17.1 Divergence And Curl Of A Vector Field
Here the important concepts of divergence and curl are defined.
Definition 17.1.1 Let f : U → ℝp for U ⊆ ℝp denote a vector field. A scalar valued function is called a
scalar field. The function f is called a Ck vector field if the function f is a Ck function. For a
C1 vector field, as just described ∇⋅ f
≡ div f
known as the divergence, is defined
Using the repeated summation convention, this is often written as
where the comma indicates a partial derivative is being taken with respect to the ith variable and ∂i denotes
differentiation with respect to the ith variable. In words, the divergence is the sum of the ith derivative of
the ith component function of f for all values of i. If p = 3, the curl of the vector field yields another vector
field and it is defined as follows.
where here ∂j means the partial derivative with respect to xj and the subscript of i in
the ith Cartesian component of the vector curl
. Thus the curl is evaluated by expanding the
following determinant along the top row.
Note the similarity with the cross product. Sometimes the curl is called rot. (Short for rotation not
This last symbol is important enough that it is given a name, the Laplacian.It is also denoted by Δ. Thus
∇2f = Δf. In addition for f a vector field, the symbol f ⋅∇ is defined as a “differential operator” in the
Thus f ⋅∇ takes vector fields and makes them into new vector fields.
This definition is in terms of a given coordinate system but later coordinate free definitions of the curl
and div are presented. For now, everything is defined in terms of a given Cartesian coordinate system. The
divergence and curl have profound physical significance and this will be discussed later. For now it is
important to understand their definition in terms of coordinates. Be sure you understand that for f a
vector field, div f is a scalar field meaning it is a scalar valued function of three variables. For a scalar field
f, ∇f is a vector field described earlier. For f a vector field having values in ℝ3,curlf is another vector
Example 17.1.2 Let f
k. Find div f and curlf.
First the divergence of f is
Now curlf is obtained by evaluating