1. Consider the vector fieldF(1, y, 2) = (x – 2, e – a*, z – sin(ry:).%3D(a)Sketch the vector field F(x, y, z) for (r, y, 2) in the set{(0, -2, 0), (0, –1, 0), (0,0,0), (0, 1,0), (0, 2,0)}.(b)Compute the divergence of F(r, y, z) at the point (0,-1, 3).(c)Compute the curl of F(r, y, z) at the point (0,-1, 3).

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Answered: 1. Consider the vector field F(r, y, 2)…

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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Verify Green 's theorem for the vector field F (x, y) = (x^2y, -xy^2) in the triangle OAB defined by the points O (0,0), A (1,0) and B (0.1).
Let A(t) = f(t)i + f₂(1) and B(t) = g₁(t)i + g₂(t)},t = [0, 1], where f₁f2, 81, 82 are continuous functions. If A(t) and B(t) are non-zero vectors for all and 7(0) = 2î +3ĵ, A(1) = 6î +2), B(0)=31 +23 and 6i- A B(1) = 27 +6, then show that A(1) and B(t) are parallel for some t.
1. (Section 17.1) Match the vector fields with their sketches below by placing the letter of the function in the corresponding blank: (I) F(x, y) = (II) F(x, y) = (III) F(x, y) = (IV) F(x, y) = Vector Field: (B) Vector Field: (A) (C) Vector Field (D) 0 Vector Field:
4) Let V be the vector space of polynomials of degree <2 and let p(x) = ax² + bx + c. Find the adjoint operator (T*@)(p) for T:V → V and o:V → R given by d) Tp(x) = x²p(x) + x³p'(x), @(p) = p"(1).
1. Consider two vector fields: F1(x, y) = -yi+ xj and F2(7) = F. (a) Evaluate F and F, at the given points. (x, y) (1,0) (0, 1) (-1,0) (0, –1) F(r, y) F(1, y) (т, у) (1,1) (-1,1) (-1,–1) (1, –1) F(1, y) F(x, y) On the grids shown below, sketch above vectors of vector fields F (b) and F. F;(x, y) = -yi + xj F2(F) = F.
19. Consider the vector field V(r, y, 2) = 1-y |x²+(y-1)² > x²+(y-1)² (y: H) and the curves C1 : x² + (y – 1)2 = 1, 62 : x² + (y – 2)² = 4 and 63 : x² + (y – 4)² = 1. (a) Compute V• dr.
If a vector field F (x, y, z) = (yz, x² z² , x³y³), which of the following is curl F (0, –1, 1)? (0, 1, 1) (1, 0, 0) (0, –1, 1) сurl F (0, —1, 1)
3. (12) Let V = R². Define vector addition and scalar multiplication as follows: (u1, uz)O(v1, v2) = (u1V1, U2V2) and c(u1,u2) = (u,C, u2^), c > 0 (a) Does V have an additive identity? If so, what would it be? (b) Does the distributive axiom c(uOv) = cu@cv hold? (Show why or why not). 4
What are V-F and VX F for the vector field F(x,y,z) = 3x²zi-ye*j - y²zk? OV.F = x²exy 2 VXF = x²i-3e²j+yk OF=3x²-xe* + 2yz VXF = 2i-3j-k V-F = 3x²-x-z VXF= -2yi+ 3xj + zk V-F = 6xz-e* - y² VXF = -2yzi + 3x²j-ye*k
Let A(t) = f(t)i + f₂(1) and B(t) = g₁(t)i + g₂(t)},t = [0, 1], where f₁f2, 81, 82 are continuous functions. If A(t) and B(t) are non-zero vectors for all and 7(0) = 2î +3ĵ, A(1) = 6î +2), B(0)=31 +23 and 6i- A B(1) = 27 +6, then show that A(1) and B(t) are parallel for some t.

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