2. (a) Prove that K = (Z/3Z)[x]/(x3 + 2x + 1) is a field. (b) Find...
Find the flux of the field F(x,y,z)=z² i +xj - 3z k outward through the surface cut from the parabolic cylinder z=1-yby the planes x = 0, x=2, and z=0. The flux is (Simplify your answer.)
Problem 4. (15 points each) Let F(x, y, z) = (0, x, y) G(x, y, z) = (2x, z, y) + (x, y, z) = (3y, 2x, z). (a) For each field, either find a scalar potential function or prove that none exists. (b) For each field, either find a vector potential function or prove that none exists. (c) Let F(t) = (2, 2t, t2). For which of these vector fields is ñ a flow line? Justify your answer.
(1 point) Suppose f() (2-3z)(4 (a) The roots of f(x) are - (b) As z -oo, f(a) -» (b) As -oo, f(a) - - 5)2 help (numbers) (If a root is repeated, enter it only once.) help (numbers) help (numbers)
12: Find a basis B for R', such that the matrix for the linear transformation T: R' R', T(x,y,z)-(2x-2z,2y-2z,3x-3z) relative to B is diagonal.
12: Find a basis B for R', such that the matrix for the linear transformation T: R' R', T(x,y,z)-(2x-2z,2y-2z,3x-3z) relative to B is diagonal.
Use Gaussian elimination to find the complete solution to each system (X-3y + z = 1 -2x + y + 3z =-7 x-4y + 2z = 0 A ((2t + 4, t + 1, t)h 0-B. {(2t + 5, t+2,t)) OC. (1t + 3, t + 2, t) D. ((3t 3, t+ 1, t))
Short answer, explain your reasoning (a) Find the ged in R[x] of x3 – 2x – 1 – 2 and x2 – – 2. (b) How many elements in F41 are squares? Explain a systematic way to describe them all? (c) Does C[x] have an irreducible polynomial of degree 100? Explain. (d) Does R[2] have an irreducible polynomial of degree 100? Explain. (e) Does Q[x] have an irreducible polynomial of degree 100? Explain. (f) Does F19(2) have an irreducible polynomial...
D Question 11 12 pts to Consider the vector field F (x, y, z) =< 2x – yz, 2y – az,2z – xy>. a) (3) Is this vector field conservative? Justify your answer. b) (9) Find the amount of work done by this vector field in moving a particle along the curve (t) =< 3cost, cos’t, cos” (2t) > from t = 0 tot = 1
Solve 3x + 2y – z = 1x – 2y + z = 02x + y – 3z = -1
Line Integral & Path Independency Problem 1 Prove that the vector field = (2x-3yz)i +(2-3x-2) 1-6xyzk is the gradient of a scalar function f(x,y,z). Hint: find the curl of F, is it a zero vector? Integrate and find f(x,y,z), called a potential, like from potential energy? Show all your work, Then, use f(x,y,z) to compute the line integral, or work of the force F: Work of F= di from A:(-1,0, 2) to B:(3,-4,0) along any curve that goes from A...
SOLVE THE FOLLOWING SYSTEM OF EQUATIONS BY THE CRAMER'S METHOD 3X+5Y+3Z-12 2X+5Y-2Z-6 3x+6Y+3Z-3 a) X Y b) CHECK YOUR RESULTS. (USE MATRICE FUNCTIONS, PRESS F2. AND THEN PRESS CTRL+SHIFT+ENTER) 3IF Y-SINC) EXPOO. INTEGRATE Y FROM X-0 Tox-1. COMPARE WITH REAL VALUE IF DX-0 a) INT b) INT ,IF DX- 005 REAL VALUE 3) Plot sin x letting maco c/ Prepave hese cuves 4) SOLVE THE FOLLOWING SYSTEM OF EQUATIONS BY INVERSE METHOD 3 X+3Z-13 2X +5 Y-2Z-2 3 X+6Y+2Z-3 Z-...