Vector Analysis

Phase 2: Calculus in Space

Syllabus Goal

Master vector calculus: Differential operators (Gradient, Divergence, Curl) and Integral theorems (Green’s, Divergence, Stokes’).

Differential Operators

• Gradient ($\nabla f$): Vector pointing in direction of steepest ascent.
• Divergence ($\nabla \cdot \mathbf{v}$): Outward flux density.
• Curl ($\nabla \times \mathbf{v}$): Circulation density.

Integral Theorems

1. Divergence Theorem: $\iiint_V (\nabla \cdot \mathbf{F}) dV = \iint_S \mathbf{F} \cdot d\mathbf{a}$.
2. Stokes’ Theorem: $\iint_S (\nabla \times \mathbf{F}) \cdot d\mathbf{a} = \oint_C \mathbf{F} \cdot d\mathbf{l}$.
3. Green’s Theorem: Special case of Stokes’ theorem in a plane.

Physics Applications

• Electromagnetism (Maxwell’s Equations).
• Fluid Dynamics.