the following questions the domain of all vector fields will be taken as n = R². how that the curves r(t) = (R sint, R cost) for a fixed R≥ 0 are integral curves for the vector field = (y, -x). Sketch these curves on a diagram, indicating any critical points of X and indicating with arrow the direction of increasing parameter for each sketched integral curve.

Q: In words, describe the curve r(t) = (2 sint,e',2 cost) for 0<= t <=6 . Be sure to a -- give enough…

A: To Explain: The parametric curve is r(t)=(2sint,et,2cost) for 0≤t≤6π , so that its graph can be…

Q: Draw the slope field (direction field) for the following DE: y'=y-t

A: Given DE is Make table as follows:…

Q: 4. (a) Determine the directional derivative of f(x, y, z) = zety at the point (1, −1, 1) in the…

A:

Q: Find the domain of r(t) and the interval(s) on which the vector- valued function is continuous. r…

A: We need to find the domain and the interval on which the vector-valued function is continuous.

Q: The temperature function on a circular plate R = {(r, y) |r² + y? < 36} is given by T(r, y) = 200 –…

A: Consider the temperature function given by  T(x,y)=200-x2-y2  on a circular plate…

Q: The smooth vector-valued function P(t) satisfies P(2)= (-4, 8,-7), P′(2)= (0,-5,2), and P'(2)= (3,…

A:

Q: T = [ Cos(a)+Sin(a).Cos(b) ] A-) Find function's gradient vector B-) Find function's Laplace…

A: We have to solve given problem:

Q: 1. In your own words, describe what the vectors in a vector field tell us about the function. Each…

A: A vector field associated with a function provides a visual representation of how the function's…

Q: Graph

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Q: Suppose that over a certain region of space the electrical potential V is given by the following…

A:

Q: 2. Consider the vector-valued function R(t) = Show that is continuous at t = 1. (e¹-², 1-1², (2-1)),…

A: We use the sequential definition of continuity to show that $R(t)$ is continuous at t=1. The…

Q: 3. Let F = (x - y, xy) and C be arc of the circle x? + y? = 4 traversed counter-clockwise from (2,0)…

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Q: Find an expression for a unit vector normal, no, to the surface x = (4 – cos (v)) cos (u), y = (4 –…

A: The complete solutions are given below

Q: Q2// B :- If u+v3e"cos y and u- v=e' sin y find the Jacobain of the function alu,v at dx.yl (x.y)…

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Q: Consider the function g defined by 1 g(x, y) = sin (TVry) + log3(x – 3y) Do as indicated. 1.…

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Q: Find r'(t), r(t,), and r'(t,) for the given value of to- r(t) = (5 + t)i + t°j, to = 1 r'(t) = r(t,)…

A: To find r'(t) and to sketch the curve represented by a vector-valued function and sketch the…

Q: 4. Find the unit tangent vector T(t) to the curve r(t) = (In(cos t), cost, sin t) at time t. (a)…

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Q: Let F(t) be a vector valued function whose derivative is F'(t) = G(t) = (6t, 3t, −6t). 1. If F(2) =…

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Q: Q4 (a) Given a cylinder x' +y² = 1 and a plane x +z = 1. (i) Find a vector-valued function r(t) that…

A: we have,  A cylinder x2+y2=1 and a plane x+z=1 i x2+y2=1x2+y2=cos2t+sin2tx2+y2=cost2+sint2⇒x=cost,…

Q: II. Let F and G be vector-valued F(1) = (1,0,2), F'(1) = (1,1,2), G(1) = (2,1,3), G'(1) = (-1,…

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Q: 8. Given the space curve C as the image of the vector valued function r(t) = 2ti+t³j+-t-¹ k OR r(t)…

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Q: 2. Find the length of the curve r(t) = e'i + e' sin 2tj + e' cos 2tk, 0st< 2n.

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Q: If the given three vectors are coplanar, then find the value of x. A = i-2j + 3k, B = xj + 3k, C =…

A: A, B, C are coplanar.So

Q: Given the vector-valued function r (t) = e-t i +t j +(tan t) k Find: a) r'(t) b) r"(t) c) r'(t) .…

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Q: 1-x² - y² -3 For the function f(x,y) that has a positive component. = " find a unit tangent vector…

A: The function is For finding the unit…

Q: Find a tangent vector of the curve7(t) = (t², 2 sin(t), 2 cos(t)) at (0,0, 2) (0,0,1) (1,0,0) None…

A: Here, Find it’s derivative This is the tangent at any point on the curve.