Partial Derivative Approximations

Numerical Analysis · Axiom Academy

LESSON Partial Derivative Approximations Extending finite differences to multiple dimensions For a function f(x, y), the partial derivatives are approximated just like 1D derivatives: The key: move in only one direction (x or y) at a time. For functions on a grid, we use stencil notation . The central point is (i,j), and neighbors are accessed by offsets: Second partials use the same 1D formulas in each direction: Mixed partials like ∂²f/∂x∂y require differencing in both directions: 5 The Laplacian: A Key Operator The Laplacian ∇²f = ∂²f/∂x² + ∂²f/∂y² appears throughout physics. Its 5-point stencil is:

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