Divergence (𝛻· | 𝛻·𝑓)

Divergence - Definition

The divergence of a vector-valued function 𝑣̅:

• $div\overline{v}=\nabla \cdot v=\frac{𝜕v_1}{𝜕x}+\frac{𝜕v_2}{𝜕y}+...$

where:

• $v_1,v_2,...\text{ are the components of }\overline{v}$

Divergence - Properties

The divergence operator 𝛻· is a linear operator such that it follows for any vector-valued functions 𝑓₁ and 𝑓₂, and scalar constants 𝛼 and 𝛽:

• $𝛻\cdot (𝛼f_1+𝛽f_2)=𝛼(𝛻\cdot f_1)+𝛽(𝛻\cdot f_2)$

Divergence - Use Cases

• Laplacian Operator (𝛻·𝛻 | 𝛻·𝛻𝑓)

Resources

• Steve Brunton’s The Divergence of a Vector Field: Sources and Sinks
• https://www.khanacademy.org/math/multivariable-calculus/multivariable-derivatives/divergence-grant-videos/v/divergence-intuition-part-1