Vector Calculus Operators
Source lecture(s): PC604 Lec1-2
Physical Meaning
Gradient, divergence, and curl are the three fundamental differential operators for scalar and vector fields.
Variable Definitions
| Symbol | Meaning |
|---|---|
| \(\nabla\) | del operator |
| \(\phi\) | scalar field |
| \(\vec{A}\) | vector field |
Assumptions
Fields are sufficiently smooth (continuous second derivatives).
Derivation
Apply Cartesian definitions; in curvilinear coordinates insert scale factors \(h_i\).
Applications
Electrostatics (\(\nabla\phi=-\vec{E}\)), magnetostatics (\(\nabla\cdot\vec{B}=0\)), fluid flow.
Connections to Other Equations
- divergence-theorem
- stokes-theorem