A vector field oriented outward is given by 3 F(x,y, 2) = -y'i- 2xyj + yzk Suppose that the surface of the plane x +y + z = 1 in the first octant contained within triangle C with vertices (1, 0, 0), (0, 1, 0) and (0, 0, 1) transverse in anticlockwise direction. (i) Sketch the surface of the plane. (11) Evaluate $. F dr by Cusing Stoke's theorem. (ii)

Q: Compute the upward flux of the vector field Ĥ(x, y, z) = (x, 0, 2z) across the portion of the…

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Q: Find the flux of the vector field F = (y, z, x) across the part of the plane z = 4+ x + 3y above the…

A: The surface integrals compute the flux of fields across surfaces,                   So,…

Q: Confirm that the force field F(, y) = 2xy°i+ 3x²y² j is conservative in some open connected region…

A: Solution:- The given function is  F(x,y)=2xy3i+3x2y2jpoint P(-9,8) and Q(6,7)we have to find the…

Q: Find the flux of the vector field F = (y, – z, x) across the part of the plane z = 4 + x + 2y above…

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Q: 4. Find the positively oriented flux over the surface created by the paraboloid z = 16 – x² – y2…

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Q: Use Stokes' theorem to evaluate the work done by the force field F = ((f(x, y, z), 4.xy, x + yz) ,…

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Q: Compute the work done by the vector field F(x, y, z) = (,2, 2yz) when moving a particle: 1+? > a)…

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Q: A particle starts at the point (-1, 0), moves along the x-axis to (1, 0), and then along the…

A: Consider the provided information, The work done by force F in moving particle along the path C is…

Q: Compute the flux of the vector field F(x, y, z) = (2y, 3, –2xz) through the surface S, which is the…

A: Given vector field is, Fx,y,z=2y,3,-2xz. The given surface is, y=x2+z2  and  x2+z2≤1. Now for the…

Q: Calculate the flux of the vector field F(x, y, z) = 5i + 2j+ zk through the closed circular cylinder…

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Q: Find the work done by vector field F(x, y, z) = xi + 9xyj - (x + z)k on a particle moving along a…

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Q: .) Compute the flux of the vector field F = 4x + 4yj through the surface S, which is the part of the…

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Q: b) Show the force field M UTM UTM F(r, y, z) = (r²y + In(yz)) i+ UTM 3 + y cos(2z) ) j+ tial…

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Q: Compute the flux of the integral of the vector field F(z,y, 2) = (2,y, 2) through the half cylinder…

A: Given vector field is, Fx,y,z=x,y,z. It can be written as,  Fx,y,z=xi^+yj^+zk^. Flux of vector field…

Q: Find the work done by the vector field F = in moving a particle around the circle x2 +y2 = 4 from…

A: The given vector field is F=x-y, xy. The particle is moving around the circle x2+y2=4 from 2,0 to…

Q: Find the flux of the vector field F = (y, – z, x) - 2, - across the part of the plane z = 2 + 3x +…

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Q: 4. The vector field F(x, y) = (x² – y²)i+ 10-Rj-2zrk represents the velocity of a gas flowing in…

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Q: Find the flux of the vector field F = (y, - z, x) across the part of the plane z = 4+ 2x + 3y above…

A: The given vector field is, F→=y, -z,  x. The given part of the plane is z=4+2x+3y And the rectangle…

Q: Calculate the outward flux of the vector field F→ (x,y,z) = 2xi→ + yi→ +zk→ along the boundary of…

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Q: Find the work done by the force field F(x, y, z) = (x - y2, y – z2, z - x2) on a particle that moves…

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Q: |Compute the flux (just another name for the surface integral) of the vector field F = zk through…

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Q: Verify the Divergence Theorem in the calculation of the flux of the vector field: through the…

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Q: 1. Verify the Divergence Theorem in the calculation of the flux of the vector field: through the…

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Q: Find the flux of the following vector field across the given surface with the specified orientation.…

A: Let r be the vector on the surface of  the cylinder y=6x2 for 0 ≤ x ≤ 4​, 0 ≤ z ≤ 2​.Let's…

Q: Find the work done by the force field F(x, y, z) = (x − y², y - z², z - x²) on a particle that moves…

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Q: Consider the vector field F = (x²,xy, tan(x²y)) and the surface of the tetrahedron with vertices…

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Q: Why is the upward flux of a vertical vector field F = ⟨0, 0, 1⟩ across a surface equal to the area…

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Q: Calculate the work done in moving a particle in the force field F = 3x2î+ (2xz-y)ĵ+ z k along a…

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Q: 15. Verify Stokes' Theorem for the vector field F = 2xyi + xj + (y+z)k and the surface z = 4x² - y²,…

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Q: Compute the flux of the vector field F = xi + yj + zk through the surface S, which is a closed…

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Q: Compute the flux of the vector field F(x, y, z) = 3x + 3yk through the surface S, where S is the…

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Q: Q3 (a) Calculate the divergence of curl of vector F = e2z+y î + sin yz² j+cos yzx k, V· (V × F). (5…

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Q: Find the flux of the vector field F = across the part of the plane | z = 2 + x + 3y above the…

A: Given: F¯=y,-z,x z=2+x+3y The formula to determine the flux is: ∫S∫F·nds=∫D∫(-agx-bgy+c)dA Here…

Q: 15. Verify Stokes' Theorem for the vector field F = 2xyi + xj + (y + z)k and the surface z = 4 – x²…

A: please comment if you need any clarification. If you find my answer useful put thumbs up. Thank you.

Q: Find the flux out of the curve bounded by the top half of = arccos(√√²+² ) and y=0 with vector field…

A: As per the question we are given a bounded curve C and a 2d vector field F And we have to find the…

Q: Let F be the vector field (x², 2r, 2²). Let C be the boundary of Lv **---- the surface S given by…

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Q: Compute the flux of the vector field F = xi + yj through the surface S, which is the part of the…

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Q: Calculate the work done by the force field F(x,y, z) = [x² + 10z² i+(y² + 14x² ]j+ z² + 12y² ]k +…

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Q: The outward flux of the vector field F=(x,0, z) across the cylinder x + y = 1, for 0<z<1, is %3D…

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Q: The motion of a liquid in a cylindrical container of radius 2 is described by the velocity field…

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Q: Compute the flux of the vector field F(x,y,z)=5xj+2yk→through the surface S, where S is the part of…

A: Flux problem...

Q: If the vector field A = âx3x² + âyy – â̟5z³ is given, express à in cylindrical and spherical…

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Q: Find the surface integral F. dS of the vector field F(x, y, 2) = (-æ, -y, yz®) along the part of the…

A: The complete solutions are given below

Q: Use Stokes' theorem to evaluate the work done by the force field F = ((f(x, y, z), 5xy, x+ yz),…

A: As per the question we have to find the work done, which is basically a line integral of the vector…

Q: A vector field is given as F = â„2r² + âgr²Coso + âz5z. Verify Stokes theorem for this vector field…

A: Verify Stokes Theorem for the given function:

Q: A particle at (5, 0) traverses the upper semicircle x² + y² = 25 and then closes the loop along the…

A: Formula: The work done by the force F→ is given by, ∮CF· dr, where C is the path along the region.…

Q: A hot air balloon is shaped like a truncated sphere like the figure. The gases escape through the…

A: Given a hot air balloon has a truncated spherical shape. The velocity vector field  Vx,y,z=∇×Fx,y,z…

Q: Find the flux of the vector field F = (y,- z, z) across the part of the plane z = 4 + 3z + 2y above…

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Q: Compute the flux of the vector field F = 2xi + 2yj through the surface S, which is the part of the…

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Q: Find the flux of the vector field (y, x, z) in the negative z direction through the part of the…

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Statements and Logic

Logic is the analysis of general patterns of thoughts without regard to any specific sense of context.