Find the line integrals of F = 3yi + xj + 3zk from (0,0,0) to (1,1,1) over each of the following paths. a. The straight-line path C₁: r(t) = ti + tj + tk, 0st≤1 b. The curved path C₂: r(t) = ti + t²j+tªk, 0sts1 c. The path C3UC4 consisting of the line segment from (0,0,0) to (1,1,0) followed by the segment from (1,1,0) to (1,1,1) (0, 0, 0) C₁ (1, 1, 1) C₁

Q: Let j=8 k=3 let A (0, 7-k), B(6, -k+5). Find all points P on the line y = 9-k such that AP I BP.

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Q: Let P denote the path that travels along the graph of y = x2 from (0,0) to (1,1). Let f(x,y) = √1 +…

A: Let P denote the path that travels along with the graph of y=x2 from (0, 0) to (1, 1). Let f(x,…

Q: Let F = (y – x, ry) and let C be the line segment from (2, –3) to (1, –1). %3D Evaluate F. dr.

A: We wil first equation of parametric equation of line segment

Q: 3. Compute the value of F.dr if F = -yi + xj and C is a semicircle from (-1,1) to (1,-1).

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Q: Find the exact value of F. dr using any method. b. F(x, y, z)= C is the line segment from (1, 0,…

A: The given vector field is Fx, y, z=yz, xz, xy+2z. We have to find ∫CF·dr.

Q: 2. For F(x, y, z) = (6xsin y+10x=' )i + (3x² cos y-4ye* −2y²z³)j + (-2y³e² +15x²2²)k, find a…

A: Given Data: Let us consider the given data, Fx,y,z=6x sin y +10xz3i+3x2 cos y -4yeyz-2y2zeyzj…

Q: F. dr Evaluate So

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Q: a. Find a function (f(x,y,z)) such that (F = Vf(x,y,z))? b. Use part (a) to evaluate (F. dr ) along…

A: We have to evaluate the ∫CF.→dr→.

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Q: Show that F is conservative and find a function f such that F= Vf, and use this result to evaluate F…

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Q: 2. Evaluate [ F C F = (xz, 0, yz) and the path C is the line segment from point (-1,2,0) to (3,0,1)…

A: Given that, Now,

Q: If ů, = |8|,02 = 5,03 show that {v1, v2, v3} is linearly dependent in R', and determine the linear…

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Q: Find the length of the line given by r(t) = ⟨t, 2t⟩, for a ≤ t ≤ b.

A: To determine: The length of the line given by rt=<t,2t> for a≤t≤b

Q: Which of the following subsets of functions of the set {eª, e²ª, 2eª, 2} is linearly dependent on R?…

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Q: If F(x, y, z) = (sin x + 2) i+2xz j+ (e-x) k and the path C is the line segment from A(1,2, 1) to…

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Q: Find the length of the curve r(t) (1,t²,), 0sts1 %3D «3 Ost<1

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