Show that a Dirichlet problem (see Chapter 13, Section 3 ) for Laplace’s equation in a finite region has a unique solution; that is, two solutions
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- Prove the half of Theorem 3.3 (e) that was not proved in the text.arrow_forwardp(x, ¤1, ..., an, B1, . . , Bn), which constitute a general solution of the canonical system (82). 94 CANONICAL FORM OF THE EULER EQUATIONS CHAP. 4 PROBLEMS 1. Use the canonical Euler equations to find the extremals of the functional SVZI + y? VT + ya dx, and verify that they agree with those found in Chap. 1, Prob. 22. Hint. The Hamiltonian is H(x, y, p) = - V + yª – p², and the corresponding canonical system dy %3D has the first integral p - ya = C?, where C is a constant. 2. Consider the action functional = (mx² – xx*) dt corresponding to a simple harmonic oscillator, i.e., a particle of mass m acted upon by a restoring force -xx (cf. Sec. 36.2). system of Euler equations corresponding to J[x], and interpret them. Calcu- late the Poisson brackets [x, p), [x, H] and (p, H). Is p a first integral of the canonical Euler equations? Write canonical 3. Use the principle of least action to give a variational formulation of the problem of the plane motion of a particle of mass m attracted…arrow_forward1.5.17 Suppose the solution set of a certain system of linear equations can be described as x, = 5+. 4x3, X2 = - 6 – 7x3, with x3 free. Use vectors to describe this set as a line in R°. %D Geometrically, the solution set is a line through || parallel toarrow_forward
- The goal of this problem is to fit a trigonometric function of the form f(t) = co + c1 sin(t) to the data points (0, –9.5), (5,–11.5), (7, –9.5), (*,-1.5), using least squares. (a) The problem is equivalent to finding the least squares solution to the system Xc= y where X = У — and c = (c1, c2]" (b) Find the coefficients of the best fit by finding the least squares solution to the system in part (a) Co C1arrow_forwardPart 3&4 needed to be solved correctlyarrow_forwardDerive the equation x² = 4py in Figure 10.4.6.arrow_forward
- Consider a damped undriven oscillator whose equation of motion is given by * = -w²x-yi Assume that the system is critically damped, such that r = solution is given by r(t) = (A + Bt)ert/2 2wo. When this is the case, a general where A and B are constants. A) Express the general solution as a complex function. [Hint: This is a trick question.] B) Assume the oscillator is initially at rest at the origin. At t = 0, it is given an impulse T (i.e., a sudden blow), causing it to launch with initial positive velocity vo. Determine specific expression for both x(t) and v(t) in terms of known values. C) Determine both the time when the velocity is zero and the time when the velocity starts to increase (i.e., when the acceleration changes sign). D) Extending from parts b and c, sketch both x(t) and v(t), making sure to clearly indicate the relevant values (e.g., the maximum displacement, etc...). [Hint: The velocity is zero when the object is at the maximum displacement.arrow_forward-3 6. 8. Let ū = 2 i = and uw = Is it possible to find a scalar t such 1 -3 that ū + tữ is parallel to w.arrow_forward2 {:40 %VV II. 92bb6954-6e3e-4940-a... LoJ a. Find XX, (XX)-¹, Xy, yy. b. Find B, cov (B), Var (Bo), Var (B₁), cov (Bo, B₁). c. Find SSE. 1 3] 3. For the model y = XB + E,, B = Yij = μ + α₁ + ßj + εij твот B₁ B₂ LB3- 4. For the additive (no-interaction) model Ho: 200-3B1 + B₂ = 5B₂-B0-2B3 = B3 + 2B₁ = 0. Find matrix C to express Ho in the form Ho: CB = 0. a. Find designed matrix X and XX. b. Determin rank of X and XX. X we test i = 1,2,3,4 and j = 1,2 If we wright the 8 observations in the matrix form μ α₁ α₂ y = XB + ε when ßa3 α4 [ε4] >arrow_forward
- Linear Algebra: A Modern IntroductionAlgebraISBN:9781285463247Author:David PoolePublisher:Cengage Learning