Use the Divergency Theorem to evaluate F. Nds and find the outward flux of F through the surface of the solid bounded by the graphs of the equations: F(x, y, z) = (xy² + cosz)i + (x²y + sinz)j + e²k; S: z = √√x² + y², z = 2.

Q: Find the points (x, y) and the unitary directions w = (u, v) in which the directional derivative…

A: The directional derivative is maximum in the direction of gradient.That is DW(x,y) will be maximum…

Q: Evaluate f. F.dr, where F(r, y, z) = z?i+rj and C is the boundary of the square {(r, y, z)|0 < x <…

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Q: Determine the flux of F(x, y, z) = across the surface with an upward orientation. Let the surface…

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A: The flux of this vector field across the faces of cube on the coordinate planes (S1,S2,S3) is zero.…

Q: Let F(x, y) = 2x 2yi + 3xyj, and C be the boundary of the square with corner points (±1, ±1),…

A: The objective of this question is to find the flux of the vector field F across the boundary C of…

Q: Let F(x, y) = 2x 2yi + 3xyj, and C be the boundary of the square with corner points (±1, ±1),…

A: The objective of this question is to find the flux of the vector field F across the boundary C of…

Q: Let C be the circle centered at the origin with radius 4 oriented clockwise, and let F(x, y) = (2x +…

A: Given: The vector field across the circle C centered at the origin with radius 4 oriented clockwise…

Q: Consider the differentiable function f (x, y) = x2+ y2. Find an equation for a plane that •is…

A: Solution:  Consider the differentiable function f (x, y) = x2+ y2. Find an equation for a plane that…

Q: Use Divergence theorem to find the outward flux of F = (7x², et², cos³ (y² + x)) of the cube in R³…

A: Given that  and the cube  bounded by the planes The Divergence Theorem

Q: 1) Find the outward flux of F across the region D where F= (6x² + 2xy, 2y + x²z, 4x² y³ ) and the…

A: The vector field is,F¯=6x2+2xy, 2y+x2z, 4x2y3The region D is cut from the first octant by the…

Q: Q2: Find the component ofü = 2i +j-3k that orthogonal to v = 31 - ), then vernfy your result. Q3:…

A: As per the guidelines I am supposed to do first question among multiple questions. So I am solving…

Q: a) let f(x, y) = c, ceR and N = (3,5) x (1,4). calculate the area %3D of the graph of f :N→R. a): b)…

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Q: Use a parameterization to find the flux SF-n do of F=2²i+xj-3zk in the outward direction (normal…

A: Draw the surface and parametrize the surface:

Q: Use the Divergence Theorem to find the flux of F(x, y, z)=z³ i-x³j+y³ k across the sphere x² + y² +…

A: This is a problem of vector calculus.

Q: Let F = (2z, - x, 3y): Find the flux of F, that is, - dS, where S is the graph of the function z =…

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Q: Let C' be the circle centered at the origin with radius 3 oriented clockwise, and let F(x, y) = (−7x…

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Q: A scalar function F(x,y,z) = xy2 + x2z – z2y. Find the gradF at (1,1,1).

A: A scalar function is given. We have to find the gradient at a specified point.

Q: Use the Divergence Theorem to find the flux of F = (xy², x²y, y) outward through

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Q: If F(x, y, z) = –2i – 2æj – yk and G(x, y, z) = –2xi – yj – 2zk, find curl(F × G).

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Q: 1- a) c) −2ñi + (2ñ + 1)j b) 2πi + (2π − 1)j - 4пi – (2n + 1)j 2лi + (2π + 1)j 4ñi + (2ñ + 1)j Leave…

A: Given:The given function is .To Find:We have to determine the gradient vector of  at the point…

Q: Calculate the gradient, the divergence and the curl for the following fields, when possible: ₁ = e/y…

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Q: Let F = –9zi+ (xe*z – 2xe*² ) }+ 12 k. Find f, F •dÃ, and let S be the portion of the plane 2x + 3z…

A: Given, F→=-9zi^+xexz-2xex2j^+12k^ and S be the portion of the plane 2x+3z=6 0≤y≤4

Q: Use the divergence theorem to calculate the flux of F = (x - 9y)i + (y – 5z)j + (z – 3x)k out of the…

A: The divergence theorem mathematically states that:

Q: Find the flux of F= (y, z-y, z). across the tetrahedron with vertices (0,0,0), (1,0,0), (0, 1,0),…

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Q: The complex potential of a line vortex of strength k centred on the origin is W=iklnz. Determine the…

A: The objective of this question is to find the complex potential of two line vortices of the same…

Q: Let C be the circle centered at the origin with radius 1 oriented clockwise, and let F(x, y) = ( −6x…

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