It was in PS 126 where I wasfirst introduced to the concepts of the gradient, the divergence, and the curl.It was also there where the Divergence Theorem and Stokes’s Theorem were firstintroduced. At first, all I knew were their mathematical definitions. Later throughthe discussion, however, we started to discuss how these concepts are appliedin physics.The Del operator, , is a vector operator that operates ondifferentiable functions 1.
It can act in three ways. First, the gradient is theDel operator acting on some scalar function f 2. In three-dimensionalCartesian coordinates, grad f, or f, is a vector quantity with three components2. Each component is the partial derivative of f with respect to x, y, and z,respectively. The gradient gives the direction of maximum change of thefunction f 1,2. If the gradient is equal to zero at some point (x, y, z), thatpoint is considered an extremum of the function 2. The gradient is also asurface normal vector 3.
That is, the gradient is perpendicular to all surfaces,f (x, y, z)= constant 1. This is important when we know that some vectorfields can be obtained from a scalar field; that is, grad f = F 3. As an example, we know that theelectrostatic field is the negative gradient of the electrostatic potential, E = -V 1. Thus, moving in the direction of E is moving along the direction ofdecreasing potential 4. Thus, the gradient is an important concept inelectrostatics.Second, the divergence is the Deloperator acting on some vector function Fthrough the dot product 2.
In three-dimensional Cartesian coordinates, div F is the sum of the partial derivativesof F with respect to x, y, and z.Thus, the divergence of a vector function is a scalar. The divergence is alsoknown as the flux density 1. It measures how much the vector spreads out froma point 3. When divergence is positive, the point is a source, or “sink.”When divergence is negative, the point can be called a sink, or “faucet.” Asthe divergence is related to flux, the divergence is important in fluid flow3.
The divergence is used in the continuity equation of a compressible fluidflow and also in the condition for incompressibility 3.Third, the curl is the Del operatoracting on some vector function F throughthe cross product 2. From the definition of the Del operator and the crossproduct, the curl can be constructed, and this results to a vector.
The curlmeasures how much the vector rotates around a point 2. If curl F is zero, then the field isirrotational and is also conservative 3. This is also important in fluid dynamics,to know if the fluid is rotational or not.From these, we also learned about the DivergenceTheorem, which transforms triple integrals to surface integrals over theboundary surface of a region in space using the divergence 3. The Divergence Theoremhas many applications. It can be used in fluid flow to help characterize thesources and sinks, in heat flow for the heat equation, in potential theory togive properties of solutions to Laplace’s equation, and in more 3. It is alsoused in electrostatics and gravity to give the differential form of Gauss’s Law1.
We also discussed Stokes’s Theorem, whichtransforms surface integrals over a surface to line integrals over the boundarycurve of the surface using the curl 3. Stokes’s Theorem may be used in fluidmotion to show the circulation of the flow and may also be used in showing thework done by a force 3.As discussed, the gradient, thedivergence, the curl, and the theorems can be applied to different fields ofphysics, such as electrostatics, gravity, fluid dynamics, and heat flow. This showshow fundamental these concepts and theorems are to studying physics. Forcertain, there are more applications of all of these in other fields of physics.More specifically, these may be used in atmospheric dynamics, which describesthe fluid motions of the atmosphere 5.
The divergence of the wind field maybe computed using divergence, and the vorticity of geophysical flows may be computedusing the curl 6. Relative vorticity also shows clockwise or counterclockwiserotation 6. Using Stokes’s Theorem, the circulation is related to vorticity7, and sea breeze circulation may be studied 8.