Verify Stokes' theorem for the given surface S and boundary S, and vector fields F. S = {(x, y, z) : z = 1 − x² - y², z ≥ 0}, (oriented as a graph) as = {(x, y) : x² + y² = 1} F = zi + xj + (3zx + 6xy)k JS (V x F). ds = Jas F. ds =

Q: Consider the following statements about the surface of the equation z = 2x / (ln (y) +1) - (y + 1) /…

A: The given function is z=2xlny+1-y+1x and the given point is P-2,1. The directional derivative of a…

Q: B 4a X A 66° 20

A: The given figure is -  We have to find the value of 'x'.

Q: 3. Find the directional derivative of the given function f at the given point P in the direction of…

A: 3. From the given problem:   fx,y,z=x2+y2+z2 ; p1,2,2 ; a=3,2,1   As we know that the expression for…

Q: Find the tangent plane to the surface of at (1, -2) and a normal vector. f(x, y) = ln(x² + y²)

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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: Verify Stoke's theorem for the vector point function yz2 (xz² .3 + ex over the surface S of the cube…

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Q: Use Stokes' theorem to evaluate Ĝ - dĩ, vhere Ĝ(1, Y, 2) = (2r, y,0) and D is any closed curve.

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Q: Let F = (y², x, z²)and S be the part of the paraboloid z = x² + y² with z < 1 and outward facing…

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Q: Let S be the portion of the plane 2x + 3y + z = 2 lying between the points (-1, 1, 1), (2, 1, −5),…

A: S is the portion of the plane 2x+3y+z=2 lying between the points (-1, 1, 1), (2, 1,-5), (2, 3,-11)…

Q: 3) Evaluate, to the nearest hundredth, 11. (V x F) dS where F = and S = {(x, y, z)| 2² − 1 = 3x² +…

A: Here we will use Stokes theorem  According to stokes theorem  ∫∫S∇×F.dS=∫CF.dr C is the  boundary…

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: the vector function r ( t ) = ⟨ 6 cos ⁡ ( t ) , 3 − 3 sin ⁡ ( t ) , 2 + 3 sin ⁡ ( t ) ⟩ 1.Find the…

A: We'll answer the first question only since we don't answer multiple question at a time. Please…

Q: H. Use the gradient to find the equation of the tangent plane to each of the surfaces at the given…

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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: 2. Let 3 – 4ry? 12 + y? if (x, y) # (0,0) f(x, y) = if (x, y) = (0,0) %3D (a) Is f differentiable at…

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Q: 2. The vector function Ř(u, v) = (sin u cos v, sin u sin v, cos u), (u, v) E [0, 7] × [0, 2m] gives…

A: Here, the parametric vector function for the surface is given by R→u,v=sin u cos v, sin u sin v, cos…

Q: 1. Find the unit normal vector to the surface z = Vx2 + y? at the point (3, 4, 5).

A: A detailed solution is given below

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: 03. a. If F = (y² + z² − x²)i + (z² + x² − y²)j + (x² + y² − z²)k, then evaluate, SSV × F·n dA…

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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: Q3: For the surface f (x, y, z) = In (xyz), find: (1) the gradient of f at (1,1,2), (2) the…

A: Given

Q: 1. Let U(x, y, z) be the scalar function: U(x,y,z) = xy/z+yz/x +zx/y a. Find VU b. curl (V U)

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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: 7. Let F(x, y,z)=-2yi +2xj. Use Stokes’ Theorem to find F dr where C is a circle (a) Parallel to the…

A: By stoke's theorem

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.