A vector field with component form F(x, y, z) = (P(x, y, z), Q(x, y, 2), R(x, y, z)) on R³ is conservative if ӘР ӘР de dy' dz de dz ƏR ƏR and provided that certain “nice" conditions are met (which we assume). Let F(x, y, 2) = (e* cos y, –e² sin y, 2). (a) Show that F is conservative.

Q: Show that the vector field F(x, y, z) = xz i + xyz j - y 2 k can’t be written as the curl of another…

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Q: Find the divergence of the vector field F <7", 22°, zy* div F -

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Q: The potential of the conservative vector field F = (y,x,1> is Of=z+xy O f=x+yz O f=y+xz Of = xyz

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Q: Compute the flux of the vector field F(x, Y, z) = 6i + 5j + 3k through the rectangular region with…

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

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Q: If a vector field F (x, y, z) = (evz-, In (x² z), sin z), which of the following values is equal to…

A: We have to find div F(-1,0,1).

Q: . Consider the vector field F(r, y, 2) = (8r sin y, 4x² cos y – e"z, -e" – sec² z). %3D | F. dR…

A: For a vector field F→, the line integral about the path C is ∫CF→·dr. Let rt be the parametric…

Q: Consider the vector field F(x, y) = yî- xî+ zR and path joining (1, 0, 1) and (-1, 0, 1) as C;:…

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Q: 3. Verify the circulation form of Green's Theorem for the vector field F(x,y)=xyi +2y and the region…

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Q: Determine if the vector field F=(yz,xz,xy) is conservative?

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Q: Is there a vector field Ğ on R³ such that curl Ġ = (xyz,–y*z³,y*z*)?

A: the given function is, G→=xyzi^-y4z3j^+y3z4k^ If there exists a vector field G then the divergence…

Q: Use the extended divergence theorem to compute the total flux of the vector field F(x, y, z) = −3x^2…

A: The function given to us is F→=-3x2+2xz-y, 7y3-6y, 9x2+4z2-3x. We are asked to find the total flux…

Q: Find the gradient vector field (F(x, y, z)) of f(x, y, z) = In(4x + y + 62) . F(1, y, 2) =

A: we have to find the gradient vector field of f

Q: Determine whether the vector field…

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Q: What is the flux of the vector field (x2,y2,z1 ) through a cuboid with x ranging from 0 to 8.3, y…

A: use divergence theorem

Q: Given the vector field F(x,y,z) = [x, y, -2z] satisfies div(F) = 0. Find a vector field G(x,y,z)…

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Q: Verify that the Divergence Theorem is true for the vector field F on the region E. Give the flux.…

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Q: Compute the flux of the vector field F 3yi + 4j – 3xzk through the surface S, which is the surface y…

A: We apply formula for finding this

Q: Let ryz , (교2 + y"), 17 F = ( x2 +y? be a vector field on U = {(x, y, z) E R³ | (x, y) # (0, 0)}.…

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Q: Find the divergence of the vector field F. F(x, y, 2) = In(9x+ 8y² )i + 72xyj + In(8y² + 5z? )k #6…

A: Solution is done in step 2 and 3

Q: Find the gradient vector field (F(r, Y, 2)) of f(x, y, z) = In(4x + 6y + 22) . F(7, y, 2) = (

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Q: Verify that the Divergence Theorem is true for the vector field F on the region E. Give the flux.…

A: I have solution for 1

Q: For the vector field in three-dimensional space F = (4x°y – sin(x – 2))i+ e² v#j– yzk Evaluate…

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Q: Let C be any smooth simple closed curve with positive orientation that lies entirely in a plane with…

A: Find ∫f.dr as follows. ∫f.dr=∬∇×f.n…

Q: 4. (a) Show that the outward flux of the postion vector field F(x,y,z) = (x)7+ (y)] + (z) through a…

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Q: If F = Pi+ Qj+Rk is a vector field on R and P, Q. and R have continuous second-order partial…

A: Given:F=Pi+Qj+Rk is a vector field  in ℝ3To show that:∇.curl F=0

Q: The divergence of the vector field F(z, y, z) = (ya°, az", zy?) is: %3D

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Q: 5. (a) Verify Gauss' Divergence Theorem for the vector field F = yî+x²j– zk and the volume V given…

A: 5.(a)The Gauss' Divergence theorem is,

Q: Show that the vector field F = 2.xyi + (x – 2 y)j is conservative and evaluate the line integral %3D…

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Q: 2. Show that F is a conservative vector field and then use that fact to evaluate SF. dr along the…

A: We need to show F is conservative vector field.

Q: Determine if the vector field F(x,y,z)= (yz,xz+2y,xy+1) is conservative. If it is, find its…

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Q: Use the extended divergence theorem to compute the total flux of the vector field F(x, y, 2) 3D…

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Q: Consider the surface integral of the vector field F= [xy², 3x², x²z]over a surface S, where S…

A: From Gauss divergence theorem, we have ∬SF→⋅dS=∭VdivF→dV   The vector field is F→=xy2,3x2,x2z.…

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: Find the divergence of the vector field F. F(x, y, z) = In 5x²+ + 10xyj + In( 2y2 + 722)k div F(x,…

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Q: Show that the vector field F = 2.xyi +(x² – 2 y)j is conservative and evaluate the line integral…

A: Given the vector field

Q: Show that the vector field F = 2xyi +(x² - 2y)j is conservative and evaluate the line integral (F•dš…

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Q: Show that the vector field F(x, Y, z) = (x² + y²)i – 2xyj is not conservative on R³.

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Q: Consider the vector field F(x,y,z)=(−3x^2,8(x+y)^2,9(x+y+z)^2). Find the divergence and curl of FF.…

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Q: Consider the curve C from (-2, 0, 1) to (6, 5, 2) and the conservative vector field F(x, y, z) =…

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Q: Show that if the vector field F = Pi + Qj + Rk is conservative and P, Q, R have continuous…

A: Given: The vector field F=Pi+Qj+Rk is conservative P, Q, R have continuous first order partial…

Q: The vector field F(r, y) = (2xy +y) dx + (æ + 2xy) dy is conservative. %3D The function f(x, y) = x²…

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Q: Verify that the Divergence Theorem is true for the vector field F on the region E. Give the flux.…

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Q: Define a vector field in space as follows: F(x, y, z) = (x + 2z, y -cos z, x³ + 2z). Suppose E is…

A: Given  F→(x,y,z) =x+2z, y-cosz, x3+2z Then  div(F→) =∇.F→=i∂∂x+j∂∂y+k∂∂z.F→…

Q: Show that the vector field F(x,y,z)=yz^2i+xz^2j+2xyzk is conservative, find a potential function f,…

A: The given vector function is F(x,y,z)=yz2i+xz2j+2xyzk The given vector field F is conservative if…

Q: F is the three-dimensional vector field defined by F(x,y,z) = (x,y,z). In other terms, P(x,y,z) =…

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Q: B) Find the work done by the conservative vector field F (x, y, z) = yzi+xzj+xyk along a smooth…

A: Since, it is given that the field is conservative.

Q: Let C be the helix that winds around the cylinder x2+y? = 1 (counterclockwise viewed from the…

A: here i used parametric equation of helix and done simple calculation like dot product and after that…

Q: 3. Determine if the vector field F(x, y, z) = e" sin yzi + ze² cos yzj + ye² cos yzk is conservative…

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Estimation

A new framework of data analysis where the new statistical methods are used for interpreting the results and analyzing the data is known as estimation in statistics.