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Curl and Divergence
In this section we are going to introduce a couple of new concepts, the curl and the divergence of
a vector
...
Given the vector field F = P i + Q j + R k the curl is defined to be,
curl F = ( Ry − Qz ) i + ( Pz − Rx ) j + ( Qx − Py ) k
There is another (potentially) easier definition of the curl of a vector field
...
This is defined to be,
∇=
∂
∂
∂
i+
j+
k
∂x
∂y
∂z
We use this as if it’s a function in the following manner
...
Note as well
∇f =
that when we look at it in this light we simply get the gradient vector
...
Facts
1
...
This is
easy enough to check by plugging into the definition of the derivative so we’ll leave it to you
to check
...
If F is a conservative vector field then curl F = 0
...
3
...
This is not so easy to verify and so we
won’t try
...
Solution
So all that we need to do is compute the curl and see if we get the zero vector or not
...
Next we should talk about a physical interpretation of the curl
...
Then curl F represents the tendency of particles at the point ( x, y, z ) to
rotate about the axis that points in the direction of curl F
...
Let’s now talk about the second new concept in this section
...
The divergence can be
defined in terms of the following dot product
...
div F =
∂ 2
∂
∂
2 2
(
x y ) + ( xyz ) + ( − x y ) = 2 xy + xz
∂x
∂y
∂z
We also have the following fact about the relationship between the curl and the divergence
...
Solution
Let’s first compute the curl
...
(
)
div curl F =
∂
∂
∂
( z ) + ( 2 yz ) + ( y − z 2 ) = 2 z − 2 z = 0
∂x
∂y
∂z
We also have a physical interpretation of the divergence
...
This can also be thought of as the tendency of
a fluid to diverge from a point
...
The next topic that we want to briefly mention is the Laplace operator
...
The final topic in this section is to give two vector forms of Green’s Theorem
...
The second form uses the divergence
...
If the curve is parameterized by
r (t ) = x (t ) i + y (t ) j
then the outward unit normal is given by,
n=
y′ ( t )
x′ ( t )
i−
j
r′ (t )
r′ (t )
Here is a sketch illustrating the outward unit normal for some curve C at various points