Question 3. Let u = (1, 1, 1) E R³ and denote U = span(u). Let W = {(x1, X2, 13) E R³ : ¤1+ x2 + x3 = 0} Is R3 = U O W? YES NO
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- 1. Chapter 15 Review 13: Sketch the domain D (in the xy-plane) and calculate sDf(x,y)dA.D = {0 ≤y ≤1, 0.5y2 ≤x ≤y2}, f(x,y) = ye1+xQuestion 7.Consider the solid inxyz-space, which contains all points (x,y,z) whosez-coordinate satisfies0≤z≤4−x2−y2. Which statements do hold?a) The solid is a sphere.b) The solid is a pyramid.c) Its volume is 8π.d) Its volume is16π31. (Section 17.7) Use Stokes’ Theorem to calculate the work done by −→F (x, y, z) = ex2ˆı − 2xzˆj + xˆk in moving a particle aroundthe closed path determined by the intersection of positively oriented surface S : x + 4y + 2z = 4 and the coordinate planes.
- this question is differential geometry Let F: IR3 →IR3 is a diffeomorphism and M is a surface in IR3 , prove that the image F(M) is also a surface in IR3.Question 7. Consider the solid in xyz-space, which contains all points (x, y, z) whose z-coordinate satisfies 0 ≤ z ≤ 4 − x2 − y2 . Which statements do hold? a) The solid is a sphere. b) The solid is a pyramid. c) Its volume is 8π. d) Its volume is 16π . 3Solve the problem for w(r, θ) in a unit disk ∆w = 0, in 0 < r < 1,w(1, θ) = x2 + y, on r = 1. Find the expression of w(r, θ) and then express it in terms of x, y.
- Where M, N, and P are the midpoints of the sides of triangle RST and C is the centroid of the triangle, it follows that RC = CM. a. True b.FalseFind the first and second fundamental form of x(u,v)=(e(u-v)/2 cos(u+v)/2, e(u-v)/2 sin(u+v)/2,(u-v)/2 ) where x is a patch of any surface.Given a ray r(t) = (0; 0; 0) + t(1; 0; 0), t ≥ 0, and a set of spheres of unitradius and centered respectively at: (1) O = (0; 0; 0), (2) O = (3; 0; 0), (3) O = (1; 1; 0), (4) O = (-3; 0; 0), (5) O = (0; 3; 0). Which of the given spheres will be intersected from outside by the ray?
- Suppose S is a smooth surface parametrized by r(u, v) over D = [0, 1] x [0, 1] with ru = <1,2,3> and rv = <-1,0,2>. Find the area of SSuppose F(x,y)=<x^2+6y,5x−7y^2>. Use Green's Theorem to calculate the circulation of F around the perimeter of the triangle C oriented counter-clockwise with vertices (12,0), (0,6), and (−12,0).Given α = cos(xz)dx ∧ dy a two-form on R3 and the constant vectorsfieldsv =(1,0,1)and w =(2,2,3)on R3 find α(v, w). Then find αp (wp,vp) for both points (1, 2, π) and (1/2,2,π)