Why does a two-dimensional vector field with zero divergence on a region have zero outward flux across a closed curve that bounds the region? Choose the correct answer below. O A. Since dx* dy = 0, PF. = 0 by the flux form of Green's Theorem. dg O B. Since = 0. PF•nds = 0 by the flux form of Green's Theorem. dy dg OC. Since of = 0. PF•dr= 0 by the flux form of Green's Theorem. O D. Since of dg = 0. PF•dr=0 by the flux form of Green's Theorem. dx' dy

Q: What would you call the vector field V = yī – xj Solenoidal Solid Cylindrical Symmetric Irrotational

A:

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

A: Given, F=x+y,x2-yℝ=x,y:y2≤x≤18-y2 By the general theorem circulation, ∮Pdx+Qdy=∬Qx-PydAWhere,…

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

A:

Q: (a) Given the vector field F = xy + 2yzý + 3zx 2, calculate the line integral around the closed path…

A:

Q: Compute the flux of the vector field F = xi + 5yj through the cylindrical surface as shown in the…

A:

Q: 301). vector Field ż F(x,y, z) Fas4xey coy),2xとこ> 5 i). Is the field conservetve. Cakalate crl of F…

A:

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: 11. Suppose that E is the solid upper hemisphere of radius a > 0, whose base is a circular re- gion…

A:

Q: Find the outward flux of the given field across the given cardioid. 5x i+ (e*+5 tan 1ly)j 1 + y2 F =…

A: The following formula from Green's theorem is used to find the outward flux, outwards flux=∮F·n…

Q: Use the Divergence Theorem to compute the net outward flux of the vector field F across the boundary…

A:

Q: Is the two-dimensional vector field V = -2y 2x a possible velocity field x²+y² 'x² +y² for an…

A: For a two dimensional incompressible for a vector field V=ui+vj. ∂u∂x+∂v∂y=0

Q: Given a scalar field f(x, y)= xy + x² and C is the curve defined by the equation y =-x². Calculate F…

A: Given scalar field  fx,y=xy+x2 and C is the curve defined by the equation y=-x2.

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

A:

Q: Given that F(x, y, z) = (ey, xey + ez, ye²) is a conservative vector field. a. Find a potential…

A:

Q: a) Is the vector field F = (a2 - yr, y?- xy) path independent (conservative)? %3D O True O False b)…

A:

Q: 8. Given the vector field G(x, y, z) = e>+2z (1 + x j + 2xk) : a) Show G(x, y, z) is conservative…

A:

Q: Consider again the vector field F = (x +y)i + (x² + y²)j_what is the downward flux across the line…

A:

Q: Use the divergence theorem to find the outward flux F ndS of the vector field F = cos(8y + 8z) i + 4…

A: …

Q: 2) Given the vector field F = (- Vx²+y² 2z2) write the integral to calculate the outward x²+y² flux…

A: We have to find the outward flux across the cylinder x2+y2=100.

Q: (a) Is the vector field F(x, y) = (xe³, ye²) conservative? Justify your answer. If it is…

A:

Q: 1. Determine the velocity field v so that the surface F(r, y.t) = (y+t)+ (z+t)? - a² = 0 to be a…

A: The given surface is - F(x,y,t) = (y+t)2 + (x+t)2-a2.......(1) The velocity field (V→) components…

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

A:

Q: Consider the semicircle r(t) = a cos(t)i + a sin(t)j with 0 0. For a given vector field F, the flux…

A: Given  rt = acosti+asintj, 0≤t≤π & a>0 for a given vector f the flux across r(t) = ∫CF.Nds…

Q: (a) Suppose that F is an inverse square force field, that is ,  F (r) =cr/|r|3 for some constant c,…

A:

Q: For the vector field: F = (x + y)i + (y + z)j + (z + x)k 1) Compute the flux out of the cube bounded…

A:

Q: Consider I = ∫CF⋅dr, where (img17) is a conservative vector field and curve C is parameterized by:…

A:

Q: For vector field F(x, y, z)=(1+ 9xry, 3x +e, ye' +2= (a) Carefully calculate curl F. (b) Find the…

A: We have to calculate curlF  and total work

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

A:

Q: Q6: Find the outward flux of the vector field F=(x²y+e'sin(y))i+ (x²+ e*cos(y))j through the right…

A: Solution

Q: Consider the vector field F = (a*y°, æ°y*) Is this vector field conservative? Select an answer If…

A:

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

A: Ans :A)

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

A:

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

A:

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

A:

Q: 1).Evaluate ∮CF•dr, whereF=⟨e^x−y^3,cosy+x^3⟩, and C is a circle of radius 2, centered at the…

A:

Q: Consider the vector field F=(x^2+y^2,10xy). Compute the line integrals ∫c_1 F⋅dr and ∫c_2 F⋅dr,…

A: Given:

Q: 2. Find the surface integral SS. ř · dS for the vector field F = xyî + yzĵ + zxk where S is part of…

A: To find Surface integral ∬SF→·dS→: The Surface S is of the form z=g(x,y) and is oriented upwards.…

Q: Consider again the vector field F = (x + y)i + (x² + y² )j. What is the downward flux across the…

A:

Q: 2) Given that the total flux across the surface, S of the tetrahedron bounded by the planes x = 0, y…

A:

Q: 10. Use the Divergence Theorem and the vector field F = to compute the outward flux from the region…

A: Here we will evaluate the outward flux.

Q: For what values of a & b such that the vector field f= (ay'e" + xz)i +(ye" +z')}+| byz' + is…

A: f is given. We find a,b so that f is conservative (irrotational).  f is conservative (irrotational)…

Q: This is a two part problem. Compute the flux of the vector field F = 2x^2y^2(k) through the surface…

A:

Q: A vector field A is given by A = V×Fwhere F = (î+ĵ+k)eVæ²+y²+z². Consider an unknown surface that is…

A:

Q: 6) Given a hemisphere bounded by S,: z = /25 – x2 – y? and S2:z = 0.Find the outward flux of the…

A: The flux of a vector-valued function F(x,y,z)=F1i+F2j+F3k over a surface S is defined as ∫SFdS.…

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: Given the following curve: 7: [0, 1] → R? kt² tHy(t) = 3t2 +2 and the following vector field: R? →…

A:

Q: Jse a CAS and the divergence theorem to compute the net outward flux for the vector field across the…

A:

Q: - Here is a graph of the vector field F(r, y) = r²i+ y°j. 2 (a) Calculate div F. (b) For which…

A:

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

A: Counter clockwise  circulation on the region R  for the vector field F=M,Nis given by…