this recent post on Maxwell’s equations.