Chapter 9, Problem 9.2P

Interpretation Introduction

Interpretation: The following equation is to be proved as $\text{<mover accent="true"> <mo>V</mo> <mo>→</mo></mover>}\text{.}\nabla$ along with the physical significance:

  ${\text{v}}_{\text{x}}\frac{\partial }{\partial \text{x}}\text{+}{\text{v}}_{\text{y}}\frac{\partial }{\partial \text{y}}\text{+}{\text{v}}_{\text{z}}\frac{\partial }{\partial z}$

Concept introduction:

The product $\text{<mover accent="true"> <mo>V</mo> <mo>→</mo></mover>}\text{.}\nabla$ is known as the dot product of the two vectors, where $\text{<mover accent="true"> <mo>V</mo> <mo>→</mo></mover>}$ is the velocity vector and $\nabla$ is the differential with respect to each direction.

The dot product of the $\text{<mover accent="true"> <mo>V</mo> <mo>→</mo></mover>}\text{.}\nabla$ is called the divergence of the vector. The divergence of the 2 vectors is always a scalar quantity.

The dot product of vectors in same direction is 1 and the dot product of vectors in different direction is zero.