7. Find the flux of F(z,y, 2) = ri+yj+zk through the portion of the paraboloid z = 1+z²+y° that is on or below the plane z = 5 with upward orientation.

Q: find the outward flux of F across the boundary of D. F =-2x i - 3y j + z k D: The upper region cut…

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Q: Let F(x, y) = 3 i + j + 2k, and let S be a square whose side lengths are 3, parallel to the…

A: Flux is F.dA Where F is the Field and A is the area

Q: Use divergence theorem to find the flux of F(x, y, z)=x²yî - xy²)+(2+2); o is the surface of the…

A: The given problem is to find the flux of the given vector field over the given surface bounded above…

Q: Compute the flux of F = 3(x + z)i +j+ 3zk through the surface S given by y = x2 + z2, with 0 0, z 2…

A: flux is determined as shown below

Q: Find the flux of F(x,y,z) = sin(xyz) i+ xy j + ze*/5 k across the part of the cylinder y? + 2 z? = 4…

A: According to question,Fx,y,z= sin xyzi +xy j + z ex5 k,equation of cylinder,y2 + 2z2 =4planes, x = 1…

Q: Integrate f(x,y) = √(64 - x2) over the smaller sector cut from the disk x2 + y2 ≤ = 64 by the rays θ…

A: Concept:   The calculus helps in understanding the changes between values that are related by a…

Q: 1. Use Green's Theorem to find the counterclockwise circulation and outward flux for the given filed…

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Q: 2. Show that the xy plane W = (x, y,0) in Rº is generated by: (i)u = and v = (ii)u : and v = 3

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Q: 3. Find the displacement u(r, t) in a vibrating circular elastic membrane of radius 1 that satisfies…

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Q: Compute the flux of F = 3(x + z)i +j+3zk through the surface S given by y = x2 + z2, with 0 0, z >…

A: From given data we see that the surface is oriented towards xz plane,  Hence,  d→s=dxdzj→

Q: Exercise 10.2 Evaluate the line integral fc(² – 2y²) ds, where C is the line segment from (0,0) to…

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Q: The outward flux of F(x, y, z) = xi + yj + zk across any unit cube is.

A: Given flux is F=xi+yj+zk. Compute div F. divF=∂∂xx+∂∂yy+∂∂zz=1+1+1=3

Q: Let S be the portion of the cylinder y² + z² = 4 with z > 0 between the planes r = 0 and a = 5…

A: To compute the flux of below function F across S F=x2,y2,z2

Q: 2. The task is to determine the flux of B across G, where G is the positively-oriented portion of…

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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: 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: For F(x,y,z) = (6y-4xz/(z2+1)2)i + (-2yz-6x/(z2+1)2)j + (-3/z2+1)k,  Calculate the flux of the disk…

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Q: Calculate the line integral F(x, y, z)=2xy^2 i+(2yx^2 +2y)j     on a path from (0, 0, 0) to(3, 1, 1)

A: We need to compute the line integral: F(x,y,z)=2xy2i+(2yx2+2y)j on a path from the point: 0, 0, 0 to…

Q: Find the outward flux of F through S. F-Ñ dS where F(z, y, z) = -3zî+y³j+yk, and S is the surface of…

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

Q: d) An open surface F is given by F = {(x,y, z) |z = 4 – x² – y² , z > 0} . Compute the flux of V…

A: V(x,y,z) = <2x+3y,2y+3x,-4z>F={ (x,y,z) : z=4-x2-y2 , z≥0}Consider, g(x,y,z) = z+x2+y2-4…

Q: Compute the flux of F = (5(z + 2), 5, 52) through the surface S given by y = ² + 2, with 0 0, z 2 0,…

A: We have to find flux.

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: 3. Calculate the rotational flow of F(x, y, z) = -j + xj + x²k, that goes through the cone given by…

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Q: EXERCISES 2.10 1. Use Stokes' Theorem to evaluate F.dr for E = (xz,-y,x²y), where C is the boundary…

A: Applying stokes theorem we solve it

Q: Find F.dS if F (x, y, z) = 2xi+2yj+2zk and S is t he part of the paraboloid z = 8-x²–-y? lying above…

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Q: Let F(x, y, z) = z tan-¹(y²)i + z³ In(x² + 3)j + zk. Find the flux of F across S, the part of the…

A: The given problem is to find the flux of vector field over the given surface of part of the…

Q: The tangent plane to the elliptic paraboloid z = 2x2 + y² at the point (1, 1, 3) is %3D 1) 8x + 6y +…

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Q: Compute the flux of F = 3(x + z)i +j+ 3zk through the surface S given by y = x2 + z2, with 0 <y<9, x…

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Q: Find the counterclockwise circulation and the outward flux of thefield F = (-sin y)i + (x cos y)j…

A: Use tangential form of Green's Theorem to find the counterclockwise circulation. Counterclockwise…

Q: 5. Find the flux of F (x,y,z) = y cos(z)i-yx sin(x) j+ xy³k through the outward oriented cube of…

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Q: Evaluate || F-n dS where S is the cylinder (including the disk on the bottom) x² + y² =1 for 0<z<1…

A: First, we plot the given surfaces.Cylinder : x2+y2=1; 0≤z≤1Paraboloid : 3x2+3y2+z = 4; 1≤z≤4.

Q: „ Compute the flux of F = xi + yj + zk through the curved surface of the cylinder x? + y² = 1…

A: Consider the vector field:

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: In R, let S be the piece of the plane 3x + 3y + 3z = 3 that lies in the first octant, oriented…

A: Given S is the piece of the plane 3x+3y+3z=3 that lies in the first octant oriented upward. The…

Q: 6. Compute the flux of F = [0, 0, z²] through the spherical surface S given as the upper hemisphere…

A: To find- Compute the flux of F→ = 0, 0, z2 through the spherical surface S given as the upper…

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: 7. Use the divergence theorem to find the outward flux of F= 3xy²i+ xe² j+ z° k across the boundary…

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Q: A cube is defined by 1 < x, y, z < 1.2. If D = 2x?ya, + 3x?y²a, -6z²a.C/m? (a) Apply Gauss's law to…

A: Consider the provided question, A cube is defined by 1&lt;x, y, z&lt;1.2 Given,…

Q: Let F(x, y, z) = z tan-(y2)i + z³ In(x2 + 9)j + zk. Find the flux of F across S, the part of the…

A: Get points of intersection of the paraboloid and z=3:  Thus, point of intersection is a circle in…

Q: 16. Use the Divergence Theorem to find the outward flux across the boundary of the region D where F…

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Q: Find the flux of F(x,y,z) = (-x,-y, z³) , across S. Where S is the part of the cone z = /x² + y²…

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Q: Let S be the cylinder x² + y² =a²2, 0≤zsh, together with its top, x² + y² sa², z = h. Let F = -7yi +…

A: Given : Surface S is the cylinder x2+y2=a2, 0≤z≤h, together with its top, x2+y2≤a2, z=h and  F→=-7y…

Q: Let F(x, y, z) = z tan-1(y²)i + z³ In(x² + 1)j + zk. Find the flux of F across S, the part of the…

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

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Q: (7) Determine the upward flux of V x F through the part of the elliptical cone C = {(x, y, z)2 =…

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Q: Let the surface Σ be the solid bounded by the paraboloid z “ 1 ´ x2 ´ y2 and the xy´plane.…

A: Use Gauss divergence theorem to compute the outward flux.

Q: Calculate the flux of F across G. F(x, y, z) = 2xi + 2yj + 2zk; G is the surface of the sphere x2 +…

A:

Q: 2. Use the Divergence Theorem to calculate the flux of F(x, y, z) = (x³ + y) i+(y³ + 2³) j+(z³ + x²)…

A: The given vector field is  F(x, y, z) = (x³ + y) i+ (y³ + z³) j+(x³ + x²) k across the surface of…

Q: Use Stokes’ Theorem to evaluate F(x, y, z) = 3zi + 4x j + 2yk; C is the boundary of the paraboloid…

A: Use Stokes theorem