Explanation Given: The equation ρ = 2 cos ϕ Key concepts applied: Standard equation of a sphere In rectangular coordinates, the equation of a sphere with center C ( h , k , l ) and radius r is given by, ( x − h ) 2 + ( y − k ) 2 + ( z − l ) 2 = r 2 Spherical coordinate system In a spherical coordinate system, a point P in space is represented by an ordered triple ( ρ , θ , ϕ ) . ρ is the distance between P and the origin, ρ ≥ 0 . θ is the same angle used in cylindrical coordinates for r ≥ 0 . ϕ is the angle between the positive z − axis and the line segment O P → , 0 ≤ ϕ ≤ π . Note that the first and the third coordinates ρ and ϕ are non-negative. To convert from rectangular to spherical coordinates (or vice versa), use the conversion guidelines as shown below, To convert a point, Spherical to rectangular x = ρ sin ϕ cos θ y = ρ sin ϕ sin θ z = ρ cos ϕ Rectangular to spherical ρ 2 = x 2 + y 2 + z 2 tan θ = y x ϕ = arccos ( z x 2 + y 2 + z 2 ) To convert a set, use any of the above-mentioned equations as necessary...

3rd Edition

ISBN: 9781464125263

Author: Jon Rogawski, Colin Adams

Publisher: W.H. Freeman & Co

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System that uses coordinates to uniquely determine the position of points. The most common coordinate system is the Cartesian system, where points are given by distance along a horizontal x-axis and vertical y-axis from the origin. A polar coordinate system locates a point by its direction relative to a reference direction and its distance from a given point. In three dimensions, it leads to cylindrical and spherical coordinates.

Chapter 13.7, Problem 78E

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The proof that the given equation is the equation of a sphere with its center on the z-axis and to determine its radius and center.

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Chapter 13.7, Problem 77E

Chapter 13.7, Problem 79E

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Chapter 13 Solutions

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Ch. 13.1 - Prob. 1PQ Ch. 13.1 - Prob. 2PQ Ch. 13.1 - Prob. 3PQ Ch. 13.1 - Prob. 4PQ Ch. 13.1 - Prob. 5PQ Ch. 13.1 - Prob. 6PQ Ch. 13.1 - Prob. 1E Ch. 13.1 - Prob. 2E Ch. 13.1 - Prob. 3E Ch. 13.1 - Prob. 4E

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The proof that the given equation is the equation of a sphere with its center on the z- axis and to determine its radius and center.

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The proof that the given equation is the equation of a sphere with its center on the z-axis and to determine its radius and center.

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Chapter 13 Solutions