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Chapter 13: Q 16. (page 1066)

What geometric conditions do you look for when you are deciding which coordinate system to use in $R^{3}$ ?

Short Answer

Expert verified

The geometric condition is region of integration.

Step by step solution

Given Information

We need to determine the geometric condition while deciding the coordinate system to be used.

Explanation

The coordinates system are rectangular, cylindrical and spherical in $R^{3}$ .

Region of integration determines the usage of coordinate systems among three above mentioned coordinate systems.

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Most popular questions from this chapter

Evaluate the sums in Exercises $23 - 28$ .

\sum_{j = 1}^{m} \sum_{k = 1}^{n} (j - k)

Let $Ω$ be a lamina in the xy-plane. Suppose $Ω$ is composed of two non-overlapping lamin $Ω_{1}$ and $Ω_{2}$ , as follows:

Show that if the masses and centers of masses of $Ω_{1}$ and $Ω_{2}$ are $m_{1}$ and $m_{2},$ and $({\bar{x}}_{1}, {\bar{y}}_{1}) and ({\bar{x}}_{2}, {\bar{y}}_{2})$ respectively, then the center of mass of $Ω$ is $(\bar{x}, \bar{y}),$ where

$\bar{x} = \frac{m_{1} {\bar{x}}_{1} + m_{2} {\bar{x}}_{2}}{m_{1} + m_{2}} and \bar{y} = \frac{m_{1} {\bar{y}}_{1} + m_{2} {\bar{y}}_{2}}{m_{1} + m_{2}}$

Use Definition $13.4$ to evaluate the double integrals in Exercises $29 - 32$ .

localid="1649936867482" $\iint_{x} x^{2} y d A$

where $R = {(x, y) ∣ 1 \leq x \leq 3 and 0 \leq y \leq 2}$

$How many summands are in \sum_{j = 3}^{13} \sum_{k = 5}^{20} \frac{j^{2}}{e^{jk}} ?$

Earlier in this section, we showed that we could use Fubini’s theorem to evaluate the integral $\int \int_{R} x^{2} y d A$ and we showed that $\int_{2}^{5} \int_{1}^{3} x^{2} y d x d y = 91$ Now evaluate the double integral by evaluating the iterated integral $\int_{1}^{3} \int_{2}^{5} x^{2} y d y d x$ .

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What geometric conditions do you look for when you are deciding which coordinate system to use in R3?

Short Answer

Step by step solution

Given Information

Explanation

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What geometric conditions do you look for when you are deciding which coordinate system to use in $R^{3}$ ?