154CHAPTER 2. CURVILINEAR COORDINATES(2.3) Prove that each of the following systemsa) Parabolic cylindrical coordinates (u, v, z) defined byx = u v, y = [u? – u²], z = zb) Paraboloidal coordinate system (u, v, 4) defined byI = UV COS¢, y = uv sin o, z =(u² – v]%3Dis orthogonal, and then find the following for each of them:The scale factors, the base unit vectors, the vector de, the elementde? and the element of volume dr.[hint: to show that the system is orthogonal prove that h12h23 = 0]h13Gradient, Divergence, Laplacian and Curl

The given question can be solved as shown in step2.

Answered: (2.3) Prove that each of the following…

Elementary Geometry For College Students, 7e

7th Edition

ISBN:9781337614085

Author:Alexander, Daniel C.; Koeberlein, Geralyn M.

Publisher:Alexander, Daniel C.; Koeberlein, Geralyn M.

ChapterP: Preliminary Concepts

SectionP.CT: Test

Problem 1CT

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Find the parametric equations for the line through the point P=(-5,4,-1) that is perpendicular to the plane 2x-1y-5z=1. Use 't' as your variable, t=0 should correspond to P, and the velocity vector of the line should be the same as the standard normal vector of the plane. a)X=? b)y=? c)z=?
1. Convert the point (1,1, v2) from Cartesian to spherical coordinates. (a) (vz. vz) (b) (1.#,v2) (o) (V2등,1) (d) (vz, 1, )
aj Jai aj lai ow consider the circle in a general position where the base vectors i & j are replaced with new vectors & j, to obtain a general vector function for a circle ř(t)=c +acos(t)i, +a sin(t) j, for 0
Express the given parametric equations of a line using bracket notation and also using i, j, k notation. (a) x = 6t, y = - 4 + 3t O (x, y) = (0, 4) +1 (6,3) ai + bj = -4j + t (6i + 3j) O (x, y) = (0, 4) + t (6, 3) ai + bj = 4j+ t (6i + 3j) O (x, y) = (0, 6) + (-4, 3) ai + bj = − 4j + t (6i + 3j) O(x, y) =(4,0) + t (3,6) ai + bj = −4i + t (3i + 6j) O (x, y) = (6,3)+1(0, - 4) ai + bj (6i + 3j) +t(-4j) (b)x= 4 + 6t,y=4+9t, z = 87t O (x, y, z) = (-4, 4, − 8) +1(-6, ai + bj + ck = 9,7) ( − 4i + 4j - 8 k) + t ( − 6i - 9j + 7k) O (x, y, z) = (4, ai + bj + ck = 4, 8) + t (6, 9, - 7) (4i − 4j + 8 k) + t (6i + 9j – 7k) O (x, y, z) = (6,9, −7) +1(4, - 4,8) ai + bj + ck = (6i + 9j − 7k) + t(4i - 4j + 8k) O (x, y, z) = (-6, 9, 7) + 1 - 4, 4, 8) ai+bj + ck = ( – 6i – 9j + 7k) + t( − 4i + 4j − 8k) O (x, y, z) = (4, 4, 8) + t (6,9,7) ai + bj + ck = (4i +4j + 8 k) + t (6i + 9j + 7k)
c) Use spherical coordinate to evaluate V1-x2 1 Vx2 + y2 + 2 dz dy dx.
Express the given parametric equations of a line using bracket notation and also using i, j, k notation. (a) x = 6t, y = - 4 + 3t O (x, y) = (0, - 4) + t (6,3) ai + bj = −4j + t (6i + 3j) O(x, y) = (0, 4) + t (6, 3) ai + bj = 4j + t (6i + 3j) ○ (x, y) = (0, 6) + t ( − 4, 3) ai + bj - 4j + t (6i + 3j) - (x, y) = (-4, 0) + t (3, 6) ai + bj = − 4i + t (3i + 6j) (x, y) = (6, 3) + t(0, − 4) ai + bj = (6i + 3j) + t ( − 4j)

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