(a) Let F= 2xz2i+j+xy³zk and f(x, y, z) = x²y. Compute the following quantities: (i) Vf (ii) curl(F) (iii) F x Vf (iv) F. Vf (b) Does there exist a vector field G such that curl(G) = (xsin y, cos y, z - xy)? (c) Suppose that F, G: R³ R³ are C¹ vector filelds and f,g: R³ Rare C2 real-valued functions. Prove the following statements: (i) div(F x G) = G curl(F) - F-curl (G) (ii) div(Vfx Vg) = 0

Q: For the vector field G = (yey + 3 cos (3x + y))i + (xeªy + cos(3x + y)), find the line integral of G…

A: We find the required integrtion using fundamental theorem of line Integral.

Q: 1. Consider the vector field F(r, y) = -y²i+(x+2)j. (a) Evaluate F(0, 1). F(0,1) = (b) Sketch the…

A: Given:

Q: # 2: For the vector field F = xi a) Plot the vector field on the xry Cartesian coordinate system b)…

A: We know that,      1.              2.

Q: Let + y³)%¸ æyz 7 4 F = x² + be a vector field on U = {(x, y, z) E R° | (x, y) + (0, 0)}. Find its…

A:

Q: a) Calculate the flux of the vector field F(x, y, z) = 275k through a sphere of radlus 4 centered at…

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Q: (b) In the (x, y)-plane, sketch the vector fields v = (y, 2y, 0) and w = (x−y, x+y, 0), clearly…

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Q: Compute the curl of the vector field F = sin(x) i+ cos(y) j+ 8z3 k

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Q: (c) For the vector fields defined in part (b), evaluate separately each term in the iden- tity given…

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Q: Find the k-component of (curl F) for the following vector field on the plane. F = (Xex)i + (8y ex)j…

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

A: Let F and G be vector fields with differentiable components.Let We will compute .The cross product…

Q: Problem #6: Consider the vector field F(x, y) = (Axy+ 4xy + By)i+ (x* + 2x2 + 8xy)j (a) Find the…

A: A vector field is said to be conservative if it is irrotational. Also, if a vector field F→ is…

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A: We will use divergence theorem to find flux through the sphere of radius 2 centered at origin .

Q: (a) Calculate the flux of the vector field F(x, y, z) = 27 - 4k through a sphere of radius 3…

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Q: Let F be the vector field (-2 sin(2x - y), sin(2x - y)). Find two non-closed curves C₁ and C₂ such…

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Q: Show that the vector field F(x, y, z) = (-4y cos(4x), 4x sin(–4y), 0) is not a gradient vector field…

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Q: 6 Find the constants a and b so that F = (ax'y°z°, x°y²z°, bx°y° z²) is conservative. You may use…

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Q: (4) Find a function f such that F = Vf. Consider the vector field F(x, y, z) = y cos(xy)i+(x cos(xy)…

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Q: Find the curl of the vector field F = (3y cos(x), 5x sin(y))

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Q: Show that the vector field F(x, y, z) = (-ycos(8x), 8x sin(-y), 0) is not a gradient vector field by…

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Q: Let F₁ (x, y, z) = (y, x-z²+2, -2yz) and F₂(x, y, z) = (x − z², -y +2, 2yz). (a) Find the curl of…

A: Given vectors  and  we need to find the curl of each vectors and then determine which vector is…

Q: 2x 2y Consider the vector field F = |x² + y2’ x² + y² (a) If C1 is the unit circle x² + y² = 1,…

A: Compute the circulation and flux using parameterization:  ∮C1Fdr=∮C12xx2+y2,2yx2+y2·dx,dy…

Q: Find the curl of the vector field F (6x sin(y), 2y cos(x)). curl F =

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Q: Find the curl of the vector field F = (5y cos(x), 4x sin(y)) curl F -

A: