What numerical calculations can I do only with differentical geometry without needing a metric

In metric-free differential geometry, numerical calculations often focus on the connectivity of a space rather than its physical dimensions. A primary example is the application of the **exterior derivative** ($d$) within the framework of **Discrete Exterior Calculus** (DEC). This calculation determines how a field changes across the elements of a mesh using only the topology of the manifold. ## 1. Defining the Simplicial Complex To perform a numerical calculation, we first represent the manifold as a **simplicial complex**, which is a collection of points (0-simplices), edges (1-simplices), and faces (2-simplices). Consider a simple directed graph consisting of two nodes and one edge: - **Node A** and **Node B** (0-simplices). - **Edge E**, which is oriented from Node A to Node B (a 1-simplex). In this pre-metric setting, we do not define the coordinates of A or B, nor the length of E. We only define their **incidence** (how they are connected). ## 2. Assigning Differential Forms We assign numerical values to these simplices, known as **discrete differential forms**. - A **0-form** ($f$) is a function that assigns a value to each node. Let $f(A) = 10$ and $f(B) = 25$. - A **1-form** ($\omega$) is a quantity that lives on edges, representing a "flux" or "circulation." The goal is to calculate the exterior derivative of the 0-form, denoted as $df$, which results in a 1-form. ## 3. The Calculation of the Exterior Derivative The discrete exterior derivative is calculated using the **generalized Stokes' Theorem**. For a 0-form, the value of $df$ on an edge is simply the difference between the values at its boundary points. The calculation for Edge E is: **Value on Edge E = $f(B) - f(A)$** Substituting our values: **$df(E) = 25 - 10 = 15$** This result, **15**, is a precise numerical value. It represents the total increase of the field $f$ along the path from A to B. ## 4. Why This Calculation is Metric-Free This calculation is considered metric-free for several reasons: - **No Distance**: We did not use a metric tensor to calculate the length of Edge E. Whether the nodes are 1 millimeter or 1 kilometer apart, the value of the 1-form $df$ on that edge remains exactly 15. - **No Coordinates**: We did not define a coordinate system ($x, y, z$). The calculation relies entirely on the **orientation** (A to B) and the **connectivity** provided by the mesh. - **Topological Invariance**: If we "stretch" the manifold without breaking the connections, the result of this specific calculation does not change. ## 5. Verification of the Coboundary Property A critical numerical check in differential geometry is ensuring that **$d^2 = 0$**. If we had a triangle (a 2-simplex) and calculated the exterior derivative of our 1-form over the three edges forming its boundary, the sum must equal zero. Numerically, if we sum the changes around a closed loop: **$(f(B) - f(A)) + (f(C) - f(B)) + (f(A) - f(C)) = 0$** This identity is satisfied exactly in discrete calculations, providing a robust foundation for simulations in electromagnetism and fluid dynamics without requiring any measurement of angles or volumes.