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Magazine Chapter 11: Problem 29
An equation is given in spherical coordinates. Express the equation in rectangular coordinates and sketch the graph. $$ \phi=\pi / 4 $$
Short Answer
Expert verified
The equation is a cone centered on the z-axis.
Step by step solution
01
Understand the Spherical Coordinate System
In spherical coordinates, a point in space is defined by three values: radius \( r \), polar angle \( \theta \), and azimuthal angle \( \phi \). The azimuthal angle \( \phi \) is measured from the positive z-axis.
02
Equation Analysis
The given equation is \( \phi = \frac{\pi}{4} \). This describes all points in space for which the angle \( \phi \), with respect to the z-axis, is \( \pi / 4 \). This means the points form a conical surface that symmetrically extends around the z-axis.
03
Transform Spherical to Rectangular Coordinates
The relationship between spherical coordinates and rectangular coordinates \((x, y, z)\) are given by: - \( x = r \sin \phi \cos \theta \) - \( y = r \sin \phi \sin \theta \) - \( z = r \cos \phi \).For our problem, since \( \phi = \frac{\pi}{4} \), we have:- \( \sin \frac{\pi}{4} = \frac{\sqrt{2}}{2} \) - \( \cos \frac{\pi}{4} = \frac{\sqrt{2}}{2} \).
04
Find the Rectangular Equation
From \( z = r \cos \phi \), we know \( = \frac{\sqrt{2}}{2}r \).So, for \( \phi = \frac{\pi}{4} \), we have:\[ z = \frac{\sqrt{2}}{2}r \]This allows us to express the equation in terms of rectangular coordinates:\[ z = \frac{x^2 + y^2 + z^2}{2\sqrt{2}} \] by expressing \( r \) in terms of \( x, y, \) and \( z \); \( r^2 = x^2 + y^2 + z^2 \).
05
Interpret and Sketch the Graph
From the derived equation \( z = \frac{x^2 + y^2 + z^2}{2\sqrt{2}} \), we understand that it represents a cone centered along the z-axis, extending infinitely in both positive and negative z-direction, with its vertex at the origin. The slope of the cone sides, relative to the z-axis, is determined by \( \phi = \frac{\pi}{4} \).
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Spherical Coordinates
Spherical coordinates provide a unique way to locate a point in 3D space. Unlike the familiar rectangular system (Cartesian coordinates) which uses \((x, y, z)\), spherical coordinates use three different values: \(r\), \(\theta\), and \(\phi\).
- \(r\) denotes the radial distance of the point from the origin.
- \(\theta\) is the polar angle, typically measured from the positive x-axis in the xy-plane.
- \(\phi\), the azimuthal angle, is measured from the positive z-axis downwards.
Rectangular Coordinates
Rectangular coordinates, or Cartesian coordinates, are the traditional system used in mathematics for describing points in space.
- Each point is defined by \((x, y, z)\) coordinates, representing its position along the principal axes originating from the center point (origin).
- They are straightforward and particularly helpful for mapping algebraic equations.
- \(x = r \sin \phi \cos \theta\)
- \(y = r \sin \phi \sin \theta\)
- \(z = r \cos \phi\)
Conical Surface Equation
A conical surface is a geometric shape that extends infinitely and symmetrically around a central axis, usually the z-axis in coordinate systems.
The given equation \(\phi = \frac{\pi}{4}\) indicates that the angle between any point on the surface and the z-axis is \(45^\circ\). This describes a cone intersecting the xy-plane in a circular manner.
The given equation \(\phi = \frac{\pi}{4}\) indicates that the angle between any point on the surface and the z-axis is \(45^\circ\). This describes a cone intersecting the xy-plane in a circular manner.
- The standard equation arising from spherical coordinates \(z = \frac{x^2 + y^2 + z^2}{2\sqrt{2}}\) represents this conical surface in rectangular form.
- The vertex of the cone is at the origin, forming symmetrical slopes radiating outwards.
- It graphically appears as a cone extending along the z-axis with openness determined by its angle \(\phi\).
