Laplacian of f:∇2f = div(grad f)=∂2f∂x21+ · · · +∂2f∂x2nTheorems for C2functionsand vector fields in R3:curl(grad f ) = 0div(curl F) = 0Parametrized Surfaces in R3: S = X(D), X : D ⊂ R2→ R3s-coordinate curve at t = t0: x(s) = X(s, t0); Tangent vector at X(s0, t0): Ts(s0, t0) =∂X∂s(s0, t0)t-coordinate curve at s = s0: x(t) = X(s0, t); Tangent vector at X(s0, t0): Tt(s0, t0) =∂X∂t(s0, t0)Standard normal vector at X(s0, t0): N(s0, t0) = Ts(s0, t0) × Tt(s0, t0)Smoothness condition at X(s0, t0): N(s0, t0) 6= 0Probability (one-variable): Event [a, b] ⊂ R: Prob(a ≤ X ≤ b) =Zbap(x) dxExpectation: E(f(X)) =ZRf(x)p(x) dxVariance: Var(X) = E((X − E(X))2)Probability (two-variable): Event D ⊂ R2: Prob((X, Y ) ∈ D) =ZZDp(x, y) dAExpectation: E(f(X, Y )) =ZZR2f(x, y)p(x, y) dAMarginal Dists: px(x) =ZRp(x, y) dy, py(y) =ZRp(x, y) dxIndependence Condition: p(x, y) = px(x)py(y)Circle (disk) of radius r: Area = πr2, Circumference = 2πrSphere (ball) of radius r: Volume =43πr2, Surface area = 4πr2Circular cylinder of radius r, height h: Volume = πr2h, Surface area∗= 2πrhCircular cone of radius r, height h: Volume =13πr2h, Surface area∗= πrpr2+ h2∗Note: the formulas above for the cylinder and cone are for “lateral” surface area; they donot include the circular