Explanation Given: ρ = 2 , ϕ = π 4 Graph: 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. When plotting spherical coordinates, plot the radius first on the horizontal axis coordinates, then rotate the point about the angle θ about the origin and then rotate the point about the angle ϕ about the z- axis. The cartesian coordinate axes are often drawn in to make the graph more readable. While this method is easy to plot points, it often requires sophisticated software to plot a given function in 3D. Here we use the online portal www.geogebra.org/3d to sketch the graphs of the given function. To do this, simply convert the given function described in cylindrical coordinates to rectangular coordinates. After this simply enter the function in the portal to obtain the sketch. 3D Sketches can also be made manually however it is tedious. To draw the graph of the given set in spherical coordinates using www.geogebra.org/3d let us first convert them into the rectangular coordinates. Description ρ = 2 , ϕ = π 4 Step 1...

ACHIEVE FOR CALCULUS 4 TERM >CSI CUSTOM<

21st Edition

ISBN: 9781319438333

Author: Rogawski

Publisher: MAC HIGHER

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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 12.7, Problem 58E

To determine

To sketch

The given set of points described in spherical coordinates

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Chapter 12.7, Problem 57E

Chapter 12.7, Problem 59E

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Chapter 12 Solutions

ACHIEVE FOR CALCULUS 4 TERM >CSI CUSTOM<

Ch. 12.1 - Prob. 1PQ Ch. 12.1 - Prob. 2PQ Ch. 12.1 - Prob. 3PQ Ch. 12.1 - Prob. 4PQ Ch. 12.1 - Prob. 5PQ Ch. 12.1 - Prob. 6PQ Ch. 12.1 - Prob. 1E Ch. 12.1 - Prob. 2E Ch. 12.1 - Prob. 3E Ch. 12.1 - Prob. 4E

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To sketch The given set of points described in spherical coordinates

To sketch The given set of points described in spherical coordinates

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Chapter 12 Solutions

To sketch

The given set of points described in spherical coordinates