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Chapter 16: Problem 3

Find an equation for the surface. The top half of the sphere \(x^{2}+y^{2}+z^{2}=1\) in cylindrical coordinates.

Short Answer

Expert verified
The equation is \(r^{2} + z^{2} = 1\) with \(z \geq 0\).

Step by step solution

01

Identify the Original Equation

The equation given is for a sphere: \(x^{2} + y^{2} + z^{2} = 1\). This equation describes a sphere with radius 1 centered at the origin.
02

Convert to Cylindrical Coordinates

In cylindrical coordinates, the relations are \(x = r \cos \theta\), \(y = r \sin \theta\), and \(z = z\). We substitute these into the sphere equation: \( (r \cos \theta)^{2} + (r \sin \theta)^{2} + z^{2} = 1\).
03

Simplify the Equation

Simplify the equation: \((r \cos \theta)^{2} + (r \sin \theta)^{2} = r^{2}\). Hence, the equation becomes \(r^{2} + z^{2} = 1\).
04

Consider the Top Half Sphere

For the top half of the sphere, we need \(z\) to be non-negative. Therefore, the condition is \(z \geq 0\).
05

Final Equation in Cylindrical Coordinates

The equation of the top half of the sphere in cylindrical coordinates is \(r^{2} + z^{2} = 1\) with \(z \geq 0\).

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Key Concepts

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

Sphere Equation
A sphere in three-dimensional space can be described using the sphere equation. This equation generally takes the form
  • \[ x^2 + y^2 + z^2 = r^2 \]
  • where \( r \) represents the radius of the sphere.
  • The coordinates \( (x, y, z) \) represent any point on the surface of this sphere.
The equation
  • \[ x^2 + y^2 + z^2 = 1 \]
represents a specific sphere in space. Here, the sphere is centered at the origin of the coordinate system with a radius of 1. Understanding the sphere equation is crucial for converting and representing it in other coordinate systems, such as cylindrical coordinates.
Coordinate Transformation
Coordinate transformation refers to changing a set of coordinates from one system to another. The primary goal is to simplify equations or make analyses more convenient
  • for the problem at hand.
When dealing with three-dimensional geometries, we often move between different coordinate systems, such as cartesian and cylindrical.To convert cartesian coordinates
  • \( (x, y, z) \)
  • to cylindrical coordinates
  • \( (r, \theta, z) \),
we use the transformations:
  • \( x = r \cos \theta \),
  • \( y = r \sin \theta \),
  • \( z = z \).
This system rotation is helpful for problems involving circular symmetry around the z-axis.
Surface Equation
When we talk about the surface equation of a geometric shape, we mean the equation that precisely describes all the points that belong to the surface of that shape. In this exercise,
  • we initially deal with the sphere's equation
  • and then transform it into a cylindrical form.
While the cartesian equation of our sphere is
  • \[ x^2 + y^2 + z^2 = 1 \],
converting it into cylindrical coordinates entails using the transformations giving us
  • \[ r^2 + z^2 = 1 \].
Since the task is limited to the top half of the sphere, we employ one more condition:\[ z \geq 0 \],which ensures that we only include the upper part in our surface description.
Multivariable Calculus
Multivariable calculus deals with functions of more than one variable. It extends the principles of calculus to higher dimensions. When studying shapes in three-dimensional space,
  • it's crucial to understand how these variables interact.
Using multivariable calculus, we can explore
  • different surfaces,
  • their properties, and
  • how they can be represented in varied coordinate systems, like cylindrical.
This branch of calculus is vital for transformations between coordinate systems, aiding us in solving geometrical problems efficiently by
  • offering tools to manipulate and redefine equations to suit our analyses.

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Most popular questions from this chapter

Two independent random numbers \(x\) and \(y\) between 0 and 1 have joint density function $$ p(x, y)=\left\\{\begin{array}{ll} 1 & \text { if } 0 \leq x, y \leq 1 \\ 0 & \text { otherwise } \end{array}\right. $$ This problem concerns the average \(z=(x+y) / 2,\) which has a one-variable probability density function of its own. (a) Find \(F(t)\), the probability that \(z \leq t\). Treat separately the cases \(t \leq 0,0

Electric charge is distributed throughout 3 -space, with density proportional to the distance from the \(x y\) -plane. Show that the total charge inside a cylinder of radius \(R\) and height \(h,\) sitting on the \(x y\) -plane and centered along the \(z\) -axis, is proportional to \(R^{2} h^{2}\)

Give an example of: An integral in spherical coordinates that gives the volume of a hemisphere.

Evaluate the integral by converting it into Cartesian coordinates: $$\int_{0}^{\pi / 6} \int_{0}^{2 / \cos \theta} r d r d \theta$$.

Let \(p(x, y)\) be a joint density function for \(x\) and \(y .\) Are the following statements true or false? $$\int_{-\infty}^{\infty} \int_{-\infty}^{\infty} p(x, y) d y d x=1$$
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