Parameterize the following surfaces in R3. Describe if the surface is open or closed. If the surface is open, give a parameterization of its boundary,
Parameterize the following surfaces in R3. Describe if the surface is open or closed. If the surf...
Problem 6. Let c > 0 and let (ar, y, z) E R3 \ {p= (,y, 2) R3: y0, 2 0} S = = Identify a parametrization d: U -> S of S (so UC R2 open so that S is part of a cone. etc.) such that d 1 is a conformal chart Suggestion: parametrize as a surface of revolution. Problem 6. Let c > 0 and let (ar, y, z) E R3 \ {p= (,y, 2) R3: y0,...
Sketch the described surfaces. Give a parametrization of each. (Be sure to include bounds of parameters as necessary): (a) A cone with its point at (1, 3, 4) and its base a circle of radius 6 on the plane z = 7, centered at (1, 3, 7) (b) A cylinder of radius 3/2 centered at the y-axis, with a base on the plane y = 0 and height going on to y = +. (c) The plane 2x+3y+z=10 restricted to...
Consider the paraboloid z=x2+y2. The plane 2x−2y+z−7=0 cuts the paraboloid, its intersection being a curve. Find "the natural" parametrization of this curve. Hint: The curve which is cut lies above a circle in the xy-plane which you should parametrize as a function of the variable t so that the circle is traversed counterclockwise exactly once as t goes from 0 to 2*pi, and the paramterization starts at the point on the circle with largest x coordinate. Using that as your...
Solve c and d Please. Find the area of the following surfaces. a) The part of the plane 3z + 2y +z 6 that lies in the first octant. b) The part of the cone zVy that lies between the plane y z and the cylinder y-x c) The surface with vector equation r(u, u) = (u . cos(v), u . sin(v), u) , where 0 u 1, 0 v S T. d) The portion of the unit sphere that...
Evaluate the surface integral F dS for the given vector field F and the oriented surface S. In other words, find the flux of F across S. For closed surfaces, use the positive (outward) orientation. F(x, y, z) -xi yj+3 k S is the boundary of the region enclosed by the cylinder x2 + z2-1 and the planes y 0 and x y 2 Evaluate the surface integral F dS for the given vector field F and the oriented surface...
Q1. Evaluate the line integral f (x2 + y2)dx + 2xydy by two methods a) directly, b) using Green's Theorem, where C consists of the arc of the parabola y = x2 from (0,0) to (2,4) and the line segments from (2,4) to (0,4) and from (0,4) to (0,0). [Answer: 0] Q2. Use Green's Theorem to evaluate the line integral $. F. dr or the work done by the force field F(x, y) = (3y - 4x)i +(4x - y)j...
F·dS for the given vector field F and the oriented surface S. In other words, find the flux of F across S. For closed surfaces, use the positive (outward) Evaluate the surface integral orientation. F(x, y, z) -x2i +y^j+z2 k S is the boundary of the solid half-cylinder 0szs V 25 -y2, 0 sxs2 Need HelpRead It Watch Talk to a Tutor F·dS for the given vector field F and the oriented surface S. In other words, find the flux...
Evaluate the surface integral F·dS for the given vector field F and the oriented surface S. In other words, find the flux of F across S. For closed surfaces, use the positive (outward) orientation. F(x, v, z)-xiyj+8 k S is the boundary of the region enclosed by the cylinderx2+2-1 and the planes y-o and xy6 Evaluate the surface integral F·dS for the given vector field F and the oriented surface S. In other words, find the flux of F across...
Through what closed, oriented surface in R3 does the vector field F = < (4x + 2(x^3) z) , (−y(x^2 + z^2 ), −(3(x^2)(z^2) + 4(y^2)z) > have the greatest flux? Through what closed, oriented surface in R3 does the vector field have the greatest flux?
Determine whether the following are true or false: A) If Sis a surface parametrized byr:DR^3, then A(S) = (double integral)D dA, where A(S) is the surface area of S. B) Let c be a boundary of a closed and bounded region D in the xy-plane. Then counterclockwise is always a positive orientation of c. c) Let Fbe a constant vector field on R^3. Then the flux of F through the unit sphere x^2 + y^2 + 2^2 = 1 is...