By Stiefel E.

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Noting that A = B + C, give a geometric interpretation of your result for the triple scalar product. (b) Compute A × (B × C). 5 The orbital angular momentum L of a particle is given by L = r × p = mr × v, where p is the linear momentum. With linear and angular velocity related by v = ω × r, show that L = mr 2 ω − rˆ (ˆr · ω) . Here rˆ is a unit vector in the r-direction. For r · ω = 0 this reduces to L = I ω, with the moment of inertia I given by mr 2 . 5 this result is generalized to form an inertia tensor.

Calculate the number of cubic centimeters of glass in the parallelepiped using the triple scalar product. 18 Parallelepiped: triple scalar product. 30 Chapter 1 Vector Analysis by direct expansion in Cartesian coordinates. 3 Show that the first step in Eq. 43), which is (A × B) · (A × B) = A2 B 2 − (A · B)2 , is consistent with the BAC–CAB rule for a triple vector product. 4 You are given the three vectors A, B, and C, A = xˆ + yˆ , B = yˆ + zˆ , C = xˆ − zˆ . Compute the triple scalar product, A · B × C.

B) If u = u(x, y) and v = v(x, y), show that the condition (∇u) × (∇v) = 0 leads to the two-dimensional Jacobian J u, v x, y = ∂u ∂x ∂v ∂x ∂u ∂y ∂v ∂y = 0. The functions u and v are assumed differentiable. 7 DIVERGENCE, ∇ Differentiating a vector function is a simple extension of differentiating scalar quantities. Suppose r(t) describes the position of a satellite at some time t. Then, for differentiation with respect to time, r(t + t) − r(t) dr(t) = lim = v, linear velocity. →0 dt t Graphically, we again have the slope of a curve, orbit, or trajectory, as shown in Fig.