The goal of this repository is to build geometric intuition for vector calculus, Fourier analysis, and PDEs through mathematics, visualization, and coding.

This repository is intentionally built step-by-step.

The focus is not only symbolic derivation, but understanding:

• what the operators mean geometrically
• how fields behave locally and globally
• how Fourier space simplifies differential operators
• how PDEs describe evolving geometric structures

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Stage 1 — Vector Fields and Local Geometry

Core intuition:

• What is a vector field?
• How do fields flow through space?
• What does local structure look like?

Topics:

• Scalar fields vs vector fields
• Direction fields
• Flow lines and trajectories
• Local vs global behavior
• Orientation in space

Coding goals:

• Visualize 2D/3D vector fields
• Plot trajectories and streamlines
• Build geometric intuition interactively

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Stage 2 — Gradient, Divergence, and Curl

Core intuition:

• Different ways fields can change locally

Topics:

• Gradient as steepest ascent
• Divergence as expansion/compression
• Curl as local rotational tendency
• Cross product geometry
• Rotation axes and orientation
• Right-hand rule intuition

Coding goals:

• Numerical derivatives
• Divergence/curl visualization
• Rotational and compressive flow simulation

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Stage 3 — Fourier Transform and Frequency Space

Core intuition:

• Decomposing fields into wave modes

Topics:

• Plane waves
• Spatial frequency
• Wavevector (k)
• Fourier basis functions
• Frequency-space interpretation
• Differential operators in Fourier space

Key transition: $$[\nabla \rightarrow ik]$$

Coding goals:

• FFT implementation
• Frequency visualization
• Spatial vs spectral representations

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Stage 4 — Helmholtz Decomposition

Core intuition:

• Separating vector fields into geometric components

Topics:

• Longitudinal vs transverse structure
• Divergence-free fields
• Curl-free fields
• Projection geometry in Fourier space
• Spectral decomposition of vector fields

Coding goals:

• Helmholtz decomposition
• Projection operators
• Longitudinal/transverse visualization

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Stage 5 — PDE Geometry

Core intuition:

• PDEs as evolving geometric fields

Topics:

• Diffusion equation
• Wave equation
• Transport equations
• Laplacian operator
• Flow and propagation
• Frequency evolution in PDEs

Geometric viewpoint:

• Diffusion smooths high frequencies
• Waves propagate structured oscillations
• PDEs evolve geometric field structure over time

Coding goals:

• PDE simulation
• Spectral PDE solvers
• Time evolution visualization
• Diffusion and wave propagation demos

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Learning Philosophy

Each concept should be explored through:

1. Geometric intuition
2. Mathematical derivation
3. Visualization
4. Numerical implementation
5. Fourier/spectral interpretation

The goal is to connect:

• geometry
• calculus
• Fourier analysis
• PDEs
• physics

into a unified understanding of fields and dynamics.