Interactive Vector Field Simulation

This project is an interactive computational tool designed to help users visualize, manipulate, and analyze two-dimensional vector fields. At its core, it combines mathematical concepts from vector calculus with intuitive graphical interaction, making abstract field properties accessible and visually engaging.

A vector field in two dimensions assigns a vector (with both magnitude and direction) to every point in a plane. Mathematically, such a field is often written as:

where U(x,y) and V(x,y) are the field's components in the x and y directions, respectively.

Initial Field Construction

The project begins by letting the user define a uniform vector field. This means every point in the grid has the same vector, determined by user-supplied constants a and b. A key feature is the ability to draw arrows directly onto the field. Each arrow is mathematically interpreted as a localized perturbation—an influence that modifies the field in its vicinity. When a user draws an arrow from (x0,y0) to (x1,y1), it represents a vector influence with direction and strength proportional to the arrow's length and orientation.

The effect of each arrow is not limited to a single point; instead, it decays with distance from its origin, affecting the entire field but most strongly near where it was drawn. This is modeled as:

The resultant field is the sum of the initial uniform field and the cumulative effects of all user-drawn arrows:

Field Analysis: Gradient, Divergence, and Curl

The tool offers further insight by enabling overlays of key vector calculus properties:

• Gradient Magnitude: Measures how rapidly the field changes in space. It is computed as the root of the sum of squares of all spatial derivatives of the field components.
• Divergence: Indicates the presence of sources (where field lines spread out) or sinks (where they converge)
• Curl: Measures the tendency of the field to induce rotation around a point:

These properties are computed numerically using finite differences and visualized as color overlays, making abstract mathematical concepts tangible.

Example:

Installation

1. Clone the repository:

   git clone https://github.com/yourusername/vector-field-visualizer.git
   cd vector-field-visualizer

2. Install dependencies:

   pip install numpy matplotlib scipy

Usage

1. Run the script:

   python vector_field_visualizer.py

2. Set Initial Field:

   • Enter the x and y components (a, b) for the initial field when prompted.

3. Draw Arrows:

   • Left panel: Click and drag to draw arrows (red) representing additional vector influences.

4. Calculate Resultant:

   • Click the "Calculate Resultant" button to update the field with your drawn arrows.

5. Analyze the Field:

   • Use the right panel buttons to:
     • Display only field: Show the resultant field.
     • Overlay gradient: Visualize the field's gradient magnitude.
     • Overlay divergence: Visualize the field's divergence.
     • Overlay curl: Visualize the field's curl.

Controls & Buttons

Button	Function
Calculate Resultant	Updates the field with user-drawn arrows
Display only field	Shows only the resultant vector field
Overlay gradient	Adds a gradient magnitude heatmap
Overlay divergence	Adds a divergence heatmap
Overlay curl	Adds a curl heatmap

Requirements

• Python 3.7+
• numpy
• matplotlib
• scipy