True or False? Brief explanation. а. If f : R? →R is differentiable at all points, and f has a local minimum at the point A, then for any unit vector u, we must have Duf(A) > 0. b. Let S be a bounded, oriented surface in R. If F is a vector field in R and F|| > 0 at all points of S, then f F dS will be positive.

Q: Let S be the part of the cylinder x2 + y? = 16 lying between the planes z = 1 and z = -1. Let F be a…

A: Since you have posted multiple questions but according to guidelines, we will solve first question…

Q: 16) Vority Stokes' Theorem for the vector field Plaiy, 2)= 2?i - 2xg+y3k taking the surfuce S half…

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Q: 13. Using Stokes' theorem, evaluate the integral , curlF ds, where S is the portion of the…

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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: Evaluate  F · N dS  for vector field F, where N is an outward normal vector to surface S. F(x, y, z)…

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Q: Attached is a multivariable calculus problem.

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Q: curl F • ÑdS, where & is that part of the sphere 8, EXAMPLE Find I = x? + y? + (z – 2)² = 8 that…

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Q: Verify Stokes' theorem for the vector field F(x, y, z) = zì + xj+ yk taken over the half of the…

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Q: (b) Verify Green's Theorem for a vector field F(r, y) = cy'i – ya'j on a disc with center (0, 0) and…

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Q: Answer the following questions considering the following 3D vector field F: (i,j,k)=unit vectors…

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Q: Verify that Strokes' Theorem is true for the given vector field F and surface S. F(x, y, z) = yi +…

A: Given : Vector field F and surface S F(x,y,z)=yi+zj+xk Hemisphere ''S'' x2+y2+z2=1,y≥0

Q: Answer the following questions considering the following 3D vector field F: F(x,y,z) =…

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Q: Select all of the following curves Cand vector fields F for which Green's Theorem could be used to…

A: The chosen pairs were correct but you failed to include the first option. The correct answers are…

Q: O Attempt 2 Calculate the line integral of the vector field F = (7xy, x, y + z) around the boundary…

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Q: Identify the theorem: If C is the boundary curve of a surface S in 3-dimensional space, then the…

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Q: HIne whether or hot thé vector field is conservative. If it is conservative, find a function f such…

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Q: Evaluate F · N dS for vector field F, where N is an outward normal vector to surface S. F(x, y, z)…

A: Surface integral of a vector field: If F→x,y,z is a continuous vector field on an oriented surface…

Q: Let vector field E =(y,z,x) across the surface S which is a cylinder x + y = 16 with 0<z<2 in first…

A: Surface integration

Q: Consider the vector field F(x, Y, z) = 3x?yi+ (x³ + 1) j+9z²k. Let C be the circle cut from the…

A: Given vector field is  F(x,y,z)=3x2yi+(x3+1)j+9z2k C be circle cut from sphere x2+y2+z2=8 by the…

Q: Q : IF (x₁y, z) = x²; + y²j Divergence theorem to Calculate the integral the closed Surface shown in…

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Q: (a) Let S be the surface z2 = x² + y? bounded between the planes z = 0 and z = 2. Evaluate - x)dS. S…

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Q: Specify the component functions of a vector field F in R2 with the following property. At all points…

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Q: Let F= Wz,0, /x] be a vector field and C be the curve given by the parametrization r(t) = [cos²(t),…

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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: Verify Green's theorem for the vector field F = (xy+y²) i+(x- y) j y=x and y= r?. region R bounded…

A: Green's theorem

Q: Verify Stokes' theorem for the vector field F(x, y, z) = zî + xj + yk taken over the half of the…

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Q: Let f(x,y, z) x² + y² + z² < v?. = ax + By? + yz and S is the boundary surface of the solid sphere…

A: The function is fx, y, z=2x+6y2+15z. S is the boundary of the solid surface x2+y2+z2&lt;225, that is…

Q: Write T if true and F is false. Explain why with solutions, counterexamples, or explanations. a) In…

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

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Q: Let a vector field F(x,y)=e* i + xy²j be given. What is the integral representing the work done by F…

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Q: Let F= [x, z2, y] be a vector field and S be the closed surface consisting of the cylinder y2+z² =…

A: To calculate the below surface integral.

Q: Q3 Vegify Stokes theorem for the vector field x* +y'+z Vesty.2tokes theorem for the vector field F=…

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Q: Also, with the picture, what would be the flux going through the surface of just the cone?

A: Consider the provided question, (a)

Q: Consider the two-dimensional vector field :). F(x, y) = %3D xy + x + ey² Evaluate the line integral…

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Q: Let R be the disk: x²+y²<1. Then, the flux of the vector field F(x,y.z)= (x, y, 2z) across S, the…

A: The flux of the given vector field is: Flux=∫∫F.ndSThe given vector field is: F=&lt;x,y,z&gt;Let the…

Q: Let F = x'yi – xy²j+ (z + 2)k be a vector field. Use the Diver- gence Theorem to evaluate ff,F·ndS…

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Q: Evaluate the surface integral      SF · dS  for the given vector field F and the oriented surface…

A: Given - F(x,y,z) = y i ^- x j^ + 2z k^  S is the hemisphere x2+y2+z2 = 4 , z≥0 , oriented downward…

Q: Verify that Stokes' Theorem is true for the given vector field F and surface S. F(x, y, z) = -yi +…

A: Concepts introduction: Let S be an oriented smooth surface that is bounded by a simple, closed,…

Q: cify the component functions of a vector field F in R2 with the following property. I points except…

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Q: Suppose S is the lower hemisphere S = {(x, y, z) x² + y² + z² = 9, z ≤ 0}. Further, define a vector…

A: The given vector function is F→(x,y,z)=&lt;-y,x,x+3yz&gt;. The given surface is S={(x,y,z) |…

Q: Let E be the union of the two surfaces , and E2 given by E1 : y? + 22 = 4, 1 2, oriented with…

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Q: 10. Verify Stokes' Theorem, fF.dr=ffcurl(F) ds, for the vector field F= (z, 2x, 2y) and the surface…

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Q: | Let F = (z-y)i+(z+x)j-(r+y)k be a vector field. Verify Stokes' Theorem by computing both a line…

A: Given: F→=z-yi+z+xj-x+yk

Q: Let F = x?y?z²i+ (x³ +y³ + z³)j – 5(x² +y² + z²)k, and S be the surface of the cube 4 < x, y, z < 5…

A: Note: We are not given the method to use. So, we have used the simplest procedure that can give you…

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

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Q: Evaluate using Stoke's theorem F.ndS where the vector field F (y-z)i+ %3D (z-r)j+(r-y)k and S is the…

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Q: Let F = [z?, y?, x²] be a vector field and S be the closed surface consisting of the cylinder x2 +z²…

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Q: 3. What is true about the value of F.dS for the vector field F (r'y,-ry,-I*yz) where S is the…

A: Solution 3: Given  F=x3y,-x2y2,-x2yz We have to calculate ∬SF.dS, where S is the surface of…

Q: # 4. Evaluate the surface integral F dS for the given vector field F and the oriented surface S. In…

A: The posted question has multiple questions but according to guidelines we will solve the first…

Q: Let F= [x, z, ybe a vector field and S be the closed surface consisting of the cylinder y+z² =…

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