A pipe-flow profile is modelled

PHYSICS · Subjective · Fermi University Physics · Mathematical Methods for Physics · Linear algebra and vector spaces · Linear algebra and vector spaces

Mathematical Methods for Physics, Ch Vector spaces, bases, linear maps and dual vectors

Problem

A pipe-flow profile is modelled, on the interval [0,1][0,1] where xx is the distance from one wall, measured along a diameter in units of the pipe diameter (so xx is dimensionless), by f(x)=x(1−x)f(x)=x(1-x). A uniform reference profile is g(x)=1g(x)=1. Using the inner product ⟨f,g⟩=∫01f(x)g(x) dx\langle f,g\rangle=\int_0^1 f(x)g(x)\,\mathrm{d}x, compute ⟨f,g⟩\langle f,g\rangle and state whether ff and gg are orthogonal on [0,1][0,1].

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