Are the following statements true or false? ? 1. If F. dA = 12 and S is a flat disk of area 47, then div(F) = 3/r. S ? v 2. If F is a vector field in 3-dimensional space satisfying div(F) = 1, and S is a closed surface oriented outward, then F. dA is equal to the volume enclosed by S. v 3. If F is a vector field in 3-dimensional space, then grad(div(F)) = 0. ? v 4. If F and G are vector fields satisfying div(F) = div(G), then F = G. ? v 5. If F is a vector field in 3-dimensional space, and W is a solid region with boundary surface S, then F. dà = div(F) dV.

Q: The gradient vector field of for,y)= In +2y) is 2. T True A False

A: The gradient vector field of a function, is defined by the formula below. , where are the partial…

Q: Draw the unit cube D = {(x, y, z2: 0 ≤ x ≤ 1, 0 ≤ y ≤ 1, 0 ≤ z ≤ 1} and sketchthe vector field F =…

A: Given:The unit cube Vector field Objective:Sketch the vector field  on the six faces of the…

Q: 3. Use Gauss' theorem to evaluate IL (* + y + 2) do, where E is the surface of the ball x? + y? + z²…

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Q: 3. Consider the region D bounded by the curves C₁, C2, C3, C4 oriented as in the following picture:…

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Q: 3. The vector field F below is in the xy-plane and looks the same in all other horizontal planes. In…

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Q: The field F = xyi+yj - yzk is the velocity field of a flow in space. Find the flow from (0,0,0) to…

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Q: Suppose S is the surface where x? + y? + 3z² = 1. Let F be the vector field F = (xy, yz, z?).…

A: suppose S is the surface where x2+y2+3z2=1 let F be the vector field Fxy, yz , z2 claim- evaluate…

Q: 3. Let F = xzi+yzj+xyk be a vector field and S be the part of the sphere x² + y² + z² = 9 that lies…

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Q: Determine the following value: F20 + F30 1. 2. F34 3. F67

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Q: v.(OF = (vo) - F+o(V .F (V) F+

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Q: 4. Show that the line integral of F = [0, z] around a closed curve in the r- y plane, oriented as in…

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Q: Given the vector function f = x²i+yxj + z³k, evaluate the following integrals after you have…

A: (a) In first part we can use divergence theorem, which. states that where S is a closed surface…

Q: a) Show that F is conservative while G is not. b) Evaluate GdR, where C3 is defined by R(t) = tî +…

A: This is a problem of vector calculus.

Q: Problem 9. Let F(r, y) = (ry,x²) be a vector field over ry-plane. Use Green's Theorem to evaluate…

A: We need to use the Green's theorem to evaluate the integral. Use the concept of Green's Theorem…

Q: Let vector field F =(x,y,z²) across the surface S which is an upper hemisphere x²+ y² + z² = 4 . 1)…

A: Let vector field F =(x,y,z²) across the surface S which is an upper hemisphere x²+ y² + z² = 4 . 1)…

Q: If a vector field F (x, y, z) = (y² + z², xe cos z, - xe³ cos z), which of the following values is…

A: Given vector field is Fx, y, z=<y2+z2, xey cosz, -xey cosz>.   To Find: div F.

Q: The vector field F is shown in the xy-plane and looks the same in all other horizontal planes (in…

A: Step 1:The vector fieldF is in xy planeThe Fis independent of z.Step 2:The divFis calculated as…

Q: - Let F: R3 → R³ be a vector field F = (G, H, F) and let w = F dxdy + G dydz + H dzdx. Show that dw…

A: Step 1: Step 2: Step 3: Step 4:

Q: P. Consider the following vector.field F(x.y).and compute the line integral over C, where C is an…

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Q: Let F be the vector field (-2 sin(2x - y), sin(2x - y)). Find two non-closed curves C₁ and C₂ such…

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Q: Let vector field F=(xz,-y,z) and surface, S is a cylinder x² + y² = 4 for 3≤z≤0 with the outward…

A: According to Stoke's theorem If S is a smooth surface  bounded by a closed, smooth boundary curve C…

Q: Compute the flux of the vector field F = across sphere a? + y² + z² = 9.

A: Let F be a vector field and S be the boundary of the surface D. To find the flux of the vector…

Q: to be careful to choose a parametrization r(u, v) that determines the correct counterclock- wise…

A: As per the question we are given a statement regarding the Stoke's Theorem and we have to say if it…

Q: For any positive number k, an immersion z: MC² defined by x(u, v) = (u, k cos v, v, k sin v). Then:…

A: THE SOLUTION IS ATTACHED BELOW. HOPE THIS WILL HELP YOU. ALL THE BEST :)

Q: (a) State True or False and proved justification for: If S is an ellipsoid and F is a constant…

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Q: Is the following a vector space? Let 'V' = 'R'. Defined for 'j', 'k' residing in 'V' and 'm'…

A: Given that V=R, defined for j,k residing in V and m residing in R for the following operations:…

Q: 1. The flow rate of a fluid is described by F in units of kg/(m²-s). It is flowing through a…

A: The given vector field is defined as Fr=5xy2+z25yy2+z25zy2+z2. Calculate the rate of mass flow in…

Q: d) Verify the Stoke's theorem for vector field E = xyx – (x² + 2y²)ỹ by evaluating the following: i.…

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Q: A vector field is given by F = (2y + 3)i + xz j + ( yz -x)k, evaluate ∫ ?. ?? c

A: Given: F→=(2y+3)i⏞+xzj⏞+(yz-x)k⏞ C is the curve given by r→(t)=2ti⏞+tj⏞+t3k⏞ where t varies from a=0…

Q: etermine the equation of your model that will pass 2,4

A: We already have the equation of polynomialand