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CalculusCalculus10th Edition

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Chapter 11.7, Problem 81E
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Calculus

10th Edition
Ron Larson + 1 other
ISBN: 9781285057095

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Calculus

10th Edition
Ron Larson + 1 other
ISBN: 9781285057095
Textbook Problem

Rectangular and Spherical Coordinates Give the equation for the coordinate conversion from rectangular to spherical coordinates and vice versa.

To determine

The equations for the co-ordinates conversion form rectangular co-ordinate to spherical co-ordinates and vice-versa.

Explanation

Takespherical co-ordinates as (ρ,θ,ϕ) and Rectangular co-ordinates as (x,y,z).

In a spherical co-ordinate system, a point P in space is represented by an ordered triple (r,θ,z) Where, ρ is the distance between P and the origin,ρ≥0 and ϕ is the angle between the positive z-axis and the line segment OP→0≤ϕ≤π.

As shown figure below:

Rectangular to spherical co-ordinate conversion,

Consider the equationto calculate first co-ordinate

r=x2+y2=ρsinϕ

Use the Pythagorean Theorem to calculate first co-ordinate which states that the square of the hypotenuse is equal to the sum of the squares of the perpendicular and the base.

This can be written as, a2=b2+c2 and known as Pythagorean equation.

As shown in figure:

So,

ρ=r2+z2=x2+y2+z2

Hence,

ρ2=x2+y2+z2

Use trigonometric functionto calculate second co-ordinate

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Probability and Statistics for Engineering and the Sciences

Find the center-to-center distance between the two holes in Illustration2. ILLUSTRATION 2

Elementary Technical Mathematics

Parallel lines l1 and l2 have slopes m1 and m2, respectively. Find m2 if m1 equals: a 45 b -15 c 3 d f+gh+j

Elementary Geometry For College Students, 7e

Navajo Culture: Traditional Hogans S. C. Jett is a professor of geography at the University of California. Davi...

Understanding Basic Statistics

For Problems 1-50, solve each equation. Objective 1 2n13n=5n73n

Intermediate Algebra

In Exercises 4756, solve the given equation for the indicated variable. 81=3x

Finite Mathematics

Guguen and Jacob (2012) asked waitresses to wear different colored T-shirts on different days for a six-week pe...

Essentials of Statistics for The Behavioral Sciences (MindTap Course List)

Finding the Sum of a Series In Exercises 59 and 60, find the sum of the series. n=0(1)n2n+132n+1(2n+1)!

Calculus: Early Transcendental Functions (MindTap Course List)

In Exercises 118, determine whether each system of linear equations has a one and only one solution, b infinite...

Finite Mathematics for the Managerial, Life, and Social Sciences

Find the exact volume of the solid of revolution that results when a semicircular region with diameter of lengt...

Elementary Geometry for College Students

Using the Alternating Series Test In Exercises 55-60, use the Alternating Series Test, if applicable, to determ...

Calculus (MindTap Course List)

It does not exist.

Study Guide for Stewart's Multivariable Calculus, 8th

Keplers Laws In Exercises 87-94, you are asked to verify Keplers Laws of Planetary Motion. For these exercises,...

Multivariable Calculus

The populations of sheep and wolves on an island is governed by a predator-prey model with constants k = 0.03, ...

Study Guide for Stewart's Single Variable Calculus: Early Transcendentals, 8th

For the following numerical variables, state whether each is discrete or continuous. a. The number of insuffici...

Introduction To Statistics And Data Analysis

Explain the distinction between experimental and nonexperimental research strategies.

Research Methods for the Behavioral Sciences (MindTap Course List)

Graph each interval on the number line. (,2)(5,)

College Algebra (MindTap Course List)

To lest for any significant difference in the number of hours between breakdowns for four machines, the followi...

Statistics for Business & Economics, Revised (MindTap Course List)

Initial public offerings (IPOs) of stocks are on average underpriced. The standard deviation measures the dispe...

STATISTICS F/BUSINESS+ECONOMICS-TEXT

Find the reciprocal of each of the fractions in Exercises 7 through 12. 12. 327

Mathematics For Machine Technology

Describe, compare, and contrast the five research strategies (descriptive, correlational, experimental, quasi-e...

Research Methods for the Behavioral Sciences (MindTap Course List)

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