Let y be the boundary of the parallelogram formed by the = (1, –1,0), C = (2,0, 2) and D = (1, 1, 2) with points A = (0,0, 0), B orientation %3D %3D А В А. Let F (x, y, z) = (2xz – Theorem, find L, F dr =? 2y, z + x + sin(y²), 2z + x² + 2y). Using Stokes'

Q: 5. Consider the surface: x(u, v) y(u, v) z(u, v) where u € [0, 1] and v € [0,2]. Calculate the area…

A: Introduction: The area is the area occupied by a two-dimensional flat surface. It has a square unit…

Q: A region Rxy, in the XY plane, is shown below: A) 1- JJRa 2r + 3y – 6 = 0 2r +3y+1 0 Rry %3D…

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Q: 4. Find the surface area of the paraboloid S defined by the equation: (r, y, z) = (u cos v, usin v,…

A: We will find surface area by using formula.

Q: |Use Green's Theorem to evaluate [(3x² +y)dx+4xydy where C is the triangular C region with vertices…

A: The solution is given as

Q: Find the Jacobian a(x, y, z) a(u, v, w) for the indicated change of variables. If x = f(u, v, w), y…

A: x=ρ sinφ cosθy=ρ sinφ sinθz=ρ cosφ

Q: Let C be a simple closed smooth curve in the plane 2x + 2y + z = 2, oriented as shown here. Show…

A: Given, The equation of the curve is 2x+2y+z=2. The closed integral is ∮C2y dx+3zdy-x dz Let…

Q: 4. Use Stokes’ theorem to evaluate fa F. dr, where F(x,y, z) = (xy,yz, zx), and C is the triangle…

A: I am going to solve the given problem by using some simple calculus to get the required result.

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Q: Sketch the region enclosed by the given curves. y = 3 cos(5x), y= 3 – 3 cos(5x), 0 < x< 1/5 y y 6- 2…

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Q: Determine a region of the xy-plane for which the DE (1+ y) y' = x²will have a unique solution whose…

A: Theorem of existence of uniqueness: If f(x,y) and dfdy are continuous on rectangular region R, then…

Q: Calculate the line integral fe F. dr, where F = (ry, x² + y² + z², yz), and C is the boundary of the…

A: Given F→=xy,x2+y2+z2,yz and C is the boundary of the parallelogram with vertices…

Q: 3.use Greens theorem S(1+x dx + 2xy dy ,where C is triangle With vertices (0,0),(1,0).(1,3) oriented…

A: "Since, you have asked multiple questions, we'll solve the first question for you. If you want any…

Q: 4. Use Green's Theorem to evaluate the line integral | 2xydx+ xy'dy, where C is the triangle with…

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Q: Integrate G(x, y, z) = y + z over the surface of the wedge in the first octant bounded by the…

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Q: 2). Find the flux of F(x, y, z) = (0, yz, z) outward through the surface S cut from the cylinder y +…

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Q: Evaluate . F dr where F(x,y,z) = (cos x, -2x, y + sin(z³)) and C is the boundary of the triangle…

A: This is a problem of surface integral.

Q: Let R be the trapezoid with vertices (1,-1), (3,3), (3,-3) and (9,9). Give a change of variables to…

A: Given that ,  R be the trapezoid with vertices (1,-1) , (3,3) ,(3,-3) and (9,9) . Let , the vertices…

Q: a) Let G be the wedge in the first octant that is cut from the cylindrical solid y² + z² ≤ 4 by the…

A: To Evaluate: ∫∫∫GxyzdV. Where G: y2+z2≤4, y=2x , x=0. As per policy, we are solving only thr first…

Q: Find the normal line for the level surface f(x,y,z) = 3x5 + cos (5y) + %3D In (5z) = c at the point…

A: Question is solved.

Q: Find the normal line for the level surface f(x,y,z) = 7x³ + cos (4y) + In (4z) = c at the point…

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Q: 2. Evaluate f, z³ /x² + y² + z² dV where E is the solid hemisphere that lies above the xy-plane and…

A: Solution:- equation of sphere having centre origin (0,0) and radius=1 x2+y2+z2=1…

Q: What is the center of gravity of the solid formed when the plane region bounded by y^2 = 4x in the…

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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 :…

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Q: Calculate F. dS where F = (e" sin y, e² cos y, yz²) S is the surface of the box bounded by the…

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Q: find volume 26. The wedge cut from the cylinder x2 + y? = 1 by the planes z = -y and z = 0

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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: The region shown is bounded by x = 3, y = 0, and y = x. y 1 1 2 3 4 If the region is rotated about…

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Q: Evaluate fF.ndS, where F-4xi-2y' j+z'k and S is the surface bounding the region x' +y' =4, and the…

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

A: Given: G(x,y,z)=xyz We have to integrate G(x,y,z) over the surface of the rectangular solidcut from…

Q: Calculate the line integral fe F. dr, where F = (ry, r? + y? + 22, yz), and C is the boundary of the…

A: Step:-1 Given that a parallelogram with vertices 0, 0, 1, 0, 1, 0, 2, 0, -1, 2, 1, -2 and F→=xy,…

Q: The solid in the first octant bounded by the coordinate planes, y+ z = 4, and x = y.

A: The objective is to find the volume for y+z=4,x=y The volume of the integral formula is,…

Q: 4. Use Green's Theorem to evaluate the line integral | 2xydx+ xy*dy, where C is the triangle with…

A: The given triangle C with the vertices (0,0),(1,0) and (1,2) is as as shown below.

Q: dA, where R is the trapezoidal region Cos 5. Use appropriate change of variables to evaluate x + y,…

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Q: Evaluate SSS z²dV, where R is the region given R by above the plane z=O and between the spheres 2. +…

A: A scalar function is a rule that is used to associate a number, or quantity with a particular point…

Q: Semiinfinite conducting planes o = 0 and o /6 are separated by an infinitesimal insu- lating gap as…

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Q: Let F(x, y, 2) = ryi+ yzj+ zak and C be the boundary of the part of the paraboloid z = 1- z² – y² in…

A: Given  F(x,y,z)... C is the boundary of the part of the parabloid.... Let us solve the given…

Q: The tangent plane and the normal line of the surface x = y? + z2 + l at (3, 1, -1) have the…

A: Solve the following

Q: Let Ci be the rectangle with vertices (0,0), (2,0), (0,1), and (2,1) oriented counterclockwise, and…

A: Let C₁ be the rectangle with vertices (0,0). (2,0), (0, 1), and (2,1) oriented counterclockwise, and…

Q: (b) Let F = (-yx², xy²). Let D be the annulus 1 < a2 + y² < 4, and let C be its boundary. Evaluate

A: Value of the given integral is

Q: 56 Consider points P and Q in the x-y plane, with P = (1,0) and Q = (0,1). The line integral %3D Q…

A: Line integrals

Q: 13. Let U be the region in C bounded by the hyperbolae {z € C: 2(Re z)? – (Im z)? = 1} and {z € C:…

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Q: #6. Use Stokes' Theorem to evaluate S. F dr, where F(x, y,2) = i+(x + yz)j+(xy-V) k and C is the…

A: Given that, Fx,y,z=i+x+yzj+xy-zk and C is the boundary of the part of the plane 3x+2y+z=1 in the…

Q: Evaluate |[A.ndS where A = 2xyi+yzj+xzk and S is the surface of the region bounded by x = 0, y = 0,…

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Q: Calculate F. dS where F = (e sin Y, cos y, yz?) S is the surface of the box bounded by the planes x…

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

A: Definition used:

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

Q: A circle C in the plane x+y+z=4 has a radius of 3 and center (3, - 1,2). Evaluate O F.dr for F=…

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Domain

In simple mathematics terms, a domain is defined as the space on which all the possible inputs for the function will lie. It can also be expressed as a set of departures for the required function.