Use Stokes theorem to evaluate V × F · dS over the surface of the paraboloid z = 5 – x² – y? above the plane z = 1, where - F = (y – 32) i+ (x² – 42) j+ (x + 3y) k %3D Hint: Stokes' theorem S Ss V × F · dS = ScF•dR. What is C in this case? 5.

Q: 2. Evaluate the triple integral //y o 2 = y, the xy-plane, and the parabolic cylinder y = 1- r2. y…

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Q: SSL. xy dV, where T is the first-octant solid bounded by the coordinate planes and the upper half of…

A: Here we have to find  ∫ ∫ ∫T xy dV , where T is the first-octant solid bounded by the coordinate…

Q: e Divergence Theorem to evaluate F. dS, where (2x + ey%)i + (y³ + x°z')j+(3y²z)k, and S is the…

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Q: Use Stokes' theorem to evaluate (V x F) - dS where F(x, y, z) = -6yzi + 6rzj + 5(x2 + y²)zk and S is…

A: Given that F(x,y,z)=-6yzi^ + 6xz j^+ 5(x2+y2)z k^ Calculating the curl,…

Q: z dv, where E is enclosed by the paraboloid z = x2 + y² and the plane z = 4. Evaluate

A: see 2nd step

Q: 2. Evaluate the triple integral ///_y dV; G is the solid enclosed by the plane z = y, the xy-plane,…

A: To evaluate ∭GydV,  where G is the solid enclosed by the plane z=y, the x-y plane, and the parabolic…

Q: 1. Let F(x, y, z) = yi+ xj+ yzk and the surface S be the part of the paraboloid z = 4 – x2 – y? that…

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Q: use a parametrization to find the flux f f s F . n dsacross the surface in the specified direction.…

A: Consider the function: Given   F=4xi+4yj+2kSurface cut from the bottom of the paraboloid z=x2+y2 by…

Q: 39. Find the surface area of the portion of the hyperbolic paraboloid r(u, v) = (u+ v) i + (u – v)…

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Q: 3) Evaluate the surface integral. I| (x²z+ y²z) ds where S is the part of the plane z = 4 + x +y…

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Q: The surface area of the part of z = 2 + V4 – x² – y² that lies inside the paraboloid z x² + y² is

A: This is a problem of vector calculus.

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

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Q: Verify Stokes' Theorem for F = xyî + yzĵ + xzk and where S is the surface of the paraboloid z = x²…

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Q: Evaluate the surface integral (2ayi – yzj + zk) · dS. Where S is the part of the paraboloid z = a2 -…

A: We have to evaluate the integral ∬S2xyi^−yzj^+zxk^⋅dS where S is the path of the paraboloid…

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

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Q: Find the flux of F (x, y, z) = 4.xi+ 4yj+ 2k outward (away from the z-axis) through the surface cut…

A: The field lines of a vector field F through surfaces with unit normal n, the angle from n to F is θ.…

Q: 6. Evaluate the triple integral 2x2z +2yz dV where S is the solid that lies below paraboloid z = -r?…

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Q: b) Given F(r, y, z) = (x³ + cosh z) i+ (2y - 3ry)j – (x² + 4y²z) k. Use %3D Gauss's theorem to…

A: 1. The divergence is an operator, which takes in the vector-valued function defining this vector…

Q: Find a parametric representation for the part of the cylinder y2 + z² = 36 that lies between the…

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Q: 2. Evaluate the triple in tegral where E is the y-2x2+2z20nd the plane y= 8. solid bounded by SI…

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Q: Calculate F- dS where F = (3zy', ze", z) S is the surface of the solid bounded by the cylinder y? +…

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Q: Evaluate /|/. x dV where E is the solid enclosed by the paraboloid x = y²+z² and the plane x = 9

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Q: Use Stokes' Theorem to evaluate F. dr where F(x, y, z) = xi + yj +Xx² + y² )k and C is the boundary…

A: Stokes theorem: ∮f→·dr→=∬S curl f→·n^ ds Given F(x, y, z)=xi + yj+9(x2+y2)k and C is the boundary of…

Q: Evaluate SſF-dš where F = yi +2x j +(z-8)k and S is the surface of the solid bounded by 4x + 2y +z =…

A: By using the divergence theorem we solve the above question

Q: Find the area of the surface. The part of the paraboloid z = 1 – x² – y2 that lies above the plane z…

A: Givenz=1-x2-y2z=-2Suppose the surface is defined byz=f(x,y) for x∈R, then the area of the surface is…

Q: Use Stokes' Theorem to evaluate curl F· dS. F(x, y, z) = x2z2i + y2z²j + xyzk, S is the part of the…

A: Given: F(x,y,z)=x2z2i+y2z2j+xyzk   and S is the part of the paraboloid z=x2+y2 that lies inside the…

Q: 3. Use polar coordinates to find the surface area of the paraboloid z = x² +y² – 5 that lies below…

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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: Find the area of the surface. The part of the paraboloid z = 1 – x2 - y? that lies above the plane z…

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Q: Use Stokes's Theorem to evaluate f. F dř where F(x, y, z) = (x- y) i+ (y-z)j+ (z - x) k and S is the…

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Q: Use Stokes' theorem to evaluate f curlF · ds. F(x, y, z) = y²zi+xzj + x²y²k S is the part of the…

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Q: É = ( 2x3 + y°, y3 + z³, 3y²z), S is the surface of the solid bounded by the paraboloid z = 1– x2 –…

A: We will find out the required flux.

Q: Use Stokes' Theorem to evaluate SSs curl F· dS. F(x,y,z) = x2z2 i + y²z² j + xyz k S is the part of…

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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: Use Stokes' Theorem to evaluate F- dr. F(x, y, z) = xyi + x*j+z?k where Cis the intersection of the…

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Q: Q– Fin the surface area of the portion of the paraboloid z= 6-2x - y that is above the x-y plane…

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Q: where F(x, y, z) = (ry, x²y, y) Use Gauss's theorem to calculate where is the paraboloid z = 4 – x²…

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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: 11. Evaluate ([(y² +z? )dS where S is the part of the paraboloid x=4-y -? that lies in front of the…

A: We will evaluate the given integral.

Q: Integrate G(x, y, z) = x2y2 + 4 over the surface cut from the parabolic cylinder y2 + 4z = 16 by the…

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Q: Jse Stokes' Theorem to evaluate F. dr where F(x,y, z) = xi + yj+ 2(x² + y²)k and C is the boundary…

A: answer is in next step

Q: The surface area of the part of the paraboloid z=x+y that lies under the plane z=2 is: Z=,

A: Surface area is given as A=∬D1+δzδx2+δzδy2dA

Q: The surface area of the part of z = 2+ /4 – x² – y? that lies inside the paraboloid z = x² + y? is

A: Given is the equation of a surface and a paraboloid. The objective is to find the surface area. The…

Q: se Stokes's Theorem to calculate (curl F) · n dS . 14) F = -2x2yi + 2xy²j + z°k; S is the portion of…

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Q: The wedge cut from the cylinder, x? + y² = 1 by the planes, z = -y and z = 0. Use a tripple integral…

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Q: 2. Evaluate the triple integral / || y dV; G is the solid enclosed by the plane z = y, the xy-plane,…

A: Solution is given below;

Q: 1.Find the area of the surface. a. The part of the plane 2x + 5y + z = 10 that lies inside the…

A: Note : As per our guidelines we can supported to answer only first three subparts

Q: Use Stoke’s Theorem to evaluate the line integral ∮C(z2−y2)dx +(x2−z2)dy +(y2−x2)dz, where the curve…

A: Given equation of the curve is ∮C(z2-y2)dx+x2-z2dy+(y2-x2)dz ,the equation of the paraboloid…

Linear Algebra

In mathematics, problems can be solved by the arithmetic operators like addition, subtraction, multiplication, and division. In algebra, we represent the numbers by any letter called a variable. If these variables are of degree 1, then it is studied under linear algebra. The first-degree equations involving algebraic expressions are called linear equations. That is, if you represent it by drawing a graph you will get a straight line.