Set-up the double integral being asked, no need to evaluate. Show complete solutions and sketch the graph. Definition: Area of a Parametric SurfaceLet S be a smooth surface defined parametrically byr(u, v) = x(u, v)i + y(u, v)ĵ + z(u, v)kwhere u and v are contained in a region IR. Then the surface area of S is given by= SS₁ || ru x rollRLet S be the surface defined by the vector functionR(u, v) = (— — 2u, v + 5, ve“)SA=dudv.for all (u, v) € R². Set - up an iterated double integral, in terms of the parameters u and v,that gives the surface area of the portion of S whose projection on the xy - plane is the triangularregion with vertices at the points (0,0), (0,5) and (1,5).

Introduction: The formula of the surface area of the parametric surface is given by, SA=∬Rru×rvdudv…

Answered: Let S be a smooth surface defined…

Let S be a smooth surface defined parametrically by r(u, v) = x(u, v)i + y(u, v where u and v are contained in a region R. Then the surface area of S is given by CC

Elementary Linear Algebra (MindTap Course List)

8th Edition

ISBN:9781305658004

Author:Ron Larson

Publisher:Ron Larson

Chapter5: Inner Product Spaces

Section5.1: Length And Dot Product In R^n

Problem 83E: Guided Proof Prove that if u is orthogonal to v and w, then u is orthogonal to cv+dw for any scalars...

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Guided Proof Prove that if u is orthogonal to v and w, then u is orthogonal to cv+dw for any scalars c and d. Getting Started: To prove that u is orthogonal to cv+dw, you need to show that the dot product of u and cv+dw is 0. i Rewrite the dot product of u and cv+dw as a linear combination of (uv) and (uw) using Properties 2 and 3 of Theorem 5.3. ii Use the fact that u is orthogonal to v and w, and the result of part i, to lead to the conclusion that u is orthogonal to cv+dw.
Proof Let V be an inner product space. For a fixed vector v0 in V, define T:VR by T(v)=v,v0. Prove that T is a linear transformation.
Flux of the radial field Consider the radial vector field F = ⟨ƒ, g, h⟩ = ⟨x, y, z⟩. Is the upward flux of the field greater across the hemisphere x2 + y2 + z2 = 1, for z ≥ 0, or across the paraboloid z = 1 - x2 - y2, for z ≥ 0?Note that the two surfaces have the same base in the xy-plane and the same high point (0, 0, 1). Use the explicit description for the hemisphere and a parametric description for the paraboloid.
(a) Show that at every point on the curve r(u) = (e^u cos u, e^u sin u, e^u ) , the angle between the unit tangent vector and the z-axis is the same. Show that this is also true for the principal normal vector. (b) Give a parametric representation of the level surface e^(xyz) = 1. (c) Find the equation of the tangent plane to the surface z = 3 e x−y at the point (4, 4, 3).
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Verifying Stokes’ Theorem Verify that the line integral and the surface integral of Stokes’ Theorem are equal for the following vector fields, surfaces S, and closed curves C. Assume C has counterclockwise orientation and S has a consistent orientation. F = ⟨y - z, z - x, x - y⟩; S is the cap of the sphere x2 + y2 + z2 = 16 above the plane z = √7 and C is the boundary of S.
Verifying Stokes’ Theorem Verify that the line integral and the surface integral of Stokes’ Theorem are equal for the following vector fields, surfaces S, and closed curves C. Assume C has counterclockwise orientation and S has a consistent orientation. F = ⟨x, y, z⟩; S is the paraboloid z = 8 - x2 - y2, for0 ≤ z ≤ 8, and C is the circle x2 + y2 = 8 in the xy-plane.
Circulation and flux Consider the following vector fields.a. Compute the circulation on the boundary of the region R (withcounterclockwise orientation).b. Compute the outward flux across the boundary of R. F = r/ | r | , where r = ⟨x, y⟩ and R is the half-annulus{(r, θ): 1 ≤ r ≤ 3, 0 ≤ θ ≤ π}
Tilted disks Let S be the disk enclosed by the curveC: r(t) = ⟨cos φ cos t, sin t, sin φ cos t⟩ , for 0 ≤ t ≤ 2π, where 0 ≤ φ ≤ π/2 is a fixed angle. What is the circulation on C of the vector field F = ⟨ -y, -z, x⟩ as a function of φ? For what value of φ is the circulation a maximum?
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Find a vector valued function that represents the intersection of a sphere w/ a radius of √68 and a plane with a trace of y=x^3 on the xy plane that extends orthogonally into the positive z. Sketch the curve created by the vector value function including the sphere and the plane. A computer generated sketch w/ annotation is acceptable.