**Gradient, Divergence and Curl: the Basics**

We first consider the position vector, **r**:

**r**** = x x + y y +
z z **,

where ** x**,

**dr = dx x +
dy y + dz z **.

The operator, **Ñ** is
defined to be (in rectangular coordinates) as:

**Ñ**** =
****¶**** /****¶****x x + **

This operator operates as a vector.

**1. Gradient**

If the **Ñ**, operates on a scalar
function, f(x,y,z), we get the **gradient**:

**Ñ****f
= (****¶****f/****¶****x) x + (**

We can interpret this gradient as a vector
with the magnitude and direction of the maximum change of the function in space.
We can relate the gradient to the differential change in the function:

**df
= (****¶****f/****¶****x) dx + (****¶****f/****¶****y) dy + (**

---------------------------

Since the

**2. Divergence**

The **divergence of a vector** is defined
to be:

**Ñ**** ****·**** A = [****¶**** /****¶****x x + **

**= (****¶****A _{x} /**

since the rectangular unit vectors are
constant, ¶** x**/¶x = 0 (etc.). This will not necessarily be true for
other unit vectors in other coordinate systems. We'll see examples of this
soon.

To get some idea of what the divergence of a
vector is, we consider Gauss' theorem (sometimes called the divergence
theorem). We start with:

**òòò**** ****Ñ**** ****·**** A dV = ****òòò**** [(****¶****A _{x} /**

**òòò**** [(****¶****A _{x} /**

We can see that each term as written in the last
expression gives the value of the change in vector **A** that cuts perpendicular through
the surface. For instance, consider the first term: **(****¶****A _{x}/**

**òòò**** ****Ñ**** ****·**** A dV = ****òò**** _{closed
surface} A**

where the vector **S** is the surface area vector. Thus
we see that the volume integral of the divergence of vector **A** is equal
to the net amount of A that cuts through (or diverges from) the closed surface
that surrounds the volume over which the volume integral is taken. Hence the
name **divergence** for **Ñ** · **A** .

**Example**
from electromagnetism: Consider a single
point charge, q, and its electric field: **E** = kq/r^{2} which points radially away
from the center. Now let’s enclose that
charge in a sphere of radius, r, with the charge initially at the center. The dS vector
will also point radially away from the center.
Since the magnitude of the electric field will remain the same at all
points on the surface (since r is constant), the dot product of **E****· dS **will
be a constant; and so the integral over the entire surface will simply give
(kq/r^{2})*(4pr^{2}) = 4pkq,
and with k = 1/(4pe_{o}), we can
write this as q/e_{o} . We recognize that the charge is the source of
all the field, so it really doesn’t matter where the charge is inside the
sphere – it doesn’t have to be at the center of the sphere. If there is no charge inside, then the integral
is equal to zero. Thus, **òòò ****Ñ ****· E dV = ****òò
_{closed surface} E**

For magnetic fields, since there are no monopoles, the magnetic pole density is always zero, so

For gravity, we get:

Due to the spherical symmetry of the fields from point charges and masses, we will have to wait until we get

**3. Curl**

The **curl of a vector** is defined to be:

**Ñ**** ****´**** A = [****¶**** /****¶****x x + **

**(****¶****A _{y} /**

**= (****¶****A _{z}/**

where we have used the fact that the unit
vectors do not change with position (**¶***x***/****¶****x = 0**)** **and the fact that (*x**´** x*=0
and

To see what the curl of a vector means, we
use Stokes Theorem. We begin with:

**òò**** _{surface}
(**

**òò**** [(****¶****A _{z}/**

**= ****òò ****[(****¶****A _{z}/**

The dS_{x} = dy*dz = dy dz, etc.
However, we must worry about direction since *x**´** y *=

**òò**** (****Ñ**** ****´**** A) ****·**** dS = **

**= ****òò ****[(****¶****A _{z}/**

Regrouping gives:

**òò**** (****Ñ**** ****´**** A) ****·**** dS = **

**òò**** [****¶****A _{x}/**

Now we note that dA_{x} = (¶A_{x}/¶x)dx + (¶A_{x}/¶y)dy + (¶A_{x}/¶z)dz . In the above
integration, x was held constant when we integrated over the other variables,
so the (¶A_{x}/¶x)dx term is zero.
Thus the above double integral becomes:

**òò**** (****Ñ**** ****´**** A) ****·**** dS = ****òò**** [dA _{x}
dx + dA_{y} dy + dA_{z} dz] = **

If the integral around a closed loop is not zero,
then that implies that there is some circulation of the vector field. Note that
if the curl of the vector is zero everywhere, then there cannot be any
circulation of the vector field anywhere in space. Hence the name of **curl **for
**Ñ**** ****´** **A** .

Another derivation of Stoke’s Theorem can be
found in Volume 2 of the Feynmann lectures at:
http://feynmanlectures.caltech.edu/II_03.html#Ch3-S6
.

We can see an immediate use for the curl if
we recall our discussion about work. If the curl of a force field is zero, then
the work done around a closed path must be zero regardless of the closed path
chosen. This means that the work done between any two points must be
path-independent! This then allows a potential energy change to be defined for
this force that depends only on the beginning and ending points.** **

**Example:
**Consider the problem
we had in the last section:

**F _{x} = ax^{3} + bxy^{2} + cz** .

F_{y} = ay^{3} + bx^{2}y

F_{z} = cx

We found that this force gave us a work that was independent of path (we tried two different paths and got the same result). Let’s look at the curl of this F. It should equal zero if the Work that this force does is independent of path.

**Ñ**** ****´**** F = [****¶**** /****¶****x x + **

**= (****¶****F _{z}/**

= (0 – 0)** x
**+ (c – c)