Chapter 22

Calculus Of Vector Fields

22.1 Divergence And Curl Of A Vector FieldHere the important concepts of divergence and curl are defined.

Definition 22.1.1 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∂

∂ z

f1 (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 ) .

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