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Chapter 13: Q 31 (page 1066)

Describe the graphs of the equations and provide alternative equations in the specified coordinate system
Change $r = 2$ to the rectangular and spherical systems

Short Answer

Expert verified

The equation in rectangular system is $x^{2} + y^{2} = 4$

The equation in spherical system is $蚁 = 2 c s c 蠒$

Step by step solution

01

Step 1:Given information

The given expression is $r = 2$

02

Simplification

The equation $r = 2$ represents a right circular cylinder on z-axis

Now to convert it into rectangular system

Let us consider a point in ${鈩�}^{3}$ with the rectangular coordinates $(x, y, z)$ and cylindrical coordinates $(r, 胃, z)$

Relation between them is

$r = \sqrt{x^{2} + y^{2}}$

$\tan 胃 = \frac{y}{x}$

$z = z$

Now substitute $r = \sqrt{x^{2} + y^{2}}$ in $r = 2$

$鈬� \sqrt{x^{2} + y^{2}} = 2$

$鈬� x^{2} + y^{2} = 4$

Hence, the equation in rectangular system is $x^{2} + y^{2} = 4$

03

Simplification

To convert $r = 2$ in spherical system, let us consider a point P in ${鈩�}^{3}$ with the cylindrical coordinates $(r, 胃, z)$ and the spherical coordinates

The relation between them is

$r = 蚁 \sin 蠒$

$胃 = 胃$

$z = 蚁 \cos 蠒$

substituting $r = 蚁 \sin 蠒$ in $r = 2$

$蚁 \sin 蠒 = 2$

$鈬� 蚁 = \frac{2}{\sin 蠒}$

$鈬� 蚁 = 2 c s c 蠒$

Hence,the equation in spherical system is $蚁 = 2 c s c 蠒$

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Most popular questions from this chapter

Use the lamina from Exercise 64, but assume that the density is proportional to the distance from the x-axis.

$How many summands are in {鈭�}_{i = 2}^{15} {鈭�}_{j = 3}^{17} {鈭�}_{k = 4}^{19} \frac{i}{j + k} ?$

Explain why ${鈭�}_{i = 1}^{l} {鈭�}_{j = 1}^{m} {鈭�}_{k = 1}^{n} i j^{2} k^{3} = {鈭�}_{j = 1}^{m} {鈭�}_{k = 1}^{n} {鈭�}_{i = 1}^{l} i j^{2} k^{3}$ .

Evaluate the sums in Exercises $23 鈥� 28$ .

${鈭�}_{j = 1}^{3} {鈭�}_{k = 1}^{2} j^{k}$

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

${鈭�}_{惟} 鈥� z 蚁 (x, y, z) d V$

See all solutions

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