1. Gradient of a Scalar Field
The gradient of a scalar field points in the direction of the greatest rate of increase of the function. It is a vector that contains all the partial derivatives.
If φ(x, y, z) is a scalar field, then:
∇φ = (∂φ/∂x) î + (∂φ/∂y) ĵ + (∂φ/∂z) k̂
2. Divergence of a Vector Field
Divergence measures the rate at which "stuff" expands or contracts from a point. It is a scalar quantity.
For a vector field F = P î + Q ĵ + R k̂:
∇·F = ∂P/∂x + ∂Q/∂y + ∂R/∂z
3. Curl of a Vector Field
Curl measures the rotation or swirling strength of a vector field. It is a vector.
For a vector field F = P î + Q ĵ + R k̂:
∇×F = |î ĵ k̂|
|∂/∂x ∂/∂y ∂/∂z|
|P Q R|
4. Physical Interpretation
- Gradient → Direction and rate of steepest increase (e.g., temperature, pressure fields)
- Divergence → Net flow out of a point (e.g., fluid source/sink)
- Curl → Rotational behavior of a field (e.g., whirlpools, vortices)