1. In each part, find polar coordinates satisfying the statedconditions for the point whose rectangular coordinates are(1,√3).(a) r≥ 0 and 0 ≤ 0 < 2(b) r ≤ 0 and 0 ≤ 0 < 22. Let P be the point in 3-space with rectangular coordinates(√2, -√2, 2√3).=(a) Cylindrical coordinates for P are (r, 0, z.)(b) Spherical coordinates for P are (p, 0, 0)=3. Give an equation of a sphere of radius 5, centered at theorigin, in(a) rectangular coordinates(b) cylindrical coordinates(c) spherical coordinates.

Answered: Image /qna-images/answer/6369f288-d0a1-4ccf-9172-1803cb0dc69a.jpg

Answered: . In each part, find polar coordinates…

Algebra & Trigonometry with Analytic Geometry

13th Edition

ISBN:9781133382119

Author:Swokowski

Publisher:Swokowski

Chapter11: Topics From Analytic Geometry

Section: Chapter Questions

Problem 18T

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Does the sphere x2+y2+z2=100 have symmetry with respect to the a x-axis? b xy-plane?
Let C be the curve of intersection of the spheres x 2 + y 2 + z2 = 3 and(x - 2)2 + (y - 2)2 + z2 = 3. Use the result of Exercise 63 to find parametric equations of the tangent line to Cat P = (l, 1, 1).
Suppose that U is a solution to the Laplace equation in the disk Ω = {r ≤ 1} andthat U(1, θ) = 5 − sin^2(θ).(i) Without finding the solution to the equation, compute the value of U at theorigin – i.e. at r = 0.(ii) Without finding the solution to the equation, determine the location of themaxima and minima of U in Ω.(Hint: sin^2(θ) =(1−cos^2(θ))/2.)
You are now allowed to assume that the half-planes determined by the line with the equation ax+by +c = 0 correspond to the points (x, y) so that ax + by + c < 0 and ax + by + c > 0, respectively. Usingthis, show that axiom B4(i) holds. (Hint. Suppose (q, r) and (s, t) are on the same side of the given lineand that (s, t) and (u, v) are on the same side of the given line. en construct the parametrized linethrough (q, r) and (u, v). Consider the mappingλ γ7→ a(q − qλ + uλ) + b(r − rλ + vλ) + c and note that it is continuous and either increasing or decreasing. Use this fact to show that, for everyλ, γ(λ) > 0 or γ(λ) < 0, depending on which half-plane the points are on.)
Show that the path given by r(t) = (cos t,cos(2t), sint) intersects the xy-plane infinitely many times, but the underlying space curve intersects the xy-plane only twice.
Give a parametric representation for the cylinder x^2+z^2=121
Compute the length of the polar curve. r = 1 + θ for 0 ≤ θ ≤ π/2
Let u(x,y) satisfy the following equationuxx+(x2+y2-1)uyy=0Find the region in which the equation is elliptic and the region in which the equation is hyperbolic.
Find the points on the hyperboloid 9x^2 - 45y^2 + 5z^2 = 45 where the tangent plane is parallel to the plane x + 5y - 2z = 7
1) Draw a picture of two spheres of different sizes being tangent to each other at exactly one point P in R 3 . Do NOT put one sphere inside the other one. 2) Suppose we have two surfaces S1 and S2 which intersect at some point P = (x0, y0, z0), and further suppose that the normal vectors n1 =‹a1, b1, c1› (for S1) and n2 = ‹a2, b2, c2›(for S2) at P are parallel (and non-zero). Prove that the tangent planes TPS1 and TPS2 are the same by transforming the scalar equation for TPS1 into the scalar equation for TPS2. Hint: This should be a one-step algebraic transformation based upon using the algebraic definition of parallel vectors.
Find the length of the curve r = 2 sin^3 (u/3), 0<=u<=3pai, in the polar coordinate plane.
Two surfaces S and S^(-) with a common point p have contact order ≥ 2 at p if there exist parametrization x(u,v) and x^(-)(u,v) in p of S and S^(-) respectively such that xu = x^(-)u, xv = x^(-)v, xuu = x^(-)uu, xuv = x^(-)uv, xvv = x^(-)vv at p. Prove the following: a. Let S and S^(-) have contact order greater than or equal to 2 at p; x:U -> S and x^(-): U -> S^(-) be arbitrary parametrizations in p of S and S^(-) respectively and f: V c R^(3) -> R be a differentiable function in a neighborhood V of p in R^(3). Then the partial derivatives of order smaller than or equal to 2 of f o x^(-): U -> R are zero in x bar^(-1)(p) iff the partial derivatives of order smaller than or equal to 2 of f o x: U -> R are zero in x^(-1) (p). b. Let S and S^(-) have contact of order smaller than or equal to 2 at p. Let z = f(x, y), z = f^(-) (x, y) be the equations in a neighborhood of p, of S and S^(-) respectively where the xy plane is the common tangent plane at p = (0, 0). Then the…

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