--- license: mit --- I'll create a comprehensive Hugging Face Model Card for the Gradient Field Analyzer project. ```markdown --- language: - en tags: - mathematics - vector-calculus - gradient-fields - potential-functions - sympy - educational - scientific-computing license: mit library_name: pyqt5 pipeline_tag: text-generation --- # Gradient Field Analyzer ## Model Description The Gradient Field Analyzer is a mathematical tool designed to analyze and construct gradient fields in two dimensions. It implements analytical methods to determine whether a given vector field is a gradient field and, if so, constructs its potential function. This model specializes in two specific cases from vector calculus: - **Case 1**: Vector fields where one component is constant - **Case 2**: Vector fields where both components are linear functions - **Developed by:** Martin Rivera - **Model type:** Mathematical Analysis Tool - **Language(s):** Python - **License:** MIT - **Finetuned from model:** N/A (Original implementation) ## Uses ### Direct Use This model is intended for: - Educational purposes in vector calculus courses - Scientific computing applications requiring gradient field analysis - Research in mathematical physics and engineering - Verification of conservative vector fields - Construction of potential functions from gradient fields ### Downstream Use The analysis methods can be extended to: - Higher-dimensional gradient fields - Numerical methods for field analysis - Physics simulations involving conservative forces - Engineering applications in electromagnetism and fluid dynamics ### Out-of-Scope Use - Non-conservative vector fields (beyond verification) - Three-dimensional or higher vector fields - Numerical optimization without symbolic analysis - Real-time physical simulations ## How to Use ### Installation ```bash pip install sympy pyqt5 ``` ### Basic Usage ```python from gradient_field_analyzer import GradientFieldFactory, NumericalExamples # Analyze Case 1: Constant component field case1_analyzer = GradientFieldFactory.create_constant_component_field('x') Vx, Vy = case1_analyzer.get_vector_field() phi = case1_analyzer.find_potential() # Analyze Case 2: Linear component field case2_analyzer = GradientFieldFactory.create_linear_component_field() Vx, Vy = case2_analyzer.get_vector_field() phi = case2_analyzer.find_potential(Vx, Vy) ``` ### PyQt5 GUI Application ```python from gradient_field_gui import GradientFieldApp import sys from PyQt5.QtWidgets import QApplication app = QApplication(sys.argv) window = GradientFieldApp() window.show() sys.exit(app.exec_()) ``` ## Mathematical Background ### Gradient Fields A vector field **F** = [P(x,y), Q(x,y)] is a gradient field if there exists a scalar function φ(x,y) such that: **F** = ∇φ = [∂φ/∂x, ∂φ/∂y] ### Case 1: One Constant Component For fields where one component is constant: - If P(x,y) = c (constant), then φ(x,y) = c·x + ∫Q(x,y)dy - If Q(x,y) = c (constant), then φ(x,y) = c·y + ∫P(x,y)dx ### Case 2: Both Linear Components For linear fields **F** = [a₁x + b₁y + c₁, a₂x + b₂y + c₂]: - The field is a gradient field **if and only if** a₂ = b₁ - Potential function takes one of two forms: - φ(x,y) = a₁x²/2 + b₂y²/2 + c₂y + x(b₁y + c₁) - φ(x,y) = a₂x²/2 + b₁y²/2 + c₁y + x(a₁y + c₂) ## Model Architecture The implementation follows an object-oriented design: ``` GradientFieldAnalyzer (ABC) ├── ConstantComponentField (Case 1) └── LinearComponentField (Case 2) ``` Key components: - **Abstract base class** with common functionality - **Factory pattern** for creating analyzers - **Robust verification** of gradient conditions - **Symbolic computation** using SymPy - **GUI interface** using PyQt5 ## Training Data This model does not require traditional training data as it implements analytical mathematical methods. The "training" consists of: - Mathematical proofs of gradient field conditions - Verification against known analytical solutions - Testing with canonical examples from vector calculus ## Performance ### Analytical Accuracy - **Case 1**: 100% accuracy (direct integration method) - **Case 2**: 100% accuracy when gradient condition satisfied - **Verification**: Built-in gradient verification ensures correctness ### Computational Performance - Symbolic computation suitable for educational and analytical use - Real-time performance for typical vector fields - Handles complex symbolic expressions efficiently ## Limitations 1. **Dimensionality**: Limited to 2D vector fields 2. **Field Types**: Only handles specific cases (constant or linear components) 3. **Symbolic Limitations**: Dependent on SymPy's integration capabilities 4. **Numerical Precision**: Symbolic computation avoids numerical errors but may have expression complexity limits ## Environmental Impact - **Hardware Type**: Standard CPU - **Cloud Provider**: N/A - **Compute Region**: N/A - **Carbon Emitted**: Negligible (educational-scale computations) ## Technical Specifications ### Model Architecture - **Framework**: Python 3.7+ - **Dependencies**: SymPy, PyQt5 - **Symbolic Engine**: SymPy for mathematical operations - **GUI Framework**: PyQt5 for user interface ### Compute Requirements - **Memory**: < 100 MB for typical use cases - **Storage**: < 10 MB for code and dependencies - **CPU**: Any modern processor ## Citation If you use this model in your research or educational materials, please cite: ```bibtex @software{gradient_field_analyzer, title = {Gradient Field Analyzer: A Symbolic Tool for Vector Calculus}, author = {Martin Rivera}, year = {2025}, url = {https://huggingface.co/TroglodyteDerivations/gradient-field-analyzer}, note = {Educational tool for analyzing gradient fields and potential functions} } ``` ## Model Card Authors Martin Rivera ## Glossary - **Gradient Field**: A vector field that is the gradient of some scalar potential function - **Potential Function**: A scalar function whose gradient equals the given vector field - **Conservative Field**: Synonym for gradient field (in physics contexts) - **Symbolic Computation**: Mathematical computation using exact symbolic expressions rather than numerical approximations ## More Information For more detailed mathematical background, usage examples, and implementation details, please refer to the [documentation](https://github.com/TroglodyteDerivations/gradient-field-analyzer/docs). ## Additional Files for Hugging Face You should also create these additional files for your Hugging Face repository: ### requirements.txt ```txt sympy>=1.8 pyqt5>=5.15 ``` ### README.md (simplified version) ```markdown # Gradient Field Analyzer A Python tool for analyzing and constructing gradient fields in two dimensions. ## Features - **Case 1 Analysis**: Vector fields with one constant component - **Case 2 Analysis**: Linear vector fields with gradient condition checking - **Potential Function Construction**: Symbolic computation of scalar potentials - **PyQt5 GUI**: User-friendly interface for interactive analysis - **Educational Focus**: Designed for vector calculus education ## Quick Start ```python from gradient_field_analyzer import GradientFieldFactory # Analyze a linear vector field analyzer = GradientFieldFactory.create_linear_component_field() Vx, Vy = 2*x + 3*y + 1, 3*x + 4*y + 2 # Example field phi = analyzer.find_potential_for_specific_field(Vx, Vy) print(f"Potential: {phi}") ``` ## Installation ```bash pip install sympy pyqt5 ``` ## Documentation See the [model card](MODEL_CARD.md) for detailed mathematical background and usage examples. ``` This model card provides comprehensive documentation for your Gradient Field Analyzer, making it suitable for sharing on Hugging Face Hub as an educational and scientific tool.