4. Let a > 0. Let o be cube, oriented by outward unit normal n, with edges of length 2a, centered at the origin, whose volume is 8a3 and whose eight vertices are given by (±a, ±a, ±a). Let F be the inverse square vector field where r = xi+yj+zk as usual. Evaluate the flux integral: SS. F · n dS using a method of your choice.

Q: Use the Divergence Theorem to compute the net outward flux of the field F = (2x,3y,2z) across the…

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Q: Consider the vector field Fˆ(x, y, z) = (y/(x^2 +y^2) i - (x/(x^2 +y^2)j (a) compute the curl (b)…

A: Hello. Since your question has multiple parts, we will solve the first three parts for you. If you…

Q: Evaluate the line integral fe F n ds, where the vector field F(x, y) = (tan-'(y) + x², x?y) and C is…

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Q: For the following vector field, compute (a) the circulation on and (b) the outward flux across the…

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Q: Given the following vector field and oriented curve C, evaluate F•Tds. C F = (x,y) on the parabola…

A: Given that, Fx,y on the parabola rt=4t,9t2 for 0≤t≤1. Obtain the r't. r't=ddt4t,9t2=4,18t

Q: Given the following vector field and orientated curve C, evaluate JF•Tds. (x.y) on the curve r(t) =…

A: Solution:

Q: Find the outward (normal away from the z-axis) flux of the field F = y2î + xzî – k across the cone z…

A: In this, we find the flux of the given function across the given cone z=3√(x^2+y^2).

Q: 2. Given that F(x, y, z) = (e", xe" + e, ye²) is a conservative vector field. a. Find a potential…

A: Here we find the potential function using the partial derivatives and integral as appropriate. Then…

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

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Q: Find the flux of the field F = xy i - z k outward (normal away from the z-axis) through the cone z =…

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Q: Given the upper hemisphere (S) with the equation: S{x² + y² + z² = 81² + 2822 (S): {{x² + y² + z² =…

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Q: I. Use the Divergence Theorem: (² F. ds = Ifv. Fav SSS E = to calculate the flux of the field F…

A: Divergence theorem uses  Double integration converts into triple integration.

Q: For the following vector field, compute (a) the circulation on and (b) the outward flux across the…

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Q: Given a scalar field ps = x2 + xy, calculate the surface integral over the region y <x², 0 < x< 1.

A: We need to find surface integral.

Q: Given that F(x, y, z) = (e", xe" + e², ye) is a conservative vector field. a. Find a potential…

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Q: Let f(x, y, z) = (x² + y² + z²)¬/2. Show that the clockwise circulation of the field F = Vf around…

A: Calculate: F=∇f   =∂∂xx2+y2+z2-1/2,∂∂yx2+y2+z2-1/2,∂∂zx2+y2+z2-1/2   =-x2+y2+z2-3/2x,y,z

Q: 12. Evaluate the surface integral JJ, F ·dS for the vector field F(x,y,z)=(-x,-y,z'), where S is the…

A: This is the problem of multivariable calculus, surface iuntegral.

Q: Let S be part of the surface z = 3+ 22+ y above the region a + y S 10, oriente %3D upward. Compute…

A: Given that S is the part of the surface z=3+x2+y2 above the region x2+y2≤10, oriented upward. Let us…

Q: Let o be sphere of radius a, centered at the origin, oriented by the outward unit normal n. Let F =…

A: Let σ be sphere of radius a , centered at the origin oriented by the outward unit normal n .Let  F→…

Q: Consider the vector of electric field E(x, y, z) = (sinh x)i – (cosh y)j – xyz k a. Compute the…

A: The given vector field is, E(x,y,z) = (sinh x) i - (cosh y) j - (xyz) k   the divergence and curl…

Q: Use the Divergence Theorem to compute the net outward flux of the field F=&lt;x,2y,-z&gt; across the…

A: From the Divergence theorem, for a positively oriented boundary surface (outward Normal) of region E…

Q: ) Verify Stokes' Theorem for the hemisphere S : x²+y²+z² = 9, z > 0, %3D its bounding circle C: 1?…

A: Stokes theorem  If a smooth surface S bounded by closed ,smooth boundary curve C oriented in…

Q: A spherical vector field is given by F(r,0,0)=2r cos(0) êr + 3r sin(0) ê +0(π- 0) ê. i) Compute the…

A: It is given that Fr,θ,ϕ=2rcosθe^r+3rsinθe^θ+θπ-θe^ϕ. Thus, F1=2rcosθ, F2=3rsinθ, and F3=θπ-θ.

Q: Evaluate the surface integral /F- ds for the veetor field F = (y. -r, 2:), where S is the hemisphere…

A: Surface integration

Q: For a line integral in two dimensions, select all of the following conditions that would indicate…

A: We have to find

Q: Use the Divergence Theorem to compute the net outward flux of the field F = (4x, y, - 2z) across the…

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Q: Consider the region D bounded above by the curve y = 3 – x2 and below by the curve y = x + 1 and the…

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Q: 12. Use a parameterization to find the flux across the surface cut from the cylinder x? + y? = 1 by…

A: Image is attached with detailed explanation.

Q: 2. Given that F(x, y, z) = (e", xe" + e*, ye*) is a conservative vector field. a. Find a potential…

A:

Q: Given the following vector field and oriented curve C, evaluate T ds. F = (x.y) on the parabola r(1)…

A: In this question we will find the line integral of F over c.

Q: Evaluate the surface integral ∬SF⋅dS for the given vector field F and the oriented surface S. In…

A: Given,

Q: For what values of a and d does the vector field F = ⟨ax, dy⟩ have zero flux across the unit circle…

A: A vector field, have zero flux. Across the unit-circle centered at the origin and oriented…

Q: Given the vector field F (x, y, z) = 2xyi + (x² + z?) j +2yzk a. Verify that the vector field…

A: Ans :A)

Q: Use the Divergence Theorem to compute the net outward flux of the field F = (x,yz) across the…

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Q: Evaluate the surface integral ff, F · dS for the given vector field F and the oriented surface S. In…

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Q: Compute the line integral of the vector field F = (4y, –4r) over the circle z2 + y? = 16 oriented…

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Q: Use the Divergence Theorem to compute the net outward flux of the field F = (-3x, y, 8z) across the…

A: Given:F=-3x,y,8zS=x,y,z:x2+y2+z2=6

Q: Given the following vector field and orientated curve C, evaluate F.Tds. (x,y) F = on the curve…

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Q: Let a vector field F(x,y)=sinx 1 + xy j be given. What is the integral representing the work done by…

A: Acoording to expert guidelines, I am allowed to answer first question only. Kindly repost the rest…

Q: Given the radial field (x, y, z) (x² + y² + z²)³/2 |r|3/2 Find the outward flux across the sphere of…

A: Image is attached with detailed solution.

Q: Use the Divergence Theorem to compute the net outward flux of the field F = (x,y,z) across the…

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Q: Compute the line integral of the vector field F = (4y, –4x) over the circle x² + y? = 36 oriented…

A: Given: F= &lt;4y,-4x&gt;          x2+y2=36

Q: For the following vector field, compute (a) the circulation on and (b) the outward flux across the…

A: Green's theorem(Circulation form): LetC is simple piecewise smooth curve oriented counterclowise…

Q: Evaluate the surface integral JJS ² F. ds for the given vector field F and the oriented surface S.…

A: Consider the vector function F = x i+y j+8 k Here surface S is the boundary of the region enclosed…

Q: Consider the surface S given by y=8−x2−z2, y≥0 oriented in the positive direction of the y axis. be…

A: We will find out the required value.

Q: Find the work done by the force field F(x, y) = (4 e* – y³ î + (2 cos(y) + x³ )j on a particle that…

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Q: Consider the 3-dimensional solid E in octant one bounded by = 2- y, y = 1, and x y.S is the surface…

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Q: | Let o be the surface of a cube whose sides have length two. Let o have vertices at the endpoints…

A: V→=xyi+2yzj+3zxkV→=xy, 2yz, 3zxTo evaluate flux integral, ∬σV→.ndS

Q: Let o be the sphere of radius a, centered at the origin, oriented by outward unit normal n. Let F=…

A: According to the divergence theorem, ∫∫SF.n^ dS=∫∫V∫divV dV Hence, the flux integral will be: Let…

Q: Use the Divergence Theorem to compute the net outward flux of the field F = (-x,4y,z) across the…

A: