Iind the unit tangent vecior T, Hhe unit normal vector N, omd The aebe binor mal Vector B for curve r ar the point C3,0,0). rCt)> <3 cos C+), 3 sin (t),3In Ccos (t))> NE ? Find the equarion of the oscularing plane at the point (3,0,0). escu laring plane:?

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: Consider the helix r(t)=(cos(3t), sin(3t), 3t). Compute, at t=π/6 The unit normal vector N

A: The given vector of helix is,

Q: r(t) = (x(t), y(t), z(t)) where = - 4t? – 5t – 3 æ(t) - 5t2 + 5t + 3 y(t) = - 5t? z(t) : || Since…

A: Given a curve, rt=xt,yt,zt, where,  xt=-4t2-5t-3yt=5t2+5t+3zt=-5t2 Curvature of a curve rt, κ is…

Q: What is the normal vector, n for S is a surface of x +y² +z² = 36 in the first octant? %3D

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Q: Find the surface area of the helicoid (spiral ramp) with vector equation T(u, v) = (u cos v, u sin…

A: We need to find the area of the surface:r→u,v=ucos v, u sin v, v ; 0≤u≤1, 0≤v≤3π

Q: Find the surface area of the helicoid (spiral ramp) with vector equation F(u, v) = (u cos v, u sin…

A: The area of given helicoid is given by using double integral as follows

Q: Consider the plane: 2.x + 2y + z = 3 Find a unit normal vector.

A: Answer

Q: Find the work Fdr done by the force F i+ yj+k around the curve C that is formed by the intersection…

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Q: Let S be the disk enclosed by the curve C: r(t) = (cos q cos t, sint, sin p cos t), for 0sts2a,…

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Q: Let F = < rY, 3z, 4y Use Stokes' Theorem to evaluate | F . dī, where C is the curve of intersection…

A: Here we have, F→=xy, 3z, 4y Using the stoke theorem, we have to evaluate the ∫CF→·dr→=? Where C is…

Q: For F = xy?a, + yz²a, – x²za,, use divergence theorem to evaluate ff, F. dS where S is the sphere of…

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Q: Find the arc length of the space curve given by the vector function t3 i + t2 j + (2t + 1) k from…

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Q: Starts to move with initial A particle Velocity of i-j+rek and a cce leration Vector fn : aCH)- e i+…

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Q: The principal unit normal vector on the curve r(t)= (2 sint,2cost) %3D (a) always points directly…

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Q: Q4. For F = xy?a, + yz²a, – x²zaz, use divergence theorem to evaluate ff, F. ds where S is the…

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Q: Find a vector tangent to the curve of intersection of the two cyclinders a? + y? = 8 and y? + 22 = 8…

A: Let fx,y,z=x2+y2 and gx,y,z=y2+z2 The gradient of the functions give the perpendiculars to the…

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: Show that for a plane curve the torsion is t(s)=0.

A: For a plane curve the torsion is t(s)=0.

Q: Find the unit tangent vector T, the principal unit normal vector N, and the unit binormal vector B…

A: Consider the curve     Ct: rt =cost,sint,t (i) unit tangent vector                  The unit tangent…

Q: 2. Calculate the curvature, unit binormal vector and torsion for r(t) (e'cost)i+(e'sint)j +5k

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Q: Let S be the disk enclosed by the curve C: r(t) = (cos cos t, sint, sin p cos t), for 0sts2r, where…

A: We need to find circulation.

Q: Suppose that C has length 5. What is the value of F. dr if: (a) F(P) is normal to C at all points P…

A: Given: Length of C is 5. (a) F(P) is normal to C at all points P on C. ∫CF→·dr→=∫CFdrcos90°=0…

Q: Evaluate F· dS , where F = and S is the helicoid with vector equation r(u, v) upward orientation.…

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Q: The vector tell f (xxz) = (z=+,2oxy"6) igven. Calkulate, using the stores theorem, the surface…

A: Given that  Fx,y,z=z2−1, z+xy3,6 To evaluate the integral ∫∫S∇×FndS using the stokes theorem.

Q: A particle is moving along the curve whose position vector is given: R(t) = (cost + tsint)i + %3D…

A: R(t)→=cost+t sinti+sint-t costj+4kFind the unit normal vectorSome…

Q: Find a vector function r(t) that describes the curve of intersection between the elliptical cylinder…

A: given, elliptical cylinder 4y2+z2=4  and the plane x+y=1 we have to find a vector function r(t) that…

Q: The principal unit normal vector on the curve r(t) = (2sint,2 cost) %3D (a) always points directly…

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Q: A particle is moving along the curve whose position vector is given: R(t) = (cost + tsint)i + (sint…

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Q: A partidle is moving along the curve whose position vector is given: R(t) = (cost + tsint)i + (sint…

A: As we know that the simple form of the position vector is:   r→=xi+yj+zk   From the given problem:…

Q: Use Stoke 's Theorem to evaluate [ . dr, where v = y²i + xyj + xzk, and c is the bounding curve of…

A: We have to solve the given problem by using stokes theorem.

Q: Verify Stokes's theorem over a quarter circular disk with a radius, r = 4 in the 1* quadrant for any…

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Q: Find the area of the helicoid (or spiral ramp) with vector equation r(u, v) = ucos(v) i + usin(v) j…

A: The area of the Helicoid with vector equation r(u,v)=(ucosv)i^+usinvj^+vk^ in limits a≤u≤b  and…

Q: Find a vector-valued function whose graph is the indicated surface. The plane z = 7y r(u, v) =

A: Given the given surface is z=7y

Q: Find the surface area of the helicoid (spiral ramp) with vector equation r(u, v) = (u coS v, u sin…

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Q: Use Stoke 's Theorem to evaluate [ V. dr, where v= y²i + xyj + xzk, and c is the bounding curve of…

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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: Find the value of a at which the plane curve traced out by the vector function E + bt> is not…

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Q: Consider the ray r(t)=o+ td with origin o=(1, 1, 0) and direction d = (1, 1, 1), and the plane N-x=…

A: To find the point of intersection of the ray and the plane. Given: equation of ray is: r(t)=o+td…

Q: ☺ Find the unit fenpent and unit normal vectors to the space )curve (circulur helix) determined by…

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Q: A particle is moving along the curve whose position vector is given: R(t) = (cost + tsint)i + (sint…

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Q: Evaluate F.dr, where F (x, y, z) = - yi + xj + z*k and C is the curve of intersection of the plane y…

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Q: The scalar component of a vector A normal to cylindrical surface of radius 2 at point p A = p'a, +…

A: Given that, A=pBAap+sinϕaϕ+2zaz And the cylinder  surface of radius 2 is, p=2p-2=0 Now the normal…

Q: Let r(t ) = ti + cost j + t? k be a space curve. Which of the following vectors is parallel to the…

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Q: Find a vector-valued function whose graph is the indicated surface. The plane z = 4y r(u, v) =

A: Solution ) the plane      z = 4y   Let's make substitution  x =u , y = v  Hence  z = 4v   r ( u, v)…

Q: Find the surface area of the helicoid (spiral ramp) with vector equation F(u, v) = (u cos v, u sin…

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Q: Let C" be a simple closed piecewise-smooth curve that lies in a plane with unit normal vector n =…

A: Given: C is a simple closed piecewise-smooth curve that lies in a plane with unit normal vector n =…

Q: moving along the Curve RCH) = (In Sec E)r +tj Ests Find Apoint given as %3D where (1) The speed (2)…

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