Curl and Divergence
Two fundamental operations that tell us how a vector field spins and expands.
Introduction
In 3D vector calculus, the derivative of a field isn't just a single thing. We split it into two concepts:
- Curl: Measures the local rotation or "spin" of the field.
- Divergence: Measures the local expansion or "outflow" of the field.
Curl (Rotation)
Formula
The Curl is a Vector. It is calculated as the cross product of the del operator with the field:
If Curl is zero everywhere, the field is Irrotational (and Conservative).
Divergence (Expansion)
Formula
The Divergence is a Scalar. It is the dot product of the del operator with the field:
If Div is zero, the fluid is Incompressible.
Visualizing Fields
Rotational Field (High Curl)
Explosive Field (High Divergence)
Worked Examples
Example 1: Computing Curl
Find the curl of .
1. Setup Matrix:
2. i-component:
.
3. j-component:
.
4. k-component:
.
Result: .
Example 2: Computing Divergence
Find the divergence of .
1. Differentiate Components:
.
.
.
2. Sum:
.
The field is uniformly expanding everywhere!
Example 3: Useful Identity
Show that for any smooth vector field.
1. Write components of Curl:
.
2. Take Divergence:
.
.
3. Cancel (Clairaut's Theorem):
, , etc.
Everything cancels to 0. A rotational field has no source or sink.
Practice Quiz
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