consider it as a vector function and answer the following items sin t 1-cos²t 2021 + 1) t +0 F(t) t (1,0, 1), t = 0 lim F(t) a) calculate if possible t+0 b)F is continuous at t=0 and at t#0

Q: Let M be the space curve given with the vector valued function R(t) = (1 + 2 sin(2πt))i + (ln t)J +…

A: The equation of tangent line of M at t=2 with  the vector valued function Rt=1+2sin2πti+ln tj+cos…

Q: Consider the vector field F = F . dī along the curve C: F(t) = 7 cos(t)i + 7 sin(t)3, 0 <t <…

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Q: Consider the force field F = (2xzez — 4, cos³ z, x²eª²z – 3y cos² z sin z Find the work done by F in…

A: Given: The force field is F→=2xzex2z-4,cos3z,x2ex2z-3ycos2zsinz and the given given curve is…

Q: If a vector function v(t) = (t, In t , t¯²) is given for a scalar function Ø = x In z + Jy– cos(x +…

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Q: Show that F is a conservative vector field and use this fact to evaluate F. dr along the given curve…

A: Consider the given function. Fx,y=x2i+y2j The compare the form of the function with the general…

Q: Find an expression for the vector field F = ⟨x - y, y - x⟩ (in terms of t) along the unit circle…

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Q: For the vector field F(x,y,z)=ya'1+x1+2xyz k a) show that F is conservative vector field; b) find a…

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Q: Evaluate F. dr for the vector field F(x,y) = xy²i+ x²yj where C is given by r(t) = cos (t)i + 2 sin…

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Q: Consider the vector field F So Evaluate F · dī along the curve C: C F(t) = 4 cos(t)i + 4 sin(t)j, 0…

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Q: Question attached thanks!

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Q: Find the open interval(s) on which the curve given by the vector-valued function is smooth. (Enter…

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Q: Let the vector field F be given by F(x,y,z)=(y,xz,=z^2).  Evaluate its integral along the curve C…

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Q: Find the open interval(s) on which the curve given by the vector-valued function is smooth. (Enter…

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Q: A vector field v is expressed in spherical coordinates as v(r, 0, 0) = 2r sin cos o e, +r cos o cos…

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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: Suppose the vector-valued function r(t) = (f(t),g(t),h(t)) is smooth on an interval containing the…

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Q: Consider the vector field F and the curve C below. F(x, y) = (9 + 4xy²)i + 4x?yj, C is the arc of…

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Q: A vector field v is expressed in spherical coordinates as v(r, 0, 6) = 2r cos 0 e, – r sin 0 eg.…

A: For the solution follow the next step.

Q: Evaluate the line integral fF. dr, where F(x, Y, z) = 4xi – 3yj + 3zk and C is given by the vector…

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Q: Consider the following: the vector fie ld F = (cos (xy 2) – xy 2sin (xy 2))i – 2x 2y sin (xy 2)j %3|…

A: Given vector field F→=cosxy2−xy2sinxy2i^−2x2ysinxy2j^ and curve C with rt→=et,et+1, −1≤t≤0. a We…

Q: If r(t) is a vector-valued function that is continuous on the interval [a, b), then r(t) dt is a…

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Q: ii) Evaluate the vector line integral of F, F. ds, where y is an oriented curve defined as follows:…

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Q: Compute the divergence of the vector field F(r, y, z) = (5x + 12ry)i+ (y + e" sin z)j + (5z + e" cos…

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Q: Find the curl of the vector field F = (4x sin(y), 7y cos(x)). curl F - Dretieur

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Q: consider it as a vector function and answer the following items F(t) = {( +1, Eat) t+0 (1,0, 1), sin…

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Q: Show that F is a conservative vector field and use this fact to evaluate  F. dr along the given…

A: For a vector field of the form F=Pi^+Qj^ to be a conservative vector field, it should satisfy the…

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: 2. Consider the following vector function: r(t) = = a cos wti + a sin wtj + bwtk, where a, b and w…

A: We need to find radius of curvature.

Q: 3. Let C be the space curve given with the vector valued function R(t) = (1+2 sin(2nt))i+ (In t)J +…

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Q: dr = wa x b dt r X

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Q: Consider the vector-valued function f: R → R³ given by: f(t) = (tv4t2 +1, sin(t) cos (t), for t e R…

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Q: I. Consider the vector-valued function In(2t + 1) Ŕ(t) = ²₁+ î+ t 1. Determine the domain of R. 2.…

A: This is a problem of Vector Calculus.

Q: Show that F is a conservative vector field and use this fact to evaluate F. dr along the given curve…

A: A vector field F is conservative if Curl F =0 that is  ∇×F=0∇=∂∂xi+∂∂yj+∂∂zk where ∇→the operator is…

Q: Let R be a vector-valued function defined by VA -1, , In(t + 1) sin 3t if t+0 (2, 3, 0) if t= 0.…

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Q: Consider the curve r(t) = (t, 3 cos t, 3 sin t) for t 2 0. Is the integral o r(t)dt a number or a…

A: Given :   r(t) = ( t , 3cost , 3 sint )

Q: A vector field is defined in spherical coordinates by F(r, 0, 4) = re,r +r sin Oe, + r²es. Select…

A: In this problem, we have to use the result of divergence in spherical coordinates which is explained…

Q: . Consider the vector field F(r, y, z) = (8r sin y, 4x² cos y – e"z, –e" – sec² z). 1. Show that F…

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Q: Consider the vector-valued function t cot t t +0 - Csc t Ř(t) VI+t lnt' (-2, 0, 0), t = 0. Is R(t)…

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Q: If g is the vector-valued function defined by g (t) = (sin(2t), cos(3t)), then g' (t) = %3D

A: Given : gt = sin2t, cos3t

Q: Consider the vector field F and the curve C below. F(x, y) = x2y³i + x³y?j, C: r(t) = (t2 – 2t, t2 +…

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Q: Let r be a di¡erentiable vector function of t. Show that if r # (dr/dt) = 0 for all t, then  r  is…

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Q: If a vector function v(t) = (t, ln t,t) is given for a scalar function Ø = x lnz+√√y − cos(x + 1)…

A: Given vector function is v(t)=t,lnt,t-2                    (1) scalar function is ϕ=xlnz+y-cosx+1,…

Q: . Consider the vector field F(x, y) =⟨2xy3,1 + 3x2y2⟩ in R2. (a) Show that F is conservative. (b)…

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Q: Consider the vector field F (x, y) = i-zj and the real-valued function = tan-(#). One has x²+y² f(x,…

A: There is no error(s). All calculations are right. Since a vector field  F¯ is conservative if and…

Q: Let F = (t+ 4, sin(4t) , t°) Find the indefinite vector integral F(t) dt = ( +C1, +C2, +C3)

A: Given, F = (t+4, sin (4t), t³) ∫F(t) dt = ∫(t+4, sin (4t), t³) dt ∫F(t) dt =( ∫(t+4)dt , ∫sin(4t)dt…

Q: (b) Use the corresponding formula to evaluate the line integral F. ds where F(x, y, z) = (xz, -yz,…

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Q: 3. Let F = (2y² + z)i + 4.ryj + rk. (a) Show whether F is conservative. (b) Using the fundamental…

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Q: Find the open interval(s) on which the curve given by the vector-valued function is smooth r(0) = 4…

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Q: Evaluate F Î ds for the vector field F(x, y) = yì – xj counterclockwise along the circle x² + y² = 1…

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