Find the flux F(x, y, z) = (xy²,x²y, ln(x + y)) directed outwards through the surface of the solid limited or bounded by the graphs of x2 + y? = 4 and –1 < z < 1, that is, calculate fS FNds

Q: 6. Find the flux of F (x,y, z) = y cos(z) i – y x sin(x)j+xy³k through the outward oriented cube of…

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Q: Let v = (x²z, 1 – 2xyz – 3y + x²y, 3z - x²z) be the velocity field of a fluid. Compute the flux of v…

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Q: Let S be the closed cylinder bounded by the surface r2 + 2² = 4, and the planes y = 1 and y = 4. Use…

A: Let's find.

Q: The outflow of the field F(x,y,z)=(e^y−xz^2,x−yx^2,y−zy^2), across the edge of the solid contained…

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Q: Let i = (3 – 2æyz - - xe cos y, yʻz, e² cos y) be the velocity field of a fluid. Compute the flux of…

A: Given that V=3-2xyz-xezcosy,y2z,ezcosy And the surface x+y2+z2=1 It is bounded. Then the flux is,…

Q: Use the divergence theorem to calculate the surface integral      SF · dS;  that is, calculate the…

A: We use the divergence theorem to solve the integral

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: Let F (x, y, z) = and G be the solid | Let below the paraboloid % = 4 – x* – Y° and above the…

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Q: Use the Divergence Theorem to calculate the surface integral F. dS; that is, calculate the flux of F…

A: By the Divergence theorem ∬sF·ds=∭EF·ds The given function is, Fx,y,z=x2z3i^+2xyz3j^+xz4k^ Now…

Q: Let W be the three dimensional region under the graph of f(x, y) = exp[-2(x² + y)] and over the…

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Q: Use the Divergence Theorem to calculate the surface integralI F. dS; that is, calculate the flux of…

A: To use the divergence theorem to calculate the flux of F across the surface S. F(x, y, z) = 3xy2i +…

Q: Use the Divergence Theorem to evaluate SF. F. N dS and find the outward flux of F through the…

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Q: This means we can use the Divergence Theorem to find the flux  Let S be the surface of the portion…

A: Solution...

Q: 13. Calculate the flux of F across S.F = (3x, xz, z2) and S is the surface bounded by the paraboloid…

A: Given vector field is  F=3x,xz,z2 we have to find the flux through F where S is the surface bounded…

Q: Use the Divergence Theorem to find the flux of F(x, y, z) = (x – z)i + (y – x)j+ (z – y)k %3D -…

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Q: Let S be a simple closed surface in R' that encloses a solid region with volume equal to 5. Find the…

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Q: Let v = (4 xe cos y, y cos z, e cos y) be the velocity field of a fluid. Compute the flux of v - I…

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Q: Let S be the cylinder x2 + y2 = a2, 0&lt;=z &lt;=h, together with its top, x2 + y2 &lt;=a2, z = h.…

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Q: Find the flow of the velocity field F =(x + y)i - (x2 + y2)j along each of the following paths…

A: The velocity field F=(x+y) i -x2+y2j along each of the following paths from 1,0 to -1,0 in the xy…

Q: Let v (2, y sin z, cos z) be the velocity field of a fluid. Compute the flux of v across the surface…

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Q: Evaluate the flux integral F-ds where Pis the part of the paraboloid -= 4-x -y above the xy plane…

A: Any integral which is to be evaluated at the surface is called the surface integral. Divergence…

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: Let F(x, y, z) = z tan-1(y2)i + z3 In(x2 + 6)j + zk. Find the flux of F across S, the part of the…

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Q: Let F(x, y, z) = z tan-1(y2)i + z³ In(x² + 6)j + zk. Find the flux of F across S, the part of the…

A: Using divergence theorem:  ∬SF.dS=∭VdivF.dV…

Q: Use the Divergence Theorem to evaluate F.N dS and find the outward flux of F through the surface of…

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Q: Use the Divergence Theorem to find the flux of F(x, y, z) = 3x*i+ 3x?yj+ 23xyk across the surface of…

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Q: Let S be the triangular region in R³ with vertices at (1.0.0), (0.2.0). and (0.0.2) with upward…

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Q: Use the Divergence Theorem to evaluate SfF.N F. N ds and find the outward flux of F through the…

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Q: Let E be the solid region bounded by the upper half-sphere x2 + y² + z² = 4 and the plane z = 0. Use…

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Q: Let F(x, y, z) = z tan-1(y2)i + z3 In(x2 + 4)j + zk. Find the flux of F across S, the part of the…

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Q: Let F (x, y, z) = and G be the solid %3D | below the paraboloid % = 4 – x" – y° and above the…

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Q: Let  F(x, y, z) = z tan−1(y2)i + z3 ln(x2 + 10)j + zk.  Find the flux of F across S, the part of…

A: let, F(x,y,z)=ztan−1(y2)i+z3ln(x2+10)j+zk Find the flux of F across the part of the…

Q: Let F(x, y, z) = z tan¬1(y²)i + z³ In(x² + 9)j + zk. Find the flux of F across S, the part of the…

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Q: Let F(x, y, z) = (2x, y, –2z) be the velocity field of a fluid. (a) Calculate the surface integral…

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

A: Divergence Theorem states that  Where S is a closed surface, and E is the region inside that…

Q: Use the Divergence Theorem to calculate the surface integral F. dS; that is, calculate the flux of F…

A: It is already given step 1 as shown below.

Q: Let F(x, y, z) = z tan−1(y2) i + z3 ln(x2 + 8) j + z k. Find the flux of F across S, the part of the…

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Q: Find the flux of F=(x, y, z) across the solid that is enclosed by the hemi = 0. Assume an outward or…

A: The Gauss Divergence Theorem states that the volume integral of the divergence of a vector field A…

Q: The solid region E sits in the first octant, bounded above by the surface x² + y + z = 4. The vector…

A: Part (1):   The E is the solid region, now the volume is defined as dV = dz dy dx

Q: 4. Consider a soup can of height 2 and radius 3, whose base (bottom) lies in the r- y plane and is…

A: Given- Consider a soup can of height 2 and radius 3, whose base (bottom) lies in the x-y plane and…

Q: Use divergence theorem to find the flux of F(x, y, z) = x² yì – xy² j+ (z+ 2) k ; o is the surface…

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Q: JS S is the surface of the solid bounded by the cylinder x = y² and the planes x + z = 1 and z = 0.…

A: Here we will use divergence theorem to find flux of given vector field across S In  a simple solid…

Q: Find the flux of F = (3z + 1)k upward across the hemisphere x2 + y2 + z2 = a2, z&gt;= 0 (a) with the…

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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: Let E denote the solid cylinder bounded by x2 + y2 = 4, z = 0 and z = 3. Let F(x,y,z) = (x3 +…

A: Step:-1 Divergence Theorem:- ∬SF.dS=∭Ediv(F)dVNote:- S=∂E Now, given that  F(x, y, z)=x3+tanyz i…

Q: 1. Find the work done by the gradient of f (r. y) = (r+ y)? counterclockwise around the circle 2+ y?…

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Q: find the outward flux of F across the boundary of D. F = (6x + y)i - (x + z)j + 4yz k D: The region…

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Q: This question has several parts that must be completed sequentially. If you skip a part of the…

A: Given y=rcosθ and z=rsinθ

Q: Let F(x, y, z) = z tan-1(y2)i + z3 In(x2 + 6)j + zk. Find the flux of F across S, the part of the…

A: Given that    Fx,y,z=z tan-1y2i+z3Inx2+6j+2k Also, the part of paraboloid x2+y2+z=6 that lies above…

Optimization

Optimization comes from the same root as "optimal". "Optimal" means the highest. When you do the optimization process, that is when you are "making it best" to maximize everything and to achieve optimal results, a set of parameters is the base for the selection of the best element for a given system.

Integration

Integration means to sum the things. In mathematics, it is the branch of Calculus which is used to find the area under the curve. The operation subtraction is the inverse of addition, division is the inverse of multiplication. In the same way, integration and differentiation are inverse operators. Differential equations give a relation between a function and its derivative.

Application of Integration

In mathematics, the process of integration is used to compute complex area related problems. With the application of integration, solving area related problems, whether they are a curve, or a curve between lines, can be done easily.

Volume

In mathematics, we describe the term volume as a quantity that can express the total space that an object occupies at any point in time. Usually, volumes can only be calculated for 3-dimensional objects. By 3-dimensional or 3D objects, we mean objects that have length, breadth, and height (or depth).

Area

Area refers to the amount of space a figure encloses and the number of square units that cover a shape. It is two-dimensional and is measured in square units.