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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.

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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.

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