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Chapter 11: Problem 57

Convert the point from cylindrical coordinates to spherical coordinates. \((4, \pi / 4,0)\)

Short Answer

Expert verified
The spherical coordinates for the cylindrical coordinates (4, 蟺/4,0) are (4, 蟺/2, 蟺/4).

Step by step solution

01

Calculation of 蟻

蟻=鈭�(r虏+z虏). We substitute the values r = 4 and z = 0 into the formula to calculate the value of 蟻. It results in 蟻 = 鈭�16 = 4.
02

Calculation of 蠁

蠁=arctan(r/z). However, in our case, z = 0. Hence, the equation is undefined. We know that arctan is undefined at 蟺/2 (90掳). So, 蠁 = 蟺/2.
03

Value of 胃

胃=胃. In spherical coordinates, the value of 胃 doesn't change. It remains the same as in cylindrical coordinates. Therefore, the value of 胃 = 蟺/4.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Cylindrical Coordinates
Cylindrical coordinates extend the concept of two-dimensional polar coordinates into three dimensions by adding a height (z-axis) component. In this system, any point in space is described by a triple \(r, \theta, z\).
  • \(r\): This represents the radial distance from the origin to the projection of the point in the XY-plane.
  • \(\theta\): This is the counterclockwise angle measured in the XY-plane from the positive X-axis.
  • \(z\): This is the height of the point above the XY-plane.

These coordinates are particularly useful in describing geometries involving rotation or symmetry around a central axis, such as tubes or cylinders. This system simplifies calculations involving circular and cylindrical shapes, which would be cumbersome in Cartesian coordinates.
Spherical Coordinates
Spherical coordinates describe positions on a sphere using three numbers, analogous to how latitude, longitude, and altitude are used on Earth. A point in spherical coordinates is given as \(\rho, \theta, \phi\):
  • \(\rho\): The radial distance from the origin to the point. In our example, this was calculated as \( \sqrt{r^2 + z^2} \).
  • \(\theta\): This remains the same as in cylindrical coordinates, representing the angle from the positive X-axis in the XY-plane.
  • \(\phi\): The angle between the positive Z-axis and the line connecting the origin to the point. It's akin to a vertical angle.

Spherical coordinates are ideal for dealing with problems involving spherical symmetry, such as gravitational fields or antenna radiation patterns. They simplify the mathematics of these systems by aligning the coordinate axes with the symmetry of the problem.
Geometry
Geometry is the study of shapes, sizes, and the properties of space. Both cylindrical and spherical coordinates are vital in geometry because they allow for a more intuitive and simplified way to describe and analyze shapes in three dimensions.
  • Cylindrical systems often model geometries such as pipes, tunnels, and springs.
  • Spherical systems are best for objects like planets, atoms, or any round object.

Understanding how to convert between these coordinate systems, as seen in this exercise, is crucial because different problems are easier to solve in different systems.
Through conversions, complex geometric shapes can be broken down into more manageable forms, aiding in better visualization and problem-solving.

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Most popular questions from this chapter

Describe a method for determining when two planes \(a_{1} x+b_{1} y+c_{1} z+d_{1}=0\) and \(a_{2} x+b_{2} y+c_{2} z+d_{2}=0\) are (a) parallel and (b) perpendicular. Explain your reasoning.

Consider the vectors \(\mathbf{u}=\langle\cos \alpha, \sin \alpha, 0\rangle\) and \(\mathbf{v}=\langle\cos \beta, \sin \beta, 0\rangle\) where \(\alpha>\beta\). Find the dot product of the vectors and use the result to prove the identity \(\cos (\alpha-\beta)=\cos \alpha \cos \beta+\sin \alpha \sin \beta\)

Find two vectors in opposite directions that are orthogonal to the vector \(\mathbf{u}\). (The answers are not unique.) $$ \mathbf{u}=\frac{1}{2} \mathbf{i}-\frac{2}{3} \mathbf{j} $$

Find the point(s) of intersection (if any) of the plane and the line. Also determine whether the line lies in the plane.\(2 x-2 y+z=12, \quad x-\frac{1}{2}=\frac{y+(3 / 2)}{-1}=\frac{z+1}{2}\)

Find the standard equation of the sphere with center \((-3,2,4)\) that is tangent to the plane given by \(2 x+4 y-3 z=8\).
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