1. Let E be the surface in R³ given by z = e-²-y° where r? + y? < 4, oriented with the upward normal. Let F be the vector field given by F(r, y, z) = (y, –1, x²). Compute the flux F.dA. Σ

Q: Let F = 4ai + 1yj and let C be the closed path around the circle x² + y² = 9, traversed once…

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Q: If the vector field  given below section c describes the velocity of a fluid and you place a small…

A: As per the question, we will analyze the behavior of a small cork placed in a fluid with different…

Q: OSketch the vector field F(x, y) = -5i – 5j and calculate the line integral of F along the line…

A: Solution:-     Given that, F→ x,y=-5i∧-5j∧we have to calculate the line integral ∴we have to…

Q: Calculate the flux of the following vector field V = (2æz, (x+2), y (z² – 3)) through a square…

A: The general form of a vector-valued function in three dimensions is given by F=(F1,F2,F3), where F1,…

Q: Let S be the part of paraboloid z = 9 - x² - y² with z > 0. Verify Stokes' theorem for vector field…

A: Let S be the part of paraboloid z=9-x2-y2 and vector field Fx , y , z=4zi^+5xj^+2yk^. To Verify the…

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: Let F(x, y, z) = xye" – 2° Ja and P(1,0, –1). Let S be the surface defined by the equation xye" –…

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Q: 2. Use Green's Theorem to find the flux of F(x, y) = (-y², xy) across the square with vertices…

A: Green's Theorem:The circulation of vector field F=&lt;M,N&gt;in a closed path and the flux of the…

Q: Find the flux of the vector field F across the surface S in the indicated direction. F = x 2y i  -…

A: To find the flux of the vector field F across the surface S. F=x2yi−zk S is the portion of the cone…

Q: of the vector field F across the surface S in the indicated direction. F = x 2y i  - z k; S is…

A: Introduction: The flux of the vector field F is given by the formula, ∮CF.ndS=∫∫∫div(F)dA

Q: The surface S is defined by x + y² + z³ = 3. Find a vector normal to S at the point A with position…

A: Given a surface S the normal vector to the surface at the point A is given by

Q: 8. Let S be the filled-in square in the rz-plane with vertices (0,0,0), (1,0,0), (1,0, 1) and (0,0,…

A: Given vector field: F=-y,z,x Let A(0,0,0), B(1,0,0,), C(1,0,1) and D(0,0,1) be the vertices of the…

Q: 2. Let C be the curve of intersection between the cylinder r2+y = 1 and r-2y+z = 0, oriented…

A: The complete solutions are given below

Q: 4. Consider the vector function r(z, y) = (x, y, x² + 2y*). (a) Re-write this vector function as…

A: For the given vector function, PART A The surface function is given by, f(x,y)=xi^+yj^+(x2+2y2)k^…

Q: 33. Define T: R³ → R³ by y + 2z 2x +3y - z x +2y - 2z x - y a. Find all vectors in R' that are…

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Q: A normal vector to the tangent plane for the surface z = , at the point (1, 1, 1) is: y - Select…

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Q: Seven thieves try to share a hoard of gold bars equally between themselves. Unfortunately, six bars…

A: Seven thieves try to share a hoard of gold bars equally between themselves.Unfortunately, six bars…

Q: Find the flux of the vector field F = across the part of the plane 3+x+2y above the rectangle [0,…

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Q: Find the flux of the following vector field across the given surface with the specified orientation.…

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Q: Let F(x, y, z) = xzi+yzj + xyk, Surface S is the part of the sphere x² + y² + z² = 4 that lies…

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Q: Vector Field: F(x,y) = (xv, 3x, – 2y) and Surface: z = 4 - x - 3 y? Use the normal vector formula to…

A: Given surface z=4-x-3y2

Q: Suppose F 2.xz + 3y²) ay + (4yz²) az. (a) Calculate fLL F dS, where S is the shaded surface in…

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Q: Calculate the flux of the vector field F = (yz, xy, 3) through the elliptical cone S given by the…

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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: Let C be a curve defined by some vector function R with unit binormal B such that R(0) = (-2, 5, 4)…

A: Given C1 is a curve defined by some vector function R→ with unit binormal B→ such that R→(0)=-2,5,4,…

Q: 2. Find a unit normal vector to the surface x + y +2 = 5 at the point (0, 1, 2). Ans. Û + 2k)

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Q: 2. Consider the curve traced out by the vector valued function r(t) = (sin t cos t, cos? t, sin t)…

A: given that r(t)=(sintcost,cos2t,sint) we need to show that the curve lines on the surface of the…

Q: Let F(x, y) = (4x², 3y2). Compute the flux F nds of F across the boundary of the triangle with…

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Q: A vector field is given as F = âr2r“ + a¿r²CoSo + âz5z. Verify Stokes theorem for this vector field…

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Q: Find the level surfaces of the scalar field U = ear, where the vectors are: a = (2, 2, 1) and r = =…

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Q: 1. Consider the function f (x, y, z) = y- + y² +z? Determine the gradient vector field of f, F =Vf…

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Q: Calculate the flux of the vector field F (yz, ay, 3) through the elliptical cone S given by the…

A: Given : Vector Field F→=yz , xy , 3 and elliptical cone S are given by equation 25x2 +y2=z2 where…

Q: Consider the vector field F(x, y, z) = (xe² ye²³, -2e²) Let S be the surface defined by {X²+y²…

A: Answer :-

Q: Consider the function f(x, y) = exy. (a) (b) (c) Determine the vector field F = Vf. Evaluate F. dr…

A: (a) ∇f=fxi+fyj (b) A gradient field is always irrotational so its integral will be path independent.…

Q: 8. A line initiated from (1,0,2) and parallel to a vector i+j+k what is the parametric equation of…

A: we have to find equation of line in 8.

Q: Find the net flux of the curl of the vector field F(æ, y, z) = (y², z², x²) over the open surface…

A: Consider the given information: F(x,y,z)=(y2,z2,x2) And the surface defined by z=x2+y2 where y≥0 and…

Q: Suppose the vector field v(x, y, z) = (3y + 2) i + (x − 2) j+ (2z+1) k represents the velocity field…

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Q: 3. (ey + 1)²e-dx + (ex + 1)³e¯*dy = 0

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Q: A body rotates with a constant angular velocity w = w,î + w2ĵ+w3k about an axis. Let ř be the…

A: Given that a body rotates with a constant angular velocity w= w1i+w2j+w3k about an axis. Let r be…