6.Vector fields V and W are defined byV %3D (2х — Зу + z, -3х — у + 4z, 4y + z)W%3D (2x — 4y — 5z, -4х + 2у, -5х + 62)Determine which vector field is conservative and denote it by F.

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Answered: 6. Vector fields V and W are defined by…

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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Match each of the following planar vector fields with the corresponding plot. Note that the vectors are scaled to avoid overlap. F (x, y) = (√x² + y² (-3y 3 -3y F(x, y) = (√²+²√²+²) 3x 2 1 0 5 2 -3 3 2 - 0 -1 4 با 4 7 3 // -3 -2 Answer Bank -2 3y √x²+x² -1 0 0 F(x, y) = (- F(x, y) = (-3x 1 1 3x √x²+x² 2 2 /x²+x² 3 3 9 3x 22² +2² 3y /x² +1² 4 3 2 1 0 -2 -3 -4 4 3 2 N -3 -3 3 1906-28 24 ME BEIN 2143 MULTFIGHT 1220121 -1 0 ....... 3 2 3 4
Suppose = [2, 1, −1]ª, ♂ = [1, −1, 1]ª and √ = [2,0, 1]T. Let W = Span{7,V}. (a) Find the best approximation to y by a vector of W. (b) Write ✅ y as the sum of a vector in W and a vector in W¹. (c) Find the distance between and W. (d) Let A = [√, √]. Note that Aỡ ‡ ☞ for all ☞ € R². Find min ||A7 J'||. TER²
11) For u = (2,3,-1) and v = (-1,01) find; La) proj,(u) b) The vector component of u orthogonal to v.
Calculate the curl of the vector = (2x + 3y)i + (y+z)j + (xy + 2zx)k
7
2. Let u = (1, 0, 1) and v = (0, 2, 2). Find a vector w that is orthogonal to both u and v. Find a vector x which is orthogonal to u and lies in the same plane defined by u and v.
Suppose that ū, ī and w are vectors in a real inner product space such that (7, w) = 16, (T, w) = 5, d(2ū, v) = v27 and ||W|| = V27. Show that 27 - 0 - w = 0.
Compute the curl of each of the following vectors: V = (yz, xz, xy) F = (x, y, z) = = ( Note: Y 2 (x² + y² +2²)³/²¹ (x² + y² + x2)³/²¹ (x² + y² + x2)³/2 • If the answer is a scalar, you can just type it in the box (using the Calcpad if you like, or using / for fractions, ^ for exponents, etc.) For multiplication, you can either leave a space, or use *. So x*x=xx = x². Note that this is not the same as xx without a space; that will get read as an entirely different variable! • If you need to enter a vector, enter an ordered list of components, so for A you can enter either A = (A₂, Ay, A₂) = {Ax, Ay, Az} . Note that the system isn't great with multiplying through by overall factors, so it's better not to write e.g. • If you need to enter O, enter it in three components, as (0, 0, 0) or {0,0,0}. V x V = ▼ xr= vx= (2A, 2A, 2A₂).

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6. Vector fields V and W are defined by V = (2x – 3y + z, –3x – y + 4z, 4y + z) W = (2x – 4y – 5z, –4x + 2y, -5x + 6z) Determine which vector field is conservative and denote it by F.