VECTOR FIELDS
AND DERIVATIVES PART 3
Dr. Muhammad Mus’ab Anas
Applied Physics Program,
USIM
2
DIVERGENCE
In the previous section, we saw that the gradient operator
gives us a way to take a function 𝑓 (𝑥, 𝑦) and get a vector
field. In this section, we explore a way to take a vector field
and get a function. Eventually, we will see that the value of
this function at a point is a measure of how much the vector
field is “spreading out” there.
3
DIVERGENCE
Suppose 𝑓 (𝑥, 𝑦, 𝑧), 𝑔(𝑥, 𝑦, 𝑧), ℎ(𝑥, 𝑦, 𝑧) is a vector field V on ℝ3 . Then we define the
divergence of V, “Div V,” to be the function
𝜕𝑓 𝜕𝑔 𝜕ℎ
+
+
𝜕𝑥 𝜕𝑦 𝜕𝑧
Note that the first term is associated with the first component of V, the second term
with the second component, and the third term with the third component. This,
and the fact that the terms are being added, should remind you of the dot product.
This gives us a purely notational way to remember how to calculate the divergence of a
vector field.
4
We let ∇ denote the “vector” of
𝜕 𝜕 𝜕
, ,
𝜕𝑥 𝜕𝑦 𝜕𝑧
This, of course, is only a vector in a notational sense. But if we
suspend our disbelief for a moment and allow such absurdities, we
can write the formula for the divergence of a vector field in a very
compact way:
𝐷𝑖𝑣 𝑽 = ∇ ・ 𝑽
Let V be the vector field
𝑥 2 𝑦, 𝑥 + 𝑦𝑧, 𝑥𝑦 2 𝑧 3
Then the divergence of V is calculated as follows:
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6
END OF
PART 3