Let C CR be the curve from (0,-7,0) to (-2,-3,-8) along the intersection of the surfaces x- y 7 and x = z. Find a parametrization a for C. a :[ |1R', with a(t) =(

Q: 1. Prove that fo r*dr + rydy = where C1 is the line segment from (0,0) to (1,0) by using…

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Q: Find the coordinates of the centroid of the curve given belov x = cos t+tsin t, y =t cos t- sin t,…

A: Given data: The x-coordinate of the curve is x=cos t+tsin t. The y-coordinate of the curve is y=tcos…

Q: Show that the curve r= Vti + Vtj + (2t – 1)k is tangent to the surface x +y² – z = 1 when t=1

A: Given that r=ti+tj+2t-1k and f=x2+y2-z-1 Take the derivative for f, ∇f=ddx,ddy,ddzf=2x,2y,-1…

Q: Evaluate the unoriented line integral (22 + y° + z²) ds (2x² along the parametric curve a : [0, 1]…

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Q: Which one of the following is the parametric equa- tions of the line that is normal to the surface…

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Q: Evaluate the line integral I of the one-form w = xy dx + 12y dy over the parametric curve a : [0,…

A: Given: One-form ω=xydx+12ydy and the parametric curve α:0, ln2→ℝ2 defined by α(t)=e2t, e-2t. To…

Q: Find the line integral with respect to arc length (8x + 9y)ds, where C is the line segment in the…

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Q: Prove that fe, r*dx + rydy = 0 where Co is the line segments from (0, 1) to (0,0) by using…

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Q: Use Green's Theorem to evaluate the line integral. Orient the curve counterclockwise. 2x + 3y dx + e…

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Q: Suppose F(x, y) = -y sin(x)i + cos(x)j. (a) Find a vector parametric equation for the parabola y =…

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Q: Let P = x2 – y², Q = -2ry and C denote the ellipse centered at (0,0) and passing through the points…

A: We will use parametric equation of ellipse to find line integral . As only one question allowed to…

Q: Let f(x, y, z) = (2x + 4z + 10)/xy + 2. %3D a. Evaluate the line integral of f along the line…

A: We will find out the required solution.

Q: Give the integral representing the length of the parametric curve x=t^3, y=t^4, 0≤t≤1.

A: Given: A parametric curve:   Now the length of the parametric curve is given by the formula:

Q: Find the coordinates of the centroid of the curve x = cos t, y = t + sin t, 0<=t <= pai.

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Q: use a parametrization to find the flux ∬S F . n dδacross the surface in the specified direction. F =…

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Q: Let f(x, y, z) = (2x + 4z + 10)/xy + 2. %3D a. Evaluate the line integral of f along the line…

A: We will evaluate the given integral.

Q: Use Stokes's theorem to evaluate o F· dr, where F(x, y, 2) = xy² i+x²yj+yz k, where C is a…

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Q: (a) 'Find the orthogonal trajectories on the surface of the sphere x' + y? + z = 0 with the…

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Q: Let S be the passage of R ^ 2 defined by the parabola with equation y = 18-x ^ 2 and by the line…

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Q: 12. Let C represent the curve of intersection between the cylinder x² + y² = 4 and the plane z = 10…

A: The plane z=10-x will cut the cylinder x2+y2=4 to give an ellipse.

Q: Given the scalar function f(x,y)=2xy,calculate the line integral along the ellipse curve from point…

A: Given data:- The given curve is, x29+y24=1 The given scaler function is, f(x,y)=2xy t E 0,π/2

Q: Evaluate the line integral F. dr, vhere F(z, y) = -yi-rj and C is the curve y = v4-a from (2,0) to…

A: We have to find the line integral

Q: Evaluate the unoriented line integral (2æ² + y² + z²) ds along the parametric curve a : [0, 1] → R³…

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Q: Suppose F(x, y) = -y sin(x)i + cos(x)j. Find a vector parametric equation for the parabola y = x²…

A: The solution is following.

Q: Exercise (2): Calculate fe xydx + 3y*dy where C is the curve in xy plane, x = y² from (0,0). to…

A: we have to following integral

Q: Find the line integral with respect to arc length ∫C(6x+4y)ds, where C is the line segment in the…

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Q: | 5y? dx +xy dy

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Q: Find a parametrization for the curve y = Vx with terminal point (0, 0) using the angle 0 in the…

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Q: Suppose F(x, y) = -y sin(x)i + cos(x)j. (a) Find a vector parametric equation for the parabola y =…

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Q: Find a parametrization for the line segment from 2₁ =1-i to Z₂ = 2i. A. 2 (t) = 1 -i + (−1+3i) t 0…

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Q: 2. Let R(t) = }sin 4t î – bcos“(21) j + 2t Ř. Find the moving %3D trihedral of the curve of R(t) at…

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Q: Find the work done by F in moving a particle once counterclockwise around the given curve = (2x –…

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Q: Compute e3z dz , Jy (z – 1)2(z+ i) where +y is any simple closed curve oriented counterclockwise…

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Q: Compute the scalar line integral xy ds where C is the part of the circle of radius 3, centered at…

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Q: Find the work done by the force field F(x,y)=5x2i−2xyj on a particle that moves once around the…

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Q: Use Green's theorem to calculate the work done by force F on a particle that is moving…

A: GivenF(x,y)=x32-8yi+5x+8yj,C: Boundary of a triangle with vertices (0,0), (5,0) and (0,5)To…

Q: Suppose F(x, y) = -y sin(x)i + cos(x)j. (a) Find a vector parametric equation for the parabola y =…

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Q: 1. Let H be the hyperbolic paraboloid given by z = r² - y, and let P be the plane given by n x = 1,…

A: Given: H be the hyperbolic paraboloid z=x2-y2   and the plane P defined as n.x=1 where…

Q: Find an equation for the tangent plane to the surface z = 3y^2 − 2x^2 + x at P(2, −1, −3). Express…

A: Let's find equation of tangent plane.

Q: Compute the line integral | (yze*v= + y° + z - 2x) dr + (xze*# + 2ry) dy + (rye#"= + x + 32²) dz,…

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Q: Evaluate the unoriented line integral +y² + z²) ds along the parametric curve a : [0, 1] → R³ with…

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Q: Evaluate the line integral: f y² dx + x² dy by parameterizing C where the curve C is a square R with…

A: According to the given information, it is required to evaluate the line integral.

Q: Compute the line integral (yze"y= + y² + z – 2x) dx + (xze"v² + 2xy) dy + (xye"y= +x+ 3z²) dz C…

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Q: Let F = (x'e* +2.xy - 2y)i+(cos y+x² +3x)j and let C be the ellipse with equation (2x - 2y) +(x+ y)°…

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Q: Find the coordinates of the centroid of the curve x = cos t + t sin t, y = sin t - t cos t, 0…

A: Given,

Q: Compute the line integral [(yze²v² + y² + 2 − 2x) dx + (xze²v² + 2xy) dy + (xye²³² +x+32²) dz, where…

A: We have to compute the line integral ∫Cyzexyz+y2+z−2xdx+xzexyz+2xydy+xyexyz+x+3z2dz, where C is the…

Q: Let C be the curve in the plane with parametrization r(t) = cos ti+ sin tj for t e [0, 4T]. Compute…

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Q: Find the line integral with respect to arc length (2x + 6y)ds, where C is the line segment in the…

A: The solution of line integral is given as follows :