Consider a rotation T(x) = Ajc in R3. (That is, A is an orthogonal 3 x 3 matrix with determinant 1.) Show that T has a nonzero fixed point [i.e., a vector v with T(v) = 5]. This result is known as Eulers theorem, after the great Swiss mathematician Leonhard Euler (1707-1783). (Hint: Consider the characteristic polynomial fA(k). Pay attention to the intercepts with both axes. Use Theorem 7.1.2.)

L16 - 10 ex. In a chemical reaction involving materials A and B resulting in product C, A and B are reactants and C is the product.T e concentration of reactant A in moles per liter is denoted [ A]. Since concentration varies during reaction, [A], [B], and [eucisfi . Average rate of reaction on 1t 2i ]s ∆[C] [C](t2) − [C](1 ) ∆t = t2− t1 ∆[C] d[C] Instantaneous rate of reaction = lim = ∆t→0 ∆t dt ex. If Q(t)sanetclhre,encrntspoued as the charge moves. On [1 ,2 ] Average current= ∆Q = Q 2 Q 1 ∆t t2− t 1 Current I at time t = t 1s the rate at which charge ﬂows through a surface, and is given by ∆Q dQ I mil = . ∆t→0 ∆t d