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Chapter 13: Q 12. (page 1066)

To convert from spherical to cylindrical coordinates:
$r = - - - -, θ = - - - -, z = - - - -$

Short Answer

Expert verified

$r = ρ \sin ϕ, θ = θ, z = ρ \cos ϕ$

Step by step solution

Given Information

Given $(r, θ, z)$ be cylindrical coordinates and $(ρ, θ, ϕ)$ be spherical coordinates.

Simplification

The relation between two is:

$r = ρ \sin ϕ, θ = θ, z = ρ \cos ϕ$

Hence,

$r = ρ \sin ϕ$

$θ = θ$

and

$z = ρ \cos ϕ$

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Most popular questions from this chapter

Evaluate each of the double integral in the exercise 37-54 as iterated integrals

$\int \int_{R} (2 - 3 x^{2} + y^{3}) d A W h e r e R = {(x, y) | - 3 \leq x \leq 2, 3 \leq y \leq 5}$

Evaluate each of the double integrals in Exercises 37–54 as iterated integrals.

$\int \int_{R} \frac{x}{x + y} d A, w h e r e R = {(x, y) | 1 \leq x \leq 4 a n d 0 \leq y \leq 3}$

Evaluate each of the double integrals in Exercises $37 - 54$ as iterated integrals.

role="math" localid="1650327788023" $\int \int_{R} y \sin x d A,$

whererole="math" localid="1650327080219" $R = \{(x, y) | 0 \leq x \leq \frac{π}{2} a n d 0 \leq y \leq 1\}$

Describe the three-dimensional region expressed in each iterated integral in Exercises 35–44.

$\int_{- 3}^{3} \int_{- \sqrt{9 - x^{2}}}^{\sqrt{9 - x^{2}}} \int_{- \sqrt{9 - x^{2} - y^{2}}}^{\sqrt{9 - x^{2} - y^{2}}} f (x, y, z) d z d y d x$

In Exercises 57–60, let R be the rectangular solid defined by

R = {(x, y, z) | 0 ≤ x ≤ 4, 0 ≤ y ≤ 3, 0 ≤ z ≤ 2}.

Assuming that the density at each point in R is proportional to the distance of the point from the xy-plane, find the moment of inertia about the x-axis and the radius of gyration about the x-axis.

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To convert from spherical to cylindrical coordinates:
$r = - - - -, θ = - - - -, z = - - - -$

Short Answer

Step by step solution

Given Information

Simplification

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Most popular questions from this chapter

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To convert from spherical to cylindrical coordinates:r=----,θ=----,z=----

Short Answer

Step by step solution

Given Information

Simplification

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Statistics

Decision Maths

Applied Mathematics

Discrete Mathematics

Pure Maths

Mechanics Maths

Study anywhere. Anytime. Across all devices.

To convert from spherical to cylindrical coordinates:
$r = - - - -, θ = - - - -, z = - - - -$