Take as given the formula from Green's Theorem: V.f dx dy = | f ·n ds, where D is a region in the plane and C its boundary; f is a vector-valued function of x and y; n is the outward normal to the boundary and ds the element of arc length on the boundary. Use it to derive Green's First Identity, du (Vv . Vu + vAu) dx dy = || ds an where u and v are twice-differentiable functions on D.

Q: Consider the helix r(t) = (cos(1t), sin(1t), −1t). Compute, at t = 픔: A. The unit tangent vector T =…

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Q: 3. Let the curve C be traced by the vector-valued function R(t) = cos(at) î + at j+ sin(rt) k. %3D…

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Q: :If the tangent at (a, b) to the curve x³ + y° = c° meet the curve again in (a,,b,). b, Prove that…

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Q: Consider the helix r(t) = (cos(-2t), sin(-2t), -4t). Compute, at t = 7: A. The unit tangent vector T…

A: We can find unit tangent , normal , binormal vector and Curvature .

Q: Find the unit tangent vector for the parameterized curve: r(t) = 2ti + 2cos tj - 2 sint k

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Q: For the curve given by the vvf r(t) = ti + 2t²j+t³k, find the unit tangent T(t), principal normal…

A: I have appended the formula used in the solution part of the question

Q: 7) Find the unit tangent vector for the parameterized curve: r(t) = cos ti+ sin tj+ sin tk, 0≤t< 2π

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Q: The curves C,:x=t, y=t, z=t, t20 and C,: x= sint, y 1+cost, z=1, 120 have intersection at the point…

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Q: 3. Which Jf the following is equivalent to Laplace transforn the Inteyral. St 3(t -7) Cos 4T d? o e…

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Q: Let C be the graph of Ŕ(t) = (t sint, t³). a. Using delta definition, show that Ŕ(t) is continuous…

A: Given that R→t=t sint,t3 To find: (a) R→t is continuous at origin.              (b) unit tangent and…

Q: 1. Compute each contour integral. The curve C is a given in the figure below: (a) f2²dz (b) fadz (c)…

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Q: What is the unit tangent vector to the curve r(t) =ť°j+tkwhen t = 2. Select one: 0 긁(2j + k) ㅇ…

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Q: Consider the helix r(t) = (cos(-3t), sin(-3t), -4t). Compute, at t = : A. The unit tangent vector T…

A: For a vector r(t) Unit tangent vector Tt=r'tr't  Unit normal vector Nt=T'tT't Binomial vector…

Q: se the given curve to do the following calculations. r(t)=2ti+ 4 costj+4 sint k, for 0 st≤2 . Find…

A: NOTE: According to guideline answer of first three subpart can be given, for other please ask in a…

Q: for the follawing , find the unit tangent vector: parameterized curve rit) =<2 sint) , 2 cos lt),5…

A: Given: r(t)=2sin(t),  2cos(t), 5cos(t),  0≤t≤π The unit tangent vector to curve r(t) is…

Q: b) Find the unit tangent vector for.. r(t) = at t=0

A: The unit tangent vector can be calculated using the formula T→(t)=r→′(t)‖r→′(t)‖. The denominator is…

Q: 2. Let C be the graph of R(t) = (t sint, t³). a. Using the definition, show that Ŕ(t) is continuous…

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Q: Consider the curve given by 7(t) = (etcost, et sint, 5e¹) Find the value of t that corresponds to…

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Q: For the curve r(t), write the acceleration in the form a+T+aNN. r(t) = (6t sin t+ 6 cos t)i + (6t…

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Q: The unit tangent vector of 7(t) = ti – j- tk at t = 1 is Select one: O A. i O B. -- k OC. i+ j- O D.…

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Q: b) Find the laplace trausform of the Runction fCE)= sint.ult - Tt) where u(t-. is a unit function.

A: Here we have use property of Laplace transform when multiplied by shifted unit function.

Q: Given the vector valued function r(1) = (2e' )i + (1 – e²")j a. Determine the unit Tangent and…

A: Given is the vector valued function r(t) = (2et) i + (1-e2t) j  The objective is to determine the…