(5) Evaluate F- dr using Stokes' Theorem, where F(1, y, 2) = -y?i + xj +2²k and C is the curve of intersection of the plane y+z = 2 and the cylinder r² + y² = 1 (Orient C to be counterclockwise when viewed from above) as shown below; -y+z=2 D OL

Q: O) Use Stokes' Theorem to evaluate F. dr where F(x, y, z) = xyi + j + xyk and C is the curve of…

A: (c) The curve, C can be easily parametrized using the following parametrization: x=cos t, y=sin t…

Q: • LF. dr. Use a computer algebra system to verify your results. C is oriented counterclockwise as…

A:

Q: Use Stokes's Theorem to evaluate Jo F. dr. In this case, C is oriented counterclockwise as viewed…

A:

Q: Consider the paraboloid z = a? + y. The plane 10x – 4y + z – 2 = 0 cuts the paraboloid, its…

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Q: 2. A surface S in R®(r,y, :) is defined parametrically in the form u + -1 2u (a) Find points on the…

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Q: An equation of the tangent plane to the surface 1+y Z = at the point p(1, 7, 4) is 1+x (A) 2(x+1) -…

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Q: 5) Consider the surface S shown in the graph, whose parametrization is given by by: r(u, v) = (v²,…

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Q: 1. Verify Stokes' Theorem for F = (2x - y)i - yz²j - y² zk, where S is the upper half surface of the…

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Q: The equation of the tangent plane to the surface f (x, y, z) = xe cosz - z -1= 0 at the point…

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Q: 4. Let II be the plane tangent to the implicit surface x²y° + e²= = 5+ 2y sin(z) at the point (-2,…

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Q: Verify Stokes’s theorem for F = , where C is the curve C: x2+ y2=4 , z = 2.

A: Stoke's theorem problem...

Q: 18. Find an equation of the tangent plane to the surface given by the parametric vector equation…

A:

Q: 4. Confirm Stokes' theorem for the vector field F(x.y.z)= (0, yz, xz) for the plane surface z =1-y…

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Q: Use Stokes' Theorem to evaluate  F · dr  where C is oriented counterclockwise as viewed from above.…

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Q: Use Stokes' Theorem to evaluate F• dr where C is oriented counterclockwise as viewed from above. Jc…

A: Use Stokes' Theorem to evaluate CF · dr where C is oriented counterclockwise as viewed from above.…

Q: Let F = Use Stokes' Theorem to evaluate F. dr, where Cis the triangle with vertices (4,0,0),…

A: Please see the explanation below

Q: 3.) Verify the conclusion of Stokes' Theorem given F(x,y,z)=(x-y,y-z,z–x) where S is the surface cut…

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Q: 4. Consider the cone z = x² + y?. (a) Show that the tangent plane to this surface at any point (xo,…

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Q: Use Stokes's Theorem to evaluate JeF - dr- . In each case, C is oriented counterclockwise as viewed…

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Q: Use Stokes's theorem to evaluate OF.dr for the given F and C. (c) F(x, y, z) = xzi+ xyj– 3xz k, C :…

A:

Q: c) Use Stokes' theorem to evaluate (zy - u)i– In z j+ (x + z²) k) dr, | where C is the intersection…

A: According To Stokes Theorem∮CF⋅dr=∬S(∇×F)⋅ndS here n is unit normal to the surface.

Q: 6. Use Stokes theorem to evaluate $. F•dr, where F = (-3y² , 4z, 6x) and C is the triangle in the…

A:

Q: 18 Verify Stokes' theorem F.dr = (curl F)· dS C S where F = (x² + y – 4, 3xy, 2xz +z?) and S is the…

A: We have

Q: 6. Use the Normal Vector to a Smooth Parametric Surface to find the equation of the tangent plane to…

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Q: (a) Use Stokes' Theorem to evaluate F. dr where F(x, y, z) = x2yi + x³j + xyk and C is the curve of…

A: Given F→x,y,z=x2yi+13x3j+xykWhere C is the curve of intersection of the hyperbolic paraboloid…

Q: 3. Use Stokes' Theorem to evaluate F.dr where C F = -yz i+ (4y+ 1) j+ ryk and C is is the circle of…

A: According to Stokes' Theorem: Let S be an oriented surface with unit normal vector field, N, and…

Q: A parametric equation of a normal line at the point A(1.2,1)to the surface f(x.y,z) z-x +2xy-y 0 is:…

A:

Q: A smooth curve is normal to a surface f(x,y,z) = c at a point of intersection if the curve's…

A: Given: A smooth curve is normal to a surface fx,y,z=c at a point of intersectionif the curve's…

Q: c) Use Stokes' theorem to evaluate $ (xy – y) i – In zj+ (x+ x²) k) · dr, where C is the…

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Q: S,F•dr, and C is the triangle, with Use Stokes Theorem to evaluate where F(x,y; =) = (x+ y² )i+ (vy+…

A: Given: Fx,y,z=(x+y2)i+y+z2j+z+x2k and C is the triangle with vertices 1,0,0 ,0,1,0 and 0,0,1 We know…

Q: [₁ F. dr where C is oriented counterclockwise as viewed from above. F(x, y, z) = 2yi + xzj + (x +…

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Q: 2. A surface S in R'(z,y, =) is defined parametrically in the form u + -1 -( ) X(u, e) = 2u 2 (a)…

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Q: Use Stokes's Theorem to evaluate F. dr. In this case, Cis oriented counterclockwise as viewed from…

A: We are given a line integral and we have to use Stoke's Theorem to evaluate that integral.

Q: (ii) Use Stokes' Theorem to evaluate SF · dr, where F(x, y, z) = x²zi+xy²j+z²k and C is the curve of…

A: Given function is F(x,y,z)=x2z i^+xy2 j^ +z2k^ . Plane: x+y+z=1 Cylinder: x2+y2=9 The objective is…

Q: 1. Compute ff(V × F) - dS, where S is given by r² + y²+ (z – 1)² = 2 and z > 0, and F(r, Y, z) =…

A: Given: S is given by x2+y2+z-12=2  and z≥0 Fx,y,z=-esinzy,x+sinz,cosx

Q: Use Stokes's Theorem to evaluate F(x, y, z) = z²i+yj + zk S: z = √√16x2 - y2 6 F. dr. In this case,…

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Q: - Prove Stokes' Theorem by verifying it for F= [y,z,x] and S the paraboloid. %3D te CEY2=(x, y) = 1…

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Q: Use Stokes's Theorem to evaluate F(x, y, z) = z²i+yj + zk S: z = √4x² - y2 F. dr. In this case, C is…

A:

Q: 7. Let F(x, y,z)=-2yi +2xj. Use Stokes’ Theorem to find F dr where C is a circle (a) Parallel to the…

A: By stoke's theorem

Q: 5. Use Stokes theorem to evaluate J[V x F-ds over the surface of the paraboloid z = 5-x² - y² above…

A: If S  be   smooth surface bounded by a  closed , positively  oriented  smooth boundary curve C with…

Q: 14. Let II be the plane tangent to the implicit surface x? + y° – e*+2y = 4 + x sin(z) at the point…

A: According to question given that implicit surface x2+y3-ex+2y=4+xsin(z)Point are=(-2,1,0 )

Q: II3. S is the part of the paraboloid z = x² +y² below the plane z = 4 oriented upward and F(x, y, z)…

A: Given the surface

Q: The equation of the tangent plane to the surface f (x,y,z) = xecosz - z - 1= 0 at the point…

A: The answer is as follows :

Q: c) Use Stokes' theorem to evaluate + (zy – y) i – In z j+ (r + x*) k) · dr, where C is the…

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Q: Use Stokes's Theorem to evaluate EF. dr. In this case, C is oriented counterclockwise as viewed from…

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Q: Verify Stokes' theorem for F = z²i – y²j and the curve C is the square in the xz-plane from (0,0, 0)…

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Q: Use Stokes' Theorem to evaluate F• dr where C is oriented counterclockwise as viewed from above.…

A: Given:  F(x,y,z)= 7yi+xzj+(x+y)k To Evaluate: Concept: Stoke's Theorem: Curl:

Q: #2. Find an equation of the tangent plane to the given parametric surface at the given point (a) x =…

A: The posted question has multiple questions but according to guidelines we will solve the first…

Q: iv) Given F (x, y, z) = -yzį + 2xj + xyzk be a vector function that passes through a surface S of z+…

A:

Arc Length

Arc length can be thought of as the distance you would travel if you walked along the path of a curve. Arc length is used in a wide range of real applications. We might be interested in knowing how far a rocket travels if it is launched along a parabolic path. Alternatively, if a curve on a map represents a road, we might want to know how far we need to drive to get to our destination. The distance between two points along a curve is known as arc length. 

Line Integral

A line integral is one of the important topics that are discussed in the calculus syllabus. When we have a function that we want to integrate, and we evaluate the function alongside a curve, we define it as a line integral. Evaluation of a function along a curve is very important in mathematics. Usually, by a line integral, we compute the area of the function along the curve. This integral is also known as curvilinear, curve, or path integral in short. If line integrals are to be calculated in the complex plane, then the term contour integral can be used as well.     

Triple Integral

Examples: