Vector Geometry & Fields
Explore coordinate vector distributions and directional field derivatives. Drag attractor pegs to distort fluid flows, and scan local Divergence or Curl.
Vector Arrows
Fluid Vector currents
A vector field maps every coordinate $(x,y)$ to a vector $\mathbf{F}(x,y) = P\mathbf{i} + Q\mathbf{j}$. Choose a field preset, adjust force factors, or drag the attractor pegs directly on the grid to create vortex currents.
Hover mouse over canvas to probe local coordinates. In Dipole mode, drag the green (+) and red (-) charges.
Active Field Equation
F(x,y) = (-y)i + (x)j
Sensor Probe Readout:
Hover over canvas to probe local field metrics.
Divergence & Curl
Mathematical Flow Indices
Divergence and Curl measure the expansion rate and local rotation rate of vector fields, respectively.
Divergence (∇ · F)
Physical Definition:
Measures the net flow rate of fluid entering or leaving a specific point.
Formulas:
Div F = ∇ · F = ∂P/∂x + ∂Q/∂y
Divergence States:
- **Div > 0 (Source)**: Fluid expands outward.
- **Div < 0 (Sink)**: Fluid compresses inward.
- **Div = 0 (Solenoidal)**: Incompressible fluid flow.
- **Div < 0 (Sink)**: Fluid compresses inward.
- **Div = 0 (Solenoidal)**: Incompressible fluid flow.
Curl (∇ × F)
Physical Definition:
Measures the rotational torque strength (circulation) of fluid around a local point.
Formulas:
Curl F = ∇ × F = ∂Q/∂x - ∂P/∂y
Curl States:
- **Curl > 0**: Counter-clockwise local vortex rotation.
- **Curl < 0**: Clockwise vortex rotation.
- **Curl = 0 (Irrotational)**: Conservative field with path-independent work integrals.
- **Curl < 0**: Clockwise vortex rotation.
- **Curl = 0 (Irrotational)**: Conservative field with path-independent work integrals.
Self Evaluation
Vector Fields Quiz
Assess your mathematical grasp of divergence, curl, and field structures.
Question 1 of 5
Score: 0/0
What does the divergence (∇ · F) of a vector field measure at a given point?
Correct Answer!
Explanation goes here.
Focus Area Vocabulary
- Irrotational: A vector field whose curl is zero everywhere.
- Solenoidal: A vector field whose divergence is zero everywhere (incompressible).
- Dipole Field: Field configured by positive source charge and negative sink charge.