Here the important concepts of divergence and curl are defined.

Definition 27.1.1Let f : U → ℝ^{p}for U ⊆ ℝ^{p}denote a vector field. A scalarvalued function is called ascalar field. The function f is called a C^{k}vector field if thefunction f is a C^{k}function. For a C^{1}vector field, as just described ∇⋅ f

(x )

≡divf

(x)

known as the divergence, is defined as

p
∇ ⋅f (x) ≡ divf (x) ≡ ∑ ∂fi(x).
i=1 ∂xi

Using the repeated summationconvention, this is often written as

f (x) ≡ ∂ f (x)
i,i i i

where the comma indicates a partial derivative is being taken with respect to the i^{th}variable and ∂_{i}denotes differentiation with respect to the i^{th}variable. In words, thedivergence is the sum of the i^{th}derivative of the i^{th}component function of f for allvalues of i. If p = 3, the curl of the vector field yields another vector field and it isdefined as follows.

(curl(f)(x)) ≡ (∇ × f (x )) ≡ εijk∂jfk(x)
i i

where here ∂_{j}means the partial derivative with respect to x_{j}and the subscript of i in

(curl(f)(x ))

_{i}means the i^{th}Cartesian component of the vectorcurl

(f)

(x)

.Thus the curl is evaluated by expanding the following determinant along the toprow.

|| i j k ||
|| ∂- ∂- -∂ ||.
||f (x∂,xy,z) f (∂xy,y,z) f (∂xz,y,z) ||
1 2 3

Note the similarity with the cross product. Sometimes the curl is called rot. (Short forrotation not decay.) Also

2
∇ f ≡ ∇ ⋅(∇f ).

This last symbol is important enough that it is given a name, the Laplacian.It is alsodenoted by Δ. Thus ∇^{2}f = Δf. In addition for f a vector field, the symbol f ⋅∇ is definedas a “differential operator” in the following way.

f ⋅∇ (g) ≡ f1 (x ) ∂g(x)-+ f2 (x ) ∂g(x)-+ ⋅⋅⋅+ fp(x) ∂g-(x).
∂x1 ∂x2 ∂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.