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5.) For the integral S(8 - x) dx (a) Show to construct Rn (right hand Riemann sum with n sub intervals) (b) Simplify Rn using your sigma skillz (c) Take limit of R, as n → to evaluate given integral (d) Compute given integral by FTC to check answer
Prove the following integral. ., xPn-1(x)P,(x)dx = 2n (2n-1)(2+1) 2 use (n + 1)Pn+1(x) – (2n +1)xPn(x) + nPn-1(x) = 0, L, Pn(x)Pm (x) dx = 0, S, Pn(x)2 dx = 2n+1
x= Jy=X Show that | * (x+3y) dy dx = [ { « + 3) dx dy + m2 L); *+ 3ydr dy (x + 3y) dx dy + (x + 3y) dx dy Jy=1 Jx=y2
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Rework Example 10.3.1 by replacing фп(x) by the conventional Legendre polynomial. Pn (x): [Pn (x)' dx = 2n + 1 Using Eqs. (10.47a), and (10.49a), construct Po, P1(x), and P2(x).
3) Relativistic Particles Consider a gas of N relativistic particles with Hamiltonian rn n= where c is the speed of light, m is the rest mass, and pn is the momentum, with pn (Pnz, Pny, pns) and p; = pn . pn-Ping +pny +Pie. Show that p2e2 = 3kgT.
PART C ONLY! Thank you.
14. Fix a non-zero vector n R". Lot L : Rn → Rn be the linear mapping defined by L()-2 proj(T), fa TER or all (a) Show that if R", Such that oandj-n -0, then is an eigenvector of L What is its cigenvaluc? (b) Show that is an cigenvector of L. What is its cigenvalue? (c) What are the algebraic and geometric multiplicities of all cigenvalues of L?
14. Fix a non-zero vector n...
Hint: Apply the rank-nullity theorem to the linear map Pn → Rn+1
that sends p ?→
(p(x0), . . . , p(xn)). Then use the fact that if polynomial of
degree ≤ n has n + 1 distinct roots, then it is the zero
polynomial.
(3 points) Application: polynomial interpolation. Let (20; yo), ..., (In; Yn) be n +1 points R2 with distinct x-coordinates. Show that there exists a unique polynomial p(t) of degree <n such that p(xi) = yi...
4. Prove the following statement: Consider the ODE x = f(x) with x : J C R → Rn and f : Rn → Rn. If a continuously differentiable real-valued function V = V(x) exists such that (a) V is defined on Bs(0) {x E Rn : Irl < δ} (b) V(x) 0 for x E Bs(0) 1 fo) (c) V 0) 1 (o then the origin is unstable. (x) >0 for rE Bs
4. Prove the following statement: Consider...
1.5. Show that in the (r, )-space Rn+l the planes (1.3) are characteristic with respect to the wave equation (1.1). Also show that the plane wave solutions (1.4) are constant on these planes. The wave equation a2u We were unable to transcribe this image
1.5. Show that in the (r, )-space Rn+l the planes (1.3) are characteristic with respect to the wave equation (1.1). Also show that the plane wave solutions (1.4) are constant on these planes.
The wave equation...
(c) Show that the value of V1 + cos x dx is less than 1.6.