Here the important concepts of divergence and curl are defined.
Definition 17.1.1Let f : U → ℝ^{p}for U ⊆ ℝ^{p}denote a vector field. A scalar valued function is called ascalar field. The function f is called a C^{k}vector field if the function f is a C^{k}function. For aC^{1}vector field, as just described ∇⋅ f
(x)
≡divf
(x)
known as the divergence, is definedas
∑p ∂fi
∇ ⋅f (x) ≡ divf (x) ≡ ∂xi (x).
i=1
Using the repeated summationconvention, this is often written as
f (x ) ≡ ∂ f(x)
i,i ii
where the comma indicates a partial derivative is being taken with respect to the i^{th}variable and ∂_{i}denotesdifferentiation with respect to the i^{th}variable. In words, the divergence is the sum of the i^{th}derivative ofthe i^{th}component function of f for all values of i. If p = 3, the curl of the vector field yields another vectorfield and it is defined as follows.
(curl(f)(x))i ≡ (∇ ×f (x ))i ≡ εijk∂jfk(x)
where here ∂_{j}means the partial derivative with respect to x_{j}and the subscript of i in
(curl(f)(x))
_{i}meansthe i^{th}Cartesian component of the vectorcurl
(f)
(x )
. Thus the curl is evaluated by expanding thefollowing determinant along the top row.
Note the similarity with the cross product. Sometimes the curl is called rot. (Short for rotation notdecay.) Also
∇2f ≡ ∇ ⋅(∇f ).
This last symbol is important enough that it is given a name, the Laplacian.It is also denoted by Δ. Thus∇^{2}f = Δf. In addition for f a vector field, the symbol f ⋅∇ is defined as a “differential operator” in thefollowing way.
f ⋅∇ (g ) ≡ f (x) ∂g-(x-)+ f (x) ∂g-(x-)+ ⋅⋅⋅+f (x) ∂g(x).
1 ∂x1 2 ∂x2 p ∂xp
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, divf 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 ℝ^{3},curlf is another vector
field.