Determine whether the following statements are true and give an explanation or counter example. a. Any point on the z-axis has more than one representation in both cylindrical and spherical coordinates. b. The sets {(r,e,z): r=z} and {(p,q,0): Q =π/4} are the same. System. B. The statement is false. The origin (x,y,z) = (0,0,0) has only one unique representation in spherical coordinates. C. The statement is false. The origin (x,y,z) = (0,0,0) has only one unique representation in cylindrical coordinates. D. The statement is true. In cylindrical coordinates, any point (x,y,z) = (0,0,z) on the z-axis is represented by (r,0,z) = (0,0,z), where 0 is arbitrary. In spherical coordinates, the point (x,y,z) = (0,0,z) can be represented by (p,q,0)=(2,0,0) or (p,q,0) = (-z,л,0) where 0 is arbitrary. b. Choose the correct answer below. ○ A. The statement is false. The set {(p,q,0): Q =π/4} describes a conical surface and the set {(r,0,z): r = z} describes a circular cylinder. O B. The statement is false. The set {(p,q,0): q =л/4} includes a conical surface centered along the negative z-axis and {(r,e,z): r = z} does not. C. The statement is false. The set {(r,0,z): r = z} includes a conical surface centered along the negative z-axis and {(p,q,0): =π/4} does not. D. The statement is true. Both sets describe the same conical surface with the vertex at the origin and extending in the positive z-direction.
Determine whether the following statements are true and give an explanation or counter example. a. Any point on the z-axis has more than one representation in both cylindrical and spherical coordinates. b. The sets {(r,e,z): r=z} and {(p,q,0): Q =π/4} are the same. System. B. The statement is false. The origin (x,y,z) = (0,0,0) has only one unique representation in spherical coordinates. C. The statement is false. The origin (x,y,z) = (0,0,0) has only one unique representation in cylindrical coordinates. D. The statement is true. In cylindrical coordinates, any point (x,y,z) = (0,0,z) on the z-axis is represented by (r,0,z) = (0,0,z), where 0 is arbitrary. In spherical coordinates, the point (x,y,z) = (0,0,z) can be represented by (p,q,0)=(2,0,0) or (p,q,0) = (-z,л,0) where 0 is arbitrary. b. Choose the correct answer below. ○ A. The statement is false. The set {(p,q,0): Q =π/4} describes a conical surface and the set {(r,0,z): r = z} describes a circular cylinder. O B. The statement is false. The set {(p,q,0): q =л/4} includes a conical surface centered along the negative z-axis and {(r,e,z): r = z} does not. C. The statement is false. The set {(r,0,z): r = z} includes a conical surface centered along the negative z-axis and {(p,q,0): =π/4} does not. D. The statement is true. Both sets describe the same conical surface with the vertex at the origin and extending in the positive z-direction.
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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Transcribed Image Text:Determine whether the following statements are true and give an explanation or counter example.
a. Any point on the z-axis has more than one representation in both cylindrical and spherical coordinates.
b. The sets {(r,e,z): r = z} and {(p,q,0): q=π/4} are the same.
system.
B. The statement is false. The origin (x,y,z) = (0,0,0) has only one unique representation in spherical
coordinates.
C. The statement is false. The origin (x,y,z) = (0,0,0) has only one unique representation in cylindrical
coordinates.
'D. The statement is true. In cylindrical coordinates, any point (x,y,z) = (0,0,z) on the z-axis is represented by
(r,0,z) = (0,0,z), where 0 is arbitrary. In spherical coordinates, the point (x,y,z) = (0,0,z) can be represented by
(p,q,0) = (z,0,0) or (p,q,0) = (-z,л,0) where 0 is arbitrary.
b. Choose the correct answer below.
A. The statement is false. The set {(p,q,0): Q =л/4} describes a conical surface and the set {(r,0,z): r = z}
describes a circular cylinder.
B. The statement is false. The set {(p,q,0): Q =π/4} includes a conical surface centered along the negative
z-axis and {(r,e,z): r = z} does not.
C. The statement is false. The set {(r,0,z): r = z} includes a conical surface centered along the negative z-axis
and {(p,q,0): =π/4} does not.
D. The statement is true. Both sets describe the same conical surface with the vertex at the origin and
extending in the positive z-direction.
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