**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, del: **Ñ**, is defined to be (in rectangular coordinates):

**Ñ**( )** = x **

Note: the unit vectors are placed here in
front of the operation, e.g., **¶( )**** /****¶****x **, to
show that the operation is not performed on the unit vector itself. After the operations, the unit vectors will
be placed after as is more usual.

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 + (**

**Note:** We will use this
relation to determine the del operator, **Ñ**, in other coordinate systems later.

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

Since the

**2. Divergence**

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

**Ñ**** ****·**** A =
[ x **

**= (****¶****A _{x} /**

since the rectangular unit vectors are constant, ¶** x**/¶x = 0 (etc.).

This will not necessarily be true for 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})** r** 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

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 **

**(****¶****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
.

**Important Use:** 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 example
problem we had in the last section:

**F _{x}**

F

F

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 **

**= (****¶****F _{z}**

= (0 – 0)** x
**+ (c – c)

To “see” both the curl and divergence of a couple
of force functions, see this word document.