3. Consider a network (G, w). Let c ER, and let m: E(G) → R such that m(e):w(e) + c for all e Є E(G).=(b) What is the analogous claim for shortest paths in (G, w) and (G,m)? Provethis claim or provide a counterexample showing that it is not true.

(b)The analogous claim for shortest paths in (G, w) and (G, m) is true. That is, if P is the…

Answered: Consider a network (G, w). Let c ER,…

Advanced Engineering Mathematics

10th Edition

ISBN:9780470458365

Author:Erwin Kreyszig

Publisher:Erwin Kreyszig

Chapter2: Second-order Linear Odes

Section: Chapter Questions

Problem 1RQ

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3. Express the following regions in the stated coordinate systems. You do not need to justify your answer but showing your though process may help with partial marks. Your answer should be a set in the following form: {(V1, V2, V3) : a ≤ v₁ ≤ b, f(v1) ≤ v2 ≤ g(v1), Ø(V1, V2) ≤ v3 ≤ (v₁, v₂)} where (v₁, V2, V3) are the three variables of the coordinate system in questions in any order, a, b R are constants, f, g are functions of v₁ and 6, are functions of v₁, V2. (a) The region outside the cylinder x² + y² ≤ r² and inside the sphere x² + y² + z² r< R. Express this in the cylindrical and spherical coordinate systems. - R² assuming (b) The region in the first octant (x, y, z ≥ 0) below the plane x + 2y + 3z = 6. Express this in the cylindrical, spherical and Cartesian coordinate system.
Call a graph H regular if there exists k ≥ 1 such that dµ(v) = k for all v € V(H). (e) Does there exist a regular graph H with V(H) = V(G) and E(H) ≤ E(G)? Justify your answer.
Find a maximum matching of G. Show your working. Show that the matching you have found is indeed a maximum matching.
3. Consider the line [x,y,z] = [3,-2,3]+1[-1,2,-3]. Find the intersection points between this line and the co-ordinate axes. is.
5. Determine whether the following pair of lines is parallel, intersects at a single point, or is skew. If the lines are parallel, determine whether they are the same line (and thus intersect at all points). If the lines intersect at a single point, determine the point of intersection: r(t) = (0, 1, 2) + t(1, 1, 1), R(s) = (-2, 1,0) + s(-1, -3, -1).
a.Find the maximum value of w = xyz on the line of intersection of the two planes x + y + z = 40 and x + y - z = 0. b. Give a geometric argument to support your claim that you have found a maximum, and not a minimum, value of w.
5. Find a and b so that the lines r¡(t) = (1+t, –1+ at, 1+3t) and r(t) = (3 – 3t, 2 – 4t, 1+ bt) : parallel. are

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Consider a network (G, w). Let c ER, and let m: E(G) → R such that m(e) = w(e)+c for all e € E(G). (b) What is the analogous claim for shortest paths in (G, w) and (G,m)? Prove this claim or provide a counterexample showing that it is not true.