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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

Explain why $\sum_{i = 1}^{l} \sum_{j = 1}^{m} \sum_{k = 1}^{n} i j^{2} k^{3} = \sum_{j = 1}^{m} \sum_{k = 1}^{n} \sum_{i = 1}^{l} i j^{2} k^{3}$ .

In Exercises 45–52, rewrite the indicated integral with the specified order of integration.

Exercise 42 with the order dy dx dz.

In Exercises 61–64, let R be the rectangular solid defined by $R = \{(x, y, z) | 0 \leq a_{1} \leq x \leq a_{2}, 0 \leq b_{1} \leq y \leq b_{2}, 0 \leq c_{2} \leq z \leq c_{2}\} .$

Assume that the density of R is uniform throughout, and find the moment of inertia about the x-axis and the radius of gyration about the x-axis.

$How many summands are in \sum_{j = j_{0}}^{m} \sum_{k = k_{0}}^{n} j^{k} ?$

$L e t R = \{(x, y, z) | a_{1} ⩽ x ⩽ a_{2}, b_{1} ⩽ y ⩽ b_{2}, c_{1} ⩽ z ⩽ c_{2}\} . P r o v e t h a t : ∭_{R} d V = (a_{2} - a_{1}) (b_{2} - b_{1}) (c_{2} - c_{1}) . W h a t i s t h e r e l a t i o n s h i p b e t w e e n R a n d t h e p r o d u c t (a_{2} - a_{1}) (b_{2} - b_{1}) (c_{2} - c_{1}) .$

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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

Recommended explanations on Math Textbooks

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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

Probability and Statistics

Pure Maths

Applied Mathematics

Theoretical and Mathematical Physics

Discrete Mathematics

Decision Maths

Study anywhere. Anytime. Across all devices.

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