The vector field F(r, y) = (2xy +y) dx + (æ + 2xy) dy is conservative. %3D The function f(x, y) = x² + 2xy+K is a functionf such that F =Vf. %3D Evaluate F dr, where C is the line segment from (-1, 1) to (2, 2).

Q: Define two vector functions F(t) = 8 sin(t)i + 6 cos(t)j + (t – 2)k ü(t) = 6 sin(t)i + 8cos(t)j + (t…

A: We have to find the dot product of r(t) and u(t)

Q: Determine whether the vector functions, x1(t)= [ et ]           [ -et ] and x2(t)= [ et ]…

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Q: Let C = (-2, -1) be the center of dilation. Let X = (1, 1), Y = (3, 0) and Z = (2,5). Our dilation…

A: Center of dilation=-2, -1X=1, 1, Y=3, 0, Z=2, 5Dilation factor, k=3

Q: Let ß be the vector-valued function 3u B(u, v) = { 32 ß: (-2, 2) × (0, 2π) → R³, 3u² + 4 COS(V) } u…

A: Since you have asked multiple questions, we will answer the questions (a) and (b) for you. If you…

Q: (a) Find the direction for which the directional derivative of the function f(x, y, z) = √√x² + y² +…

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Q: Let c1 (t) = ei + 7 sin(t)j + t*k and c2 (t) = e-Sti + 5 cos(t)j – 7t*k. (Enter your solution as a…

A: Given: Let c1(t)=e5ti+7sintj+t4k and c2(t)=e-5ti+5cos(t)j-7t4k. To Find:      ddt[c1(t)+c2(t)]

Q: Let x = (1, 1, 1, 1)T and y = (8, 2, 2, 0)T . Verify that x − p is orthogonal to p.

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Q: Let r(t)= Determine the interval(s) on which this vector valued function is continuous.

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Q: Compute ∇ƒ(-1, 2, 1), where ƒ(x, y, z) = xy/z.

A: Given,f(x,y,z)=xyz     As we know ∇f(x,y,z)=∂f∂x,∂f∂y,∂f∂z

Q: Let 7 (t) = (4t³ – 4, – 3e", – 3 sin( – 4t)) - Find the unit tangent vector T(t) at the point t = 0…

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Q: Consider the vector-valued function r(t) = 4t²i + (t5)j + tk. Write a vector-valued function u(t)…

A: Topic: vector valued function

Q: 2. Consider a function g(x, y, z) = 1 + ln(3x + 6y +9z) and a vector u = (0, -1, 1). (a) Find the…

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