Problem 6. (Divergence Theorem) Consider the region R= [0, 2] × [0, 2] × [0, 2] and its boundary surface R. Given the vector field compute the flux F = (P, Q, R) = (x² + yz, y² + tan x, z² — tan(ry)) - Vaxt F.dS= [[_Pdy ^dz + Qdz ^\ dx + Rdx / dy
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![Problem 6. (Divergence Theorem) Consider the region R= [0, 2] × [0, 2] × [0, 2] and its boundary
surface R. Given the vector field
compute the flux
F = (P, Q, R) = (x² + yz, y² + tan x, z² — tan(ry))
-
WORE
F.dS=
[Pdy ^dz + Qdz ^ dx + Rdz ^ dy](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F9d1a45f8-3ba2-4624-865f-f36d23213ebc%2F9cfba82a-05eb-4596-8caa-e1e9ea344169%2Fezyimed_processed.png&w=3840&q=75)
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- 4. Given the 2-D vector field : G(x,y) = (2y)î + (x)ĵ and the triangular curve T : line segments from point A(0, 0) to point B(4,0) to point D(4,2) and back to point A(0,0). 1 (а) Sketch the curve, and the vector field at the 9 points making a square from (-2,2) to (2, –2). -x- -3: -2 -1 1 2 3 -1 -2 -3 -y Compute W = $. G· dr parametrically. (Note : You will need to set up and evaluate three line integrals.) (b) (c) Apply Green's Theorem to compute W = f, G· dr.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.The gradient vector field of f(x,y)=y(2x2 -y3 ) is given by: O1. (2xy)i +(x2 -3y² )i O II. (4xy)i -(2x2 -3y? )i O II (4xy)i +(4x2 -3y² )i OV. (4xy)i +(2x2 -3y² )i
- Calculate vector field F = (2 · z² + 2 · x²)i + (2 · y³ + 4 · x)j + (2 · x² · z³ + 2)k . a flux from a box with the coordinates of opposite corners (1,0,0) and (2,1,1) and you will be parallel to the coordinate planes. Use Gaussian divergence theorem fF.dS = SSS V · FdV. PMQuestion 3 Consider the vector field v(x) i + Vx²+y² - r²+y² a) Calculate /v• dr, where r is the straight line from (8, 0, 8) to (0, 8, 16). b) Calculate /v• dr where C is the curve from (8, 0, 8) to (0, 8, 16) in the first quadrant given by x2 + 4y? = z², x + z = 16.Sketch the vector field F(x, y) = xj, the line segment from (1,9) to (6,9), and the line segment from (9, 3) to (9, 6). Calculate the line integral of the vector field F along the line segment from (1, 9) to (6, 9). Calculate the line integral of the vector field F along the line segment from (9, 3) to (9, 6).
- 1: Which of the following vector fields are conservative? (i) F(x, y) = (9xy³ + 6) i + (8x³y7 + 4)j (ii) F(x, y) = (8ye* + sin 6y) i + (eš* + 6x cos 6y)j (iii) F(x, y) = 7yei + (1+ 7xy) e7xy j ,8x %3D %3D (A) (i) and (iii) only (B) (i) and (ii) only (C) all of them (D) (i) only (E) none of them (F) (ii) only (G) (iii) only (H) (ii) and (iii) only7. (a) Determine the value of k for which (u, v, w) is an orthogonal coordinate system ifx = −(u2 + kv2), y = uv and z = w.(b) Given that F and G are vector fields with G a vector potential of F, prove that G is notunique.(c) Show that a vector field F =(4uv − θ3/√u2 + v2, 2u2/√u2 + v2, (lnθ − 3uθ2/uv), defined in paraboloidalcoordinate system (x = uv cos θ, y = uv sin θ, z =1/2(u2 − v2) is irrotational and hence find its scalar potential.Consider the vector field v(x)=(y/(x^2+y^2))i − (x/(x^2+y^2)) j + 4k. Calculate ∫Cv dr, where C is the straight line from (1,0,1)to (0,1,2).
- 7.4 Let F(x,y,z)=(−7xz2,9xyz,−2xy3z)F(x,y,z)=(−7xz2,9xyz,−2xy3z) be a vector field and f(x,y,z)=x3y2zf(x,y,z)=x3y2z.∇f=(∇f=( , , )).∇×F=(∇×F=( , , )).F×∇f=(F×∇f=( , , )).F⋅∇f=F⋅∇f=6. Vector fields V and W are defined by V %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.