*Q6: [Exercise 33] For each of the following vector fields, find the divergence:(i) v=xi+yj+zk,(ii) v = x² i + y² j + z² k,(iii) v=yzi+xzj+xy k,(iv) v = sin(x) i + cos(y) j - z(cos(x) = sin(y)) k.Which of these vector fields (if any) are solenoidal?

Answered: Image /qna-images/answer/c48fa7dd-c8d8-411e-aaaa-caddb4bf060c.jpg

Answered: *Q6: [Exercise 33] For each of the…

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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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.
Q.3) Graph the vector field F(x,y) 3j + zk and G(x, Y, z) = xi – yj+ zk. = -yi. Find V.(F × G) where F(x, y, z) = xi – -
Match the vector field with its graph. F(x, y) = yi 臺 -6+ + HH H + 三
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Match each of the following three vector fields to one of the four vector fields graphed below (yes, one graph does not have a match), and then explain your thinking: 1. (a) F(x, y) = (2y, 2.r). Match (circle one): I II III IV (b) F(x, y) = (x², 2y). Match (circle one): I II III IV (c) F(x, y) = (x², y²). Match (circle one): I II III IV (d) Explain your choices. Explanation:
Which of the following vector fields are conservative? (i) F(x, y) = (10x²y⁹+6)i + (9x³y² + 8) j (ii) F(x, y) = (9ye³x + cos 6y)i + (e⁹x + 6x sin 6y) j (iii) F(x, y) = 10y²e10xy i+ (10+xy) e10xy j
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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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*Q6: [Exercise 33] For each of the following vector fields, find the divergence: (i) v = xi+yj + zk, (ii) v = x² i + y² j + z² k, (iii) v=yzi+xzj+xy k, (iv) v = sin(x) i + cos(y) j - z(cos(x) — sin(y)) k. Which of these vector fields (if any) are solenoidal?