Divergence of Curl
In a smooth vector field defined in a region of space (V), the divergence of the curl of is zero, i.e.
Proof of Divergence of Curl
Vector Field : Consider a vector field with components (Fx, Fy, Fz) defined in a region (V).
Curl of : Calculate the curl of using the cross product of the del operator and
:
Use Cross-Product Identities
Apply Clairaut’s Theorem
Since mixed partial derivatives are equal , the terms cancel each other.
The result simplifies to , confirming the divergence of the curl is zero. This theorem is a consequence of the vector calculus identities and plays a crucial role in understanding the relationships between different operations on vector fields.
Divergence and Curl
Divergence and Curl are important concepts of Mathematics applied to vector fields. Divergence describes how a field behaves concerning or moving away from a point, while curl measures the rotational aspect of the field around a specific point. Divergence operators give scalar results whereas Curl operators give vector results.
In this article, we will learn about the divergence definition, curl definition, divergence of the vector field, curl of a vector field, and others in detail.
Table of Content
- What is Divergence?
- What is Curl?
- Divergence of Vector Field
- Curl of a Vector Field
- Divergence of Curl
- Equations of Divergence and Curl
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