It was in PS 126 where I was

first 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 first

introduced. At first, all I knew were their mathematical definitions. Later through

the discussion, however, we started to discuss how these concepts are applied

in physics.

The Del operator, , is a vector operator that operates on

differentiable functions 1. It can act in three ways. First, the gradient is the

Del operator acting on some scalar function f 2. In three-dimensional

Cartesian coordinates, grad f, or f, is a vector quantity with three components

2. 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 the

function f 1,2. If the gradient is equal to zero at some point (x, y, z), that

point is considered an extremum of the function 2. The gradient is also a

surface 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 vector

fields can be obtained from a scalar field; that is, grad f = F 3. As an example, we know that the

electrostatic field is the negative gradient of the electrostatic potential, E = -V 1. Thus, moving in the direction of E is moving along the direction of

decreasing potential 4. Thus, the gradient is an important concept in

electrostatics.

Second, the divergence is the Del

operator acting on some vector function F

through the dot product 2. In three-dimensional Cartesian coordinates, div F is the sum of the partial derivatives

of F with respect to x, y, and z.

Thus, the divergence of a vector function is a scalar. The divergence is also

known as the flux density 1. It measures how much the vector spreads out from

a 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.” As

the divergence is related to flux, the divergence is important in fluid flow

3. The divergence is used in the continuity equation of a compressible fluid

flow and also in the condition for incompressibility 3.

Third, the curl is the Del operator

acting on some vector function F through

the cross product 2. From the definition of the Del operator and the cross

product, the curl can be constructed, and this results to a vector. The curl

measures how much the vector rotates around a point 2. If curl F is zero, then the field is

irrotational 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 Divergence

Theorem, which transforms triple integrals to surface integrals over the

boundary surface of a region in space using the divergence 3. The Divergence Theorem

has many applications. It can be used in fluid flow to help characterize the

sources and sinks, in heat flow for the heat equation, in potential theory to

give properties of solutions to Laplace’s equation, and in more 3. It is also

used in electrostatics and gravity to give the differential form of Gauss’s Law

1.

We also discussed Stokes’s Theorem, which

transforms surface integrals over a surface to line integrals over the boundary

curve of the surface using the curl 3. Stokes’s Theorem may be used in fluid

motion to show the circulation of the flow and may also be used in showing the

work done by a force 3.

As discussed, the gradient, the

divergence, the curl, and the theorems can be applied to different fields of

physics, such as electrostatics, gravity, fluid dynamics, and heat flow. This shows

how fundamental these concepts and theorems are to studying physics. For

certain, there are more applications of all of these in other fields of physics.

More specifically, these may be used in atmospheric dynamics, which describes

the fluid motions of the atmosphere 5. The divergence of the wind field may

be computed using divergence, and the vorticity of geophysical flows may be computed

using the curl 6. Relative vorticity also shows clockwise or counterclockwise

rotation 6. Using Stokes’s Theorem, the circulation is related to vorticity

7, and sea breeze circulation may be studied 8.