Solve the problem 6 Hint- Prob Q-[0.1] x [O, 1], A-{(z, yje Q : y z)...
Problem 1. Consider the function f(x)- 3.12 show that f is Riemann integrable on [0.2] and use the definition to find .后f(x)dr Problem 2. Consider the function -2, zEQ 2, O f(r) = Show that f is not Riemann integrable on 0,1 but s Reemann integrable on this interval. Problem 3. (a) Let f be a real-valued function on a, b] such thatf()0 for all c, where c E [a, b Prove that f is Riemann integrable on a, b...
Please solve all questions 1. Let 0 : Z/9Z+Z/12Z be the map 6(x + 9Z) = 4.+ 12Z (a) Prove that o is a ring homomorphism. Note: You must first show that o is well-defined (b) Is o injective? explain (c) Is o surjective? explain 2. In Z, let I = (3) and J = (18). Show that the group I/J is isomorphic to the group Z6 but that the ring I/J is not ring-isomorphic to the ring Z6. 3....
q2 please (1) Evaluate the integral (r-1) min(a, y) dy dr, Jo Jo where min(x, y) is the minimum value of r and y. (2) Let f,g : R → R be functions of one variable such that f" and g" are continuous. Show that (f"(x)-g"(y)) dydx = f(0) + g(0)-f(2)-9(2) + 2f'(2) + 2g'(0). o Jo (3) Let a > 0. In spherical coordinates, a surface is defined by r = 2acos φ for 0 φ 1. Find the...
T'he goal of this problem is to establish the following remarkable result: Bezout's theorern. If a, be Z50, then 3x, y є Z such that gcd(a, b) = ax + by. Here ged(a, b) denotes the greatest common divisor of a and b (i.e. the largest positive integer that divides both a and b). Throughout this problem, we'll use the notation (a) Write down five numbers that live in 2Z +3Z. What's a simpler name for the set 2Z +3Z?...
e 09, 201 (6) 2 points An equation for the level curve of f(z, y) = In(z+y) that passes through the point (0, e2) is A. z + y = e2 B. I+y e C. z+y 3. D. None of the above (7) 2 points The gradient of f(z,y, z) = ep at the point (-1,-1,2) is A. (2e2,e2,2e2). B. (-e,-e,2e2). C. (-2e2,-2e2, e) D. (-2e2,-e,-e) (8) 2 points Let f be a function defined and continuous, with continuous first...
us equation, L (y(x))-0. Prove that o a solution eneous equation, C(y(z))g(z). Is a hy or why not? 1. Let C be the linear operator defined as follows. (a) Let v,.. ,n be the solutions of the homogeneous equation, D an arbitrary linear combination, ciyi+..nn is also a solution. , c(y(z)) 0, Prove that (b) Let vi,. n be the solutions of the non-homogeneous equation, Cl) ga). Is a linear combination, ciy nyn also a solution? Why or why not?...
1. Let Q be the set of polynomials with rational coefficients. You may assume that this is an abelian group under addition. Consider the function Ql] Q[x] given by p(px)) = p'(x), where we are taking the derivative. Show that is a group homomorphism. Determine the kernel of 2. Let G and H be groups. Show that (G x H)/G is isomorphic to H. Hint: consider defining a surjective homomorphism p : Gx HH with kernel G. Then apply the...
Let U = q r s tu, y, W, X, y, A A={a, s, u w. } B= 4 S. Y. A C= {v. W, X, Y. } List the elements in the set A O A. f. t. V. X, } O B. S. L. W. } 0 C. 4 5 y z} D. q, I.S. t U. V W x y z} Click to select your answer AP1 Brain-nerves....docx Ch 20-22 hervet.de
(x2 + y2 + z?)1/2, and e, = r1(x, y, z) is the unit radial vector. Let F = r"e, where n is any number, r= (a) Calculate div(F). (2+n)"-1 (b) Calculate the flux of F through the surface of a sphere of radius R centered at the origin. 4TR"+2 F. ds, where C is a closed curve that does not pass through the origin? (c) What is the value of (d) A function o satisfying Ap = 0 is...
Topology C O, 1 and be the supremum norm (a) Prove that (X || |) is a Banach space. You can assume that (X, | |) is a normed vector space (over R) |f|0supE0.1 \5(x)|.| 4. Let X C (b) Show that || |o0 that the parallelogram identity fails.] on X is not induced by any inner product. Hint: Check for all E[0, 1]. Show that {gn}n>1 (0, 1] BI= {gE X |9||<1} is a compact (c) For every 2...