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Chapter 13: Q 11. (page 1066)

To convert from cylindrical to spherical coordinates:
$ρ = - - - -, θ = - - - -, ϕ = - - - -$

Short Answer

Expert verified

$ρ = \sqrt{r^{2} + z^{2}}, θ = θ, ϕ = \tan^{- 1} \frac{r}{z}$

Step by step solution

Given Information

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

Simplification

The relation between two is:

$ρ = \sqrt{r^{2} + z^{2}}, θ = θ, \tan ϕ = \frac{r}{z}$

Hence,

$ρ = \sqrt{r^{2} + z^{2}}$

$θ = θ$

and $ϕ = \tan^{- 1} \frac{r}{z}$

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Most popular questions from this chapter

Let f(x, y, z) and g(x, y, z) be integrable functions on the rectangular solid $R = \{(x, y, z) | a_{1} ⩽ x ⩽ a_{2}, b_{1} ⩽ y ⩽ b_{2}, c_{1} ⩽ z ⩽ c_{2}\}$ . . Use the definition of the triple integral to prove that :

$∭_{R} (f (x, y, z) + g (x, y, z)) d V = ∭_{R} f (x, y, z) d V + ∭_{R} g (x, y, z) d V .$

Identify the quantities determined by the integral expressions in Exercises 19–24. If x, y, and z are all measured in centimeters and ρ(x, y,z) is a density function in grams per cubic centimeter on the three-dimensional region , give the units of the expression.

$∭_{Ω} (x^{2} + z^{2}) ρ (x, y, z) d V$

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

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

$\int \int_{R} x^{2} \cos (x y) d A, w h e r e R = {(x, y) | 0 \leq x \leq π a n d 0 \leq y \leq 1}$

Evaluate the sums in Exercises $23 - 28$ .

$\sum_{j = 1}^{3} \sum_{k = 1}^{2} j^{k}$

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To convert from cylindrical to spherical coordinates:ρ=----,θ=----,ϕ=----

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 cylindrical to spherical coordinates:
$ρ = - - - -, θ = - - - -, ϕ = - - - -$