Let f : R [0, ) be an integrable function with compact support. View f...

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Let f : R³ → [0, ∞) be an integrable function with compact support. View f as defining the mass density of a rigid body B, which is the support of f, as follows: for any compact set K CR³, the mass of Kn B is Let r = µ(K) = []], f(x, y, z) dxdydz. K (x, y, z) denote the position vector in R³. Define the inertia tensor I : R³ → R³ by I(u) = SSSB rx (ur)f(x, y, z) dxdydz. Prove that I has the following properties: (1) I is a linear transformation. (2) For all u R³, u. · I(u) ≥ 0. (3) For all u, v Є R³, u · I(v) = v · I(u). Let f : R³ → [0, ∞) be an integrable function with compact support. View f as defining the mass density of a rigid body B, which is the support of f, as follows: for any compact set K CR³, the mass of Kn B is Let r = µ(K) = []], f(x, y, z) dxdydz. K (x, y, z) denote the position vector in R³. Define the inertia tensor I : R³ → R³ by I(u) = SSSB rx (ur)f(x, y, z) dxdydz. Prove that I has the following properties: (1) I is a linear transformation. (2) For all u R³, u. · I(u) ≥ 0. (3) For all u, v Є R³, u · I(v) = v · I(u).