Chapter 29

Calculus of Vector Fields29.1 Divergence and Curl of a Vector Field

Here the important concepts of divergence and curl are defined in terms of rectangularcoordinates.

Definition 29.1.1 Let f :U →Rp for U ⊆Rp denote a vector field. A scalar valuedfunction is called a scalar field. The function f is called a Ck vector field if the function fis a Ck function. For a C1 vector field, as just described ∇ ·f (x)≡ divf (x) known as thedivergence, is defined as

∇ ·f (x)≡ divf (x)≡p

∑i=1

∂ fi

∂xi(x) .

Using the repeated summation convention, this is often written as

fi,i (x)≡ ∂i fi (x)

where the comma indicates a partial derivative is being taken with respect to the ith variableand ∂i denotes differentiation with respect to the ith variable. In words, the divergence isthe 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.

(curl(f)(x))i ≡ (∇×f (x))i ≡ ε i jk∂ j fk (x)

where here ∂ j means the partial derivative with respect to x j and the subscript of i in(curl(f)(x))i means the ith Cartesian component of the vector curl(f)(x). Thus the curlis evaluated by expanding the following determinant along the top row.∣∣∣∣∣∣

i j k∂

∂x∂

∂y∂

∂ zf1 (x,y,z) f2 (x,y,z) f3 (x,y,z)

∣∣∣∣∣∣ .Note the similarity with the cross product. Sometimes the curl is called rot. (Short for

rotation not decay.) Also∇

2 f ≡ ∇ · (∇ f ) .

This last symbol is important enough that it is given a name, the Laplacian.It is also de-noted by ∆. Thus ∇

2 f = ∆ f . In addition for f a vector field, the symbol f ·∇ is defined asa “differential operator” in the following way.

f ·∇(g)≡ f1 (x)∂g (x)

∂x1+ f2 (x)

∂g (x)

∂x2+ · · ·+ fp (x)

∂g (x)

∂xp.

Thus f ·∇ takes vector fields and makes them into new vector fields.

This definition is in terms of a given rectangular coordinate system but later coordinatefree definitions of the curl and div are presented. For now, everything is defined in termsof a given Cartesian coordinate system. The divergence and curl have profound physicalsignificance and this will be discussed later. For now it is important to understand their

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