Are the following statements true or false? ? 1. If F. dA = 12 and S is a flat disk of area 47, then div(F) = 3/r. S ? v 2. If F is a vector field in 3-dimensional space satisfying div(F) = 1, and S is a closed surface oriented outward, then F. dA is equal to the volume enclosed by S. v 3. If F is a vector field in 3-dimensional space, then grad(div(F)) = 0. ? v 4. If F and G are vector fields satisfying div(F) = div(G), then F = G. ? v 5. If F is a vector field in 3-dimensional space, and W is a solid region with boundary surface S, then F. dà = div(F) dV.

Q: b) Given the vector field F(x, y, z) = y² tan 2x i - (lnx) (sin z) j+ey k. Find i. div F ii. curl F…

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Q: 2. Sketch at least ten arrows in the vector field: F(x,y)=(-2y)i + (x+1)j

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Q: Assuming that vector fields B (r") and A (r") in the zero infinity, check the correctness of the…

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Q: S Js V is the volume enclosed x (axr)ds = 2aV, where a by the closed surface S.

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Q: 3. Let F = xzi+yzj+xyk be a vector field and S be the part of the sphere x² + y² + z² = 9 that lies…

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Q: If V and W are vector fields on R3 and f is a function on R3, show that: ∇ · (V × W) = W · (∇ × V)…

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Q: For what values of b and c does the vector field F = ⟨by, cx⟩ have zero circulation on the unit…

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Q: 33. Define T: R³ → R³ by y + 2z 2x +3y - z x +2y - 2z x - y a. Find all vectors in R' that are…

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Q: 13. For the vector fields: A = x + 2y ŷ + 3z 2; B = 3y - 2x ŷ. Check both product rule given in…

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Q: Consider the vector field F(x, Y, z) = 3x?yi+ (x³ + 1) j+9z²k. Let C be the circle cut from the…

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Q: For vector field F = (-2y, y), y > 0. Find all points P such that the amount of fluid flowing into P…

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Q: Let vector field F=(xz,-y,z) and surface, S is a cylinder x² + y² = 4 for 3≤z≤0 with the outward…

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Q: 3. Let F and G be vector fields with differentiable components. Express curl (F x G) in term of div…

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Q: 2. Let F(x, y) -yi-zj + zk be a vector field. Compute F FdS, where S is the hemisphere a² + y² +₂²…

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Q: A vector field is given as F = âr2r“ + a¿r²CoSo + âz5z. Verify Stokes theorem for this vector field…

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Q: find the solution from x = 0 to x = 2 with an increment of 0.5 by a) Euler’s Method  b) Heun’s…

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Q: Show that Vx (V.f) = 0 for any vector field f.

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Q: V(E × F) = F(V × E) – E(V × F). %3D

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Q: Let us verify Stokes' theorem using the vector field F = (x 2 - y)i + 4zj + x 2k, where the closed…

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Q: (a) State True or False and proved justification for: If S is an ellipsoid and F is a constant…

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Q: let c is a differentiable vector field then div(curl)=

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Q: b) The line integral F. dr is a scalar.

A: Check whether given statement is true or false.

Q: Q6) Given the following vector field: F = x?yi+ 2xy² j+ xyz k , Find V · F , V xF %3D and V· (V ×…

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Q: For the cubical box with side 1m surface), find the value of f V xA · ds of the cube) if the vector…

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Q: 3. Let fe L(X,Y) and let dim X> dim Y .Show that there exists nonzero vector x, in X such that…

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Q: (a) Simplify the expression (Ř· VR) · Ã + V · V × V × ((Ā × R) x (R × Ã)). (b) Assuming Ả the…

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