Let S denote the plane /2x – y + z = 1. By interpreting this plane as the graph of a function, we get the parametrization 7 (x, y) = (1, , X, Y, The surface area of S over the rectangle [1,2] × [0, 1] is A(S) VOdydz

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: Q19// A scalar filed V = xyz exists over a surface S defined by %3D x2 + y2 9 bounded by the plane…

A: given scalar field V=xyz exists over a surface defined by x2+y2=9 bounded by the plane x=0,y=0, z=0…

Q: Let S be the portion of the cylinder y = ln x in the first octant whose projection parallel to the…

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Q: Find an equation of the plane tangent to the surface xy z = 2 at the point (2, 1, -1). O A.x + 6y +…

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Q: Find the area of the surface 2x^(3/2) + 2y^(3/2) - 3z = 0 above the square R: 0<= x<= 1,…

A: find z.

Q: Let L denote the intersection line between the planes 2x-y-3z=4 and 2x+y-z=2 A) Determine a…

A: Solution of the system of equation will give the equation of the line of intersection.

Q: Find the flow of F=xzi-yk through the upper part of plane z-1 in the x +y² +z = 4sphere. %3D

A: The general form of a vector-valued function in three dimensions is given by F=F1i+F2j+F3k, where…

Q: Q2// A surface S is defined by x2 + z = 9 and the bounded by the planes y = 0, y = 1,x = 0,z = 0 in…

A: Given that V=xy2z And y=0, y=1,x=0,z=0 The given surface is x2+z=9 The integral formed is,…

Q: Integrate G(x, y, z) = xyz over the surface of the cube cut from the first octant by the planes x =…

A: Here, the surface is in the first octant. Therefore, the integration is

Q: (a) Evaluate ([ Á.ds where A=(x+y²)î-2xj+2yzk and s is the surface of the plane in the first octant…

A: For the detailed solution of the problem follow the next steps.

Q: Let S be the sphere x^(2)+y^(2)+z^(2)=4 with outward normal vector. Use the divergence theorem to…

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Q: IntegrateF(x, y, z) = z, over the portion of the plane x + y + z = 4 that lies above the square…

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Q: Consider the surfaces  z = x^2 +y^2   and z+ 2x+ 2y = 0 Find the projection of the curve on…

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Q: Evaluate the surface integral xyzdS where S is the part of the plane z = 1 + 2x + 3y that lies above…

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Q: Find the sum of maximum and minimum values of the function f(x, y, z) = 9x + 18y + 27z on the elipse…

A: Using Lagrange Multipliers with Two Constraints to find the maximum and minimum values of the given…

Q: A solid cube, 2 units on a side, is bounded by the planes x = +-1, z = +-1, y = 3, and y = 5. Find…

A: Given:

Q: Calculate the scalar line Integral where the function is f(x,y)= 1 / x²+y²+1 and the curve is the…

A: We have given function fx,y=1x2+y2+1 and the curve joining the point 0,0 1,0 1,1 0,1. Let A0,0,…

Q: Find the orthogonal trajectories on the sphere x2 + y2 + z2 = a2 of its intersection with the…

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Q: A solid G is bounded by the paraboloid z = 4 – x2 – y², the cylinder 22 + y? = 4 and the plane z =…

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Q: Let f(x, y) = 2x and C be the line segment on the plane from (0, 0) to (3, 4). Evaluate the line…

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Q: Find vohume fome above ZVy-xLy, below Z0(xy plane) inside xy/ 3.

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Q: Q3// A surface S is defined by 2x2 + z 1 and the bounded by the planes y = 0,y 2, x = 0,z = 0 in the…

A: A surface S is defined by 2x2+z=1 which is bounded by the planes y=0, y=2, x=0, z=0 in the first…

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) Between 1 and the disk x²+ (1+x² +y*) (b) Between % = 1 and the entire xy-plane. (1+x* + y?)*

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Q: Suppose that the plane tangent to the surface z = f(x, y) at (2, 7, 8) has equation z + 8x – 2y =…

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Q: Use Divergence theorem to find the outward flux of F = 2xz i - 2xy j – z2 k across the boundary of…

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Q: Find the centroid of the surface cut from the cylinder y2 + z2 = 9, z>= 0, by the planes x = 0…

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Q: Find a parametrization of the curve. The intersection of the surfaces z = x^2 − y^2 and z = x^2 + xy…

A: Given : The surfaces z = x2 - y2 and z = x2 + xy - 1.   To find the parameterization of the curve.

Q: Integrate G(x, y, z) = x - y - z over the portion of the plane x + y = 1 in the first octant between…

A: Given G(x,y,z)=x-y-z

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: Write down a parametrization r(u, v) for the portion of the plane x - y+ z = 3 that lies inside the…

A: r(u, v) = ( u cos v , u sin v,  3 - u cos v + u sin v)  Where u € [0, 1]           v€ [ 0,2π]

Q: Find the orthogonal trajectories of the conicoid (x + y)z = 1 of the conics in which it is cut by…

A: Given:The equation of conicoid is (x+y)z=1The system of the plane is x-y+z=K, where k is…

Q: use divergence theorem to find the outward flux of f =2xzi-3xyj-z^2k across the boundary of the…

A: By divergence theorem

Q: Find the average height of the hemispherical surface z = √(a2 - x2 - y2) above the disk x2 + y2 ≤ a2…

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Q: (b) Integrate (x, y,z)=(y+z)+1 the paraboloid 4x - y-z -0 by the plane x 1. over the surface cut…

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Q: Find the centroid of the polyhedron with four triangular faces in the first octant that is bounded…

A: The region is bounded by the three coordinate planes and the plane x+y+z=1. Thus, the region can be…

Q: Find an equation for the plane that is tangent to the surface z = vy-x at the point (1, 2, 1).…

A: To find an equation for the plane that is tangent to the surface at the given point (1,2 1).

Q: consider the curve formed by the intersection of the surface (x+z)^2+y^2 =9 and the plane x-2z = 0.…

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Q: 5. Compute the flux of F(r, y.2) =zi- 2yj+ 2ak across the surface of the cylinder bounded by a+y =9…

A: Solve the problem as shown below.

Q: find ⌠⌠F*dS if F(x,y,z)=2xi+2yj+2zk and S is the paraboloid z=8-x^2-y^2 lying above the rectangle…

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Q: Use Divergence theorem to find the ouward flux of F = 2xz i - 3xy j – z2 k across the boundary of…

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Q: the tangent plane of the surface [ x^2+y^2+z^2=14] at the point (1,3,2) O 3 x-6y + 4z = 28 2x+6y-4z…

A: Given problem:- the tangent plane of the surface [ x2 +y2+z2=14] at the point (1,3,2)

Q: verify Stokes § = (a%-y)i - yz'i over the uppea hazf SOD Face Of the sphere =1 bounded by the

A: Given that,  F→=(2x-y)i→-yz2j→-y2zk→ To verify Stoke's theorem: ∫CF→·dr→=∬Scurl F→·n^dS Here C is…

Q: Find a parametrization for the intersection of the surfaces: z = x2 + y2 and  x2 + 4y2 = 1

A: Find the parametric representation for the intersection of the given surfaces.

Q: Use Divergence theorem to find the ouward flux of F = 2xz i - 3xy j – z2 k across the boundary of…

A:

Q: Find the area of the part of the plane 5x + 4y + z = 20 that lies in the first octant. (Use…

A: Part of the plane equation   5x+4y+z=20⇒z=20-5x-4yPlane lies in the first octant , so x≥0 ,…

Q: A line intersects the xy plane at an x– intercept of N and a y intercept of F. Determine the volum…

A: Given:- A line intersects the xy plane at an x intercept of N and a y intercept of F.  To find :-…

Q: The intersection of the ellipsoid x2 + 4y2 + z2 = 4 with the plane z= 3x gives an ellipse C. A)…

A: Given that the ellipse C is the intersection of the ellipsoid x2+4y2+z2=4 with the plane z=3x.

Q: Find a parametrization for the line segment joining points (0, 2) and (4, 0) using the angle 0 in…

A: Given: The line segment joining (0, 2) and (4, 0). Do the construction as in below figure