What is the gradient?

The gradient is a vector operation on a scalar field that results in a vector whose direction points in the greatest rate of change.

What is the laplace operator?

The laplace operator is the divergence of the gradient of a function.

The gradient is defined as a vector operator that results in a vector that points in the direction of greatest rate of change.

The divergence is a vector operator that results in a magnitude (scalar quantity) of the outward flux density around a point in space.

So, because the laplace operator is the divergence of a vector quantity, its result is a scalar whose magnitude is proportional to the maximum rate of change.

If the gradient of a scalar field is nonzero, there exists a vector field that points in the direction of the maximum rate of change. Taking the divergence at a point in this vector field, one obtains a scalar whose magnitude represents the greatest rate of change at that point. If the magnitude of this value (the laplacian of the scalar field) is positive, the maximum rate of change is occurring in an outward direction from that point. If it is negative, the maximum rate of change is occurring in an inward direction toward that point.

What is Curl?

The curl of a vector field describes the direction and magnitude of rotation at every point in space.

What is the Wave Equation?

The wave equation is a partial differential equation describing the nature of waves in time and space.

What does the term "coupled" wave equation mean?

When solving transmission line problems, it is often of interest to solve for the instantaneous voltage and current. In doing so using a lumped element (loss-less) model, it is found that the wave equations for both the voltage and current depend on each other and are thought of as "coupled."