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Calculus II: Vector Calculus and Multivariable Analysis / Multiple Integrals

Physical Applications: Mass, Centre of Mass, and Moments of Inertia

A portfolio's expected return is the weighted average of its assets' returns — precisely the centre of mass formula applied to a discrete distribution. When returns are continuously distributed, the expected return is an integral of the return against the density — a continuous centre of mass. Mass, first moments, and second moments (moments of inertia) are the three fundamental applications of multiple integrals to distributions of matter or probability. Understanding them in physical terms builds the intuition that carries directly into factor models, risk attribution, and portfolio optimisation.
Learning objectives. By the end of this note you should be able to:
  • compute the mass of a solid with a continuous density function ρ(x,y,z)\rho(x,y,z);
  • find the first moments MxyM_{xy}, MxzM_{xz}, MyzM_{yz} and centre of mass (xˉ,yˉ,zˉ)(\bar{x},\bar{y},\bar{z});
  • compute the moment of inertia IzI_z about an axis;
  • translate each formula into a probabilistic or financial analogue.

In this lesson

    1. Mass
    1. First Moments and Centre of Mass
    1. Moments of Inertia
    1. Summary Table

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