Evaluate fs F dS, where F(r, y, z) = (2², xy, z) and S is the part of the paraboloid z = 22 + y? below the plane z = 1 with upwards orientation. 2 =

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: Calculate F· dS where F = (4x*z, 4y°z, 3:*) S is the surface of the solid bounded by the hemispheres…

A: Given

Q: Given F(x, y, z) = -yzi + xzj – xyzk be a vector function that passes through a surface S of z + x2…

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Q: a) Evaluate x²y? ds, where the curve C is the intersection between the surface z = := V1– y? and the…

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Q: Evaluate e dV where E is enclosed by the paraboloid z = 5+x +y, the cylinder a2 + y = 1, and the xy…

A: The objective is to evaluate ∫∫∫EezdV, where E is enclosed by the paraboloid z=5+x2+y2, the cylinder…

Q: e dV where E is enclosed by the paraboloid z =1+ a? + y", the cylinder a? + y = 3, E Evaluate and…

A: We will use triple integration here

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

A: Given Data: Integral: ∬SF→·dS Vector: F→=exsiny, excosy, yz2 Planes: x=2, y=4, z=3 The divergence of…

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: (a) Compute directly I = S Ss F-dS, where S is the surface of the sphere x? +y? +z? = 4 and F =…

A: X²+y²+z²=4=2² Therefore radius of sphere is 2.

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: Find the sum of maximum and minimum values of the function f(x,y, z) = 8x + 16y + 24z on the elipse…

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Q: F = (x² – y²,x, 2yz); C is the boundary of the plane z = 6 – 2x – y in the first octant.

A: Given, F=x2-y2,x,2yz Find the line integral ∮CF·dr in the following. Here C is the boundary of the…

Q: Calculate F. dS where F = (3ry, xe", z*) S is the surface of the solid bounded by the cylinder y? +…

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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: Use divergence theorem to evaluate F •dš where F(x,y, z) = (2x + yz)ī + 2yzj – z?k and S is the…

A: According to the divergence theorem ∬SF¯·ds¯=∫V∇·F¯dV, where V is the volume bounded by the closed…

Q: Find of and Duf to :- f cas,z) = In (x²+zy?+32) atp(-v34) in the direction of ã=-3i-4j-12K. -314…

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Q: Calculate F. dS where F = (4x°z, 4y*z, 32*) S is the surface of the solid bounded by the hemispheres…

A: Given that F→=(4x3z, 4y3z,3z4) and S is bounded by z=16-x2-y2, z=1-x2-y2, z=0 Now,…

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

A: Given : F=ex sin y,ex cos y,yz2x=1;y=2 ;z =5step 1:By using divergence…

Q: Calculate F. dS where F = (4r*z, 4y°z, 324) and S is the surface of the solid bounded by the…

A: Given - F¯ = 4x3z, 4y3z, 3z4 and S is the surface bounded by the hemispheres z = 16 - x2 - y2, z = 4…

Q: Let F = . Use Stokes' Theorem to evaluate curlF · dS , where S is the part of the paraboloid z = 14…

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Q: CALCULATE S SS F(x, y, z)dxdydz WHERE: T f(x, y, z) = x AND "T" IS THE SOLID DELIMITED BY THE…

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Q: Q2. Use the Stokes' Theorem to find fF.dr where F(x,y,z) = and C z=9-x²-y² on the xy plane oriented…

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Q: V9/ Compute I ,cxyz), uhere s is the part f the plane +x that lies inside the half - cyllnder…

A: Given: The Integral is ∬SxyzdS. The surface is the part of the plane S  z=y+x that is inside the…

Q: The flux of F(x, y, z) = 2³î – r³}+ y®k, where o is the sphere | x2 + y? + z2 = a² with outward…

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Q: Evaluate e dV where E is enclosed by the paraboloid z = 6 + x² + y°, the cylinder 2? + y? = 5, and…

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Q: Find the flux of F(x, y, z) = (z, 4x, y) through the surface S, when S is the part of the plane 2x +…

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Q: Find the circulation of F= (xy)i+(yz')j-(2xyz)k around the curve C in the counterclockwise direction…

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Q: Evaluate the surface integral.

A: We need to find the surface integral of

Q: Evaluate € fff.₂²². e² dV where E is enclosed by the paraboloid z = 3 + z² + y², the cylinder z² +…

A: We need to find integral.

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

A: We have to find double integral of F. ds

Q: A parametrization of the portion of a surface z=x²+y2 below the plane z= 16 is given by 7(u,v)=…

A: Since you have sent more than one question so we can just answer one of them because of our…

Q: Calculate F- dS where F = (3ry², re², z³) S is the surface of the solid bounded by the cylinder y? +…

A: The general form of a vector-valued function in three dimensions is given by F¯=F1,F2,F3, where F1,…

Q: Evaluate e dV where E is enclosed by the paraboloid z = 2 + x² + y², the cylinder æ? + y² = 5, and…

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Q: Find the sum of maximum and minimum values of the function f(x, y, z) = 8x + 16y + 24z on the elipse…

A: To determine the maxima and minima of the function f(x, y, z) with respect to the function is…

Q: Calculate F. dS where = (4e°z, 4y°z, 324) F: S is the surface of the solid bounded by the…

A:

Q: Evaluate e dV where E is enclosed by the paraboloid z = 4+x² + y°, the cylinder x² + y? = 3, and the…

A: Here we have to evaluate ∫∫∫E ez dV , where E is the enclosed by the paraboloid z=4+x2+y2 , the…

Q: 5) 2z dV, E is the solid enclosed by the cone % = Vr2 + y? and the E plane z 5

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Q: Find the sum of maximum and minimum values of the function f(x, y, z) = 4x + 8y+ 12z on the elipse…

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Q: Evaluate J| 6 xyds whiese Sis the postrom of th e intsont Of the Yz-plant S plane x fyt2=1 that lies

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Q: Let E denote the part of the plane x + y + z = 1 that is inside the cylinder r? + y? = 1. Compute…

A: We have to find the value of  ∬ F→.dS→

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

A: We need to find the surface integral. We will need basic knowledge of parametrization of vector…

Q: Calculate F. dS where F = (42*z, 4y°z, 3z*) /25 – x² – y², z : 16 — г? — у? = Z 2 S is the surface…

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Q: C is the intersection curve between the paraboloid z = x² + y? and the plane 6x + 8y – z = 21,…

A: Strokes theorem:

Q: Find the flux of F across S, where F(x, y, z) = xyi + y²j + zk and S is the part of the paraboloid z…

A: According to question given that F(x,yz,)=xyi^ + y2 j ^+z k^S is the part of the paraboloid z=x2+y2,…

Q: Use Stokes' Theorem to evaluate ∫CF⋅dr where F(x,y,z)=xi+yj+3(x2+y2)k and C is the boundary of the…

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Q: Bounded by the paraboloid z = 10 – 3x? – 3y2 and the plane z = 4 (using polar coordinates)

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Q: Evaluate e dV where E is enclosed by the paraboloid z = 5 + x + y°, the cylinder + y = 1, and the ry…

A: To Determine: evaluate ∫∫∫eZdV  where E is enclosed by the paraboloid z=5+x2+y2,the cylinder…

Q: yz ds where o is the part of the plane x + y+ z = 1 that lies in the first octant.

A: Given The given integral ∫∫σyzds over the region x+y+z=1 that lies in the first octant can evaluated…

Q: Find SS F dS where F(x, y, z) = z i+(3x – 1)j+ yk and S is the portion of the plane x + y + z = 1…

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