[Copyright](/content/comp/U2F0IEp1biAxMyAwMDowMDowMCBQRFQgMjAyNg.html) Learn how to convert between Cartesian, cylindrical and spherical coordinates. Discover the utility of representing points in cylindrical and spherical coordinates. ## Table of Contents - [What Are Coordinate Systems?](/content/academy/lesson/cylindrical-spherical-coordinates-definition-equations-examples.html#section---WhatAreCoordinateSystems/index.html) - [Cylindrical Coordinates](/content/academy/lesson/cylindrical-spherical-coordinates-definition-equations-examples.html#section---CylindricalCoordinates/index.html) - [Spherical Coordinates](/content/academy/lesson/cylindrical-spherical-coordinates-definition-equations-examples.html#section---SphericalCoordinates/index.html) - [Examples of Cylindrical and Spherical Coordinate Conversion](/content/academy/lesson/cylindrical-spherical-coordinates-definition-equations-examples.html#section---ExamplesOfCylindricalAndSphericalCoordinateConversion/index.html) - [Lesson Summary](/content/academy/lesson/cylindrical-spherical-coordinates-definition-equations-examples.html#section---LessonSummary/index.html) Show FAQActivities ### What is the method for converting rectangular coordinates to cylindrical coordinates? The set of equations used to convert between rectangular (Cartesian) coordinates and cylindrical coordinates are nearly identical to those used to convert between rectangular coordinates and polar coordinates in two dimensions, along with the trivial conversion z=z. All the equations can be derived by using trigonometry, from the observation that the radius of a circle in the plane also represents the hypotenuse of a right triangle. ### What is the use of the cylindrical coordinate system? The cylindrical coordinate system, in contrast to the Cartesian coordinate system and spherical coordinate system, is useful for modeling phenomena with rotational symmetry about a longitudinal axis. Problems in calculus and differential equations that model physical phenomena, like heat distribution in a metal rod, are easier to solve when using cylindrical coordinates, as opposed to, say, Cartesian coordinates. ### What is the method for converting cylindrical coordinates to spherical coordinates? Cylindrical coordinates can be converted to spherical coordinates by using the equations ρ=+r2+z2 and ϕ=cos−1⁡zρ. Be careful, however, to select the appropriate angle 0≤ϕ<π. ### What does r represent in cylindrical coordinates? The coordinate r represents the distance from the origin (pole) to a point expressed in cylindrical coordinates, as projected into the 2D plane. The letter r is short for "radius" here, and is also used in the polar coordinate system. That's because the Euclidean distance metric can be viewed as the radius of a circle in the plane. ### What is the method for converting to spherical coordinates? We should first specify the coordinate system from which we're converting. When converting from Cartesian coordinates to spherical coordinates, we use the equations ρ=+x2+y2+z2,θ=tan−1⁡yx,andϕ=cos−1⁡zx2+y2+z2. When converting from cylindrical coordinates to spherical coordinates, the process is slightly easier. That's because one coordinate, namely θ, remains fixed. The two relevant equations in this case are ρ=+r2+z2andϕ=cos−1⁡zρ. ## Cylindrical and Spherical Coordinates Review Topics Cylindrical and spherical coordinates are used to represent points, curves and surfaces in space if in rectangular coordinates, the description is challenging. The **cylindrical coordinates** are given by the triplet | | | --- | | | where the first two are the polar radius and angle and the third component is the same as the rectangular **z**-coordinate. The **spherical coordinates** are given by | | | --- | | | where the first component is the distance from the origin to the point, the second component is the polar angle and the third component is the altitude angle measured from the positive **z**-axis to the line connecting the origin and the point. To see the advantage of using the two systems, we will obtain the cylindrical and spherical coordinate equations of various surfaces given in rectangular coordinates, by using the following cylindrical-rectangular conversion | | | --- | | | or the spherical -rectangular convertion | | | --- | | | ### Problems 1) Given the rectangular equation of a cylinder of radius **2** and axis of rotation the **x** axis as | | | --- | | | write the equation in cylindrical coordinates. 2) Given the rectangular equation of a sphere of radius **1** and center at the origin as | | | --- | | | write the equation in spherical coordinates. 3) Given the rectangular equation of a cone as | | | --- | | | write the equation in cylindrical and spherical coordinates. ### Solutions 1) Using the rectangular-cylindrical conversion | | | --- | | | we obtain | | | --- | | | 2) Using the standard rectangular-spherical conversion we obtain | | | --- | | | 3) The given cone in cylindrical coordinates is | | | --- | | | and in spherical coordinates is | | | --- | | | LessonTranscript Click for sound ## You must cCreate an account to continue watching ### Register to view this lesson Are you a student or a teacher? 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Keep playing.](/content/academy/lesson/solving-partial-derivative-equations.html) Your next lesson will play in 10 seconds - 0:04 **Mapping 3D Space** - 1:03 **Polar Coordinate Systems** - 1:48 **Cylindrical Coordinates** - 3:54 **Spherical Coordinates** - 5:59 **Application: Celestial Maps** - 6:41 **Lesson Summary** [Quiz](/content/academy/practice/quiz-worksheet-cylindrical-spherical-coordinates.html) [Course](/content/academy/course/gre-math-subject-test-study-guide-test-prep.html) [View Video Only](/content/academy/lesson/video/cylindrical-spherical-coordinates-definition-equations-examples.html) Save Timeline 3.7K views ##### Recommended lessons and courses for you Related Lessons Related Courses \\ **How to Find the Distance Between Parallel Lines**\\ \\ 10:46](/content/academy/lesson/how-to-find-the-distance-between-parallel-lines.html) \\ **Distance From a Point to a Line \| Formula & Examples**\\ \\ 7:39](/content/learn/lesson/distance-from-point-to-line.html) \\ **Finding the Plane Equation From 3 Points \| Overview & Examples**\\ \\ 4:10](/content/academy/lesson/finding-the-equation-of-a-plane-from-three-points.html) \\ **Graphing Points & Lines in Three Dimensions**\\ \\ 5:19](/content/academy/lesson/graphing-points-lines-in-three-dimensions.html) \\ **Finding Perimeter & Areas Using Coordinates & Distance Formula**\\ \\ 7:21](/content/academy/lesson/using-coordinates-to-solve-perimeter-area-problems.html) \\ **Plotting Simple Figures on Coordinate Graphs**\\ \\ 4:27](/content/academy/lesson/plotting-simple-figures-on-coordinate-graphs.html) \\ **Distance Formula \| Overview & Examples**\\ \\ 5:27](/content/academy/lesson/how-to-use-the-distance-formula.html) \\ **Perpendicular Lines \| Definition, Equation & Graph**\\ \\ 7:27](/content/academy/lesson/what-are-perpendicular-lines-definition-meaning-quiz.html) \\ **Midpoint, Distance & Slope on Coordinated Plane \| Formula & Graph**\\ \\ 5:51](/content/learn/lesson/dimensions-coordinate-plane-midpoint-distance-slope.html) \\ **Distance Between Two Points \| Formula, Calculation & Examples**\\ \\ 4:37](/content/academy/lesson/how-to-find-the-distance-between-two-points.html) \\ **Vertical Line \| Slope & Equation**\\ \\ 2:55](/content/academy/lesson/vertical-line-equation-slope-quiz.html) \\ **Partitioning a Line Segment \| Definition, Formula & Examples**\\ \\ 5:23](/content/academy/lesson/using-slope-to-partition-segments.html) \\ **How to Find the Distance between Two Planes**\\ \\ 6:48](/content/academy/lesson/how-to-find-the-distance-between-two-planes.html) \\ **Triangles in Coordinate Planes \| Classifications & Examples**\\ \\ 4:48](/content/academy/lesson/triangles-in-coordinate-planes-proofs.html) \\ **Coordinate Graph \| Definition, Characteristics & Examples**\\ \\ 4:02](/content/learn/lesson/coordinate-graph-graphing-points.html) \\ **Maxwell's Equations \| Overview, Applications & Examples**\\ \\ 6:22](/content/academy/lesson/maxwells-equations-definition-application.html) \\ **Convolution Theorem \| Proof, Formula & Examples**\\ \\ 8:35](/content/academy/lesson/convolution-theorem-application-examples.html) \\ **Orthogonal Vectors Overview, Formula & Examples**\\ \\ 6:01](/content/academy/lesson/orthonormal-bases-definition-example.html) \\ **Field in Mathematics \| Definition, Examples & Theory**\\ \\ 6:25](/content/academy/lesson/field-theory-definition-examples.html) \\ **Inertial Frame of Reference \| Overview & Examples**\\ \\ 6:32](/content/learn/lesson/inertial-frame-of-reference-properties-examples.html) \\ **GRE Test Study Guide and Test Prep**](/content/academy/course/gre-test.html) \\ **Praxis 5165 Study Guide - 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California Basic Educational Skills Exam Prep**](/content/academy/course/cbest-test.html) **What Are Coordinate Systems?** Our first encounter with coordinate systems was likely with the Cartesian coordinate system in two dimensions. At some time in our math journey, we witnessed the merger of algebra and geometry described in the language of a coordinate system. In plain English, a **coordinate system** is a systematic way of uniquely identifying points in some space by using pieces of numerical information called **coordinates**. The Cartesian coordinate system is a remarkably effective way of capturing two- and three-dimensional space by use of rectangles and rectangular prisms, respectively. However, as we will soon see, there are alternative coordinate systems in both two- and three-dimensional space that rely on geometric figures other than rectangles and their higher-dimensional counterparts. First, it is important that we have a firm grasp of the Cartesian coordinate system in two dimensions, as this will be our springboard for exploring the other relevant coordinate systems. In two dimensions, the Cartesian coordinate system partitions space into infinitely many squares with one square unit of area. In more technical terms, we take the Cartesian product Z×Z={(a,b)\|a,b∈Z}, where Z denotes the set of integers, and represent it as two perpendicular axes that extend infinitely far in both directions. We typically call the horizontal axis the x-axis and the vertical axis the y-axis. This construction allows us to uniquely identify any pair of two integers (a,b)∈Z×Z by considering the intersection of the vertical line x=a and the horizontal line y=b. So the point (−3,2) can be identified as the intersection of the lines x=−3 and y=2. | | | --- | | | The point (-3, 2) is identified by its horizontal and vertical distance from the origin, which is (0, 0) in Cartesian coordinates. We can extend this idea of representing a Cartesian product of sets as perpendicular axes to "enlarge" our space to all pairs of rational numbers, or even to pairs of real numbers. (Note: the word "enlarge" is enclosed in quotation marks here, because the Cartesian product Q×Q={(p,q)\|p=abandq=cdfor somea,b,c,d∈Zwithb,d≠0}, where Q denotes the set of rational numbers, has the same size as Z×Z in a precise sense: namely, Q×Q≅Z×Z. However, the Cartesian product R×R={(x,y)\|x,y∈R}, where R denotes the set of real numbers, is genuinely larger than Z×Z and Q×Q, in the sense that Z×Z⊊Q×Q⊊R×R and Z×Z≅Q×Q≇R×R.) Instead of our number lines along the x\- and y-axes containing only the integers, we prefer to work with axes containing the entire continuum of real numbers so we can model continuous phenomena and smooth geometric objects. So for the remainder of this discussion, we'll be working in the two-dimensional space R×R=R2 and the three-dimensional space R×R×R=R3. It is very important that we emphasize there is a distinction between the underlying space and the coordinate system used to describe it. Later on, we will use Cartesian, cylindrical and spherical coordinates, which are distinct yet equivalent coordinate systems, to describe points in the same, underlying, three-dimensional space, R3. A useful analogy to keep in mind is this: Performing conversions between two different coordinate systems is like translating a sentence about a specific object between two different languages. What we say changes, but the underlying properties of the object in question do not. We are now ready to discuss alternative coordinate systems. ### Polar Coordinates Arguably the best-known theorem in all of mathematics is the Pythagorean theorem: Given a right triangle with legs of length a and b and hypotenuse of length c, the equation a2+b2=c2 holds. Taking the square root of both sides of the equation and rewriting yields c=a2+b2. This will act as our Rosetta Stone for translating between Cartesian coordinates and **polar coordinates**. The key idea behind the polar coordinate system is that a point in R2 can be "uniquely" identified by using a distance denoted r (for _radius_) and an angle of rotation denoted θ. (Note: The word "uniquely" is in quotation marks because polar coordinates can only be expressed uniquely up to full turns. That is, the point (r,θ) is the same point as (r,θ+2πn) for any integer n. By the same reasoning, the point (r,θ) is the same point as (−r,πθ+2πn) for any integer n. Therefore, the origin, expressed as (0,0) in Cartesian coordinates, can be written as (0,θ) for any angle θ.) Remember, θ is measured in radians, counterclockwise from the positive x-axis, by convention. Whereas the Cartesian perspective identifies points (x,y)∈R2 as vertices on a rectangle with one vertex anchored at the origin (0,0), the polar perspective identifies points (r,θ)∈R2 as lying on a circle centered at the origin. We write (x,y)≡(r,θ) to capture the equivalence of the two ways of describing the same point. We should note that other sources may use ρ in place of r, and ϕ or φ in place of θ. This is purely a stylistic difference, and we will make note of potential discrepancies as they arise. The interchangeable nature of the Cartesian coordinate system and polar coordinate system vastly simplifies certain problems in calculus with a rotational nature, as we will later see. | | | --- | | | The trigonometric relationship between Cartesian and polar coordinates. It immediately follows from our knowledge of trigonometry that the following important equations, which relate Cartesian and polar coordinates, hold: r2=x2+y2,r=+x2+y2,x=rcosθ,y=rsinθ,tanθ=yx,andθ=tan−1(yx). With this set of relationships in our tool belt, we can readily convert points in two-dimensional space between Cartesian and polar coordinates. For example, consider our point (−3,2) from before, as written in Cartesian. How do we express this same point in polar coordinates? Well, we want to write it in the form (r,θ). Since x=−3 and y=2, it follows that r=−32+22=9+4=13, which is the hypotenuse of the right triangle in Quadrant II with base 3 and height 2. Moreover, θ=tan−1⁡−23=π−tan−1⁡23≈2.55, so (−3,2)≈(13,2.55). At this point, the natural question to ask is, "How do we extend our Cartesian and polar coordinate systems to three-dimensional space?" The answer is clear in the Cartesian case: We introduce a third axis, typically called the z-axis, which is perpendicular to both the x\- and y-axes that contain all the infinitely many real numbers. This allows us to identify any point (a,b,c)∈R3 as the intersection of the lines x=a,y=b and z=c. In other words, (a,b,c) is a vertex of some rectangular prism, which is anchored at the origin, (0,0,0). The answer is less clear in the polar case, because there are actually two ways of extending our coordinate system based on circles to three spatial dimensions. There's a fork in the road here, with one path leading to the cylindrical coordinate system, the other leading to the spherical coordinate system. Let's examine the cylindrical coordinate system first. To unlock this lesson you must be a Study.com member [Create an account](/content/academy/plans.html) **Cylindrical Coordinates** **Cylindrical coordinates** extend the polar coordinate system into R3 by adding a third coordinate, z, to the familiar polar coordinates r and θ. That's the source of the term "cylindrical," as extending a circle along the z-axis creates a cylinder. | | | --- | | | The cylindrical coordinate system extends the polar coordinate system into 3D by adding a third coordinate, z. The utility of this coordinate system lies in its ability to readily convert points expressed in Cartesian -- or, as we'll soon see, spherical coordinates -- to cylindrical coordinates and vice versa. Since we haven't introduced spherical coordinates yet, we'll focus for the time being on converting between Cartesian coordinates and cylindrical coordinates. Fortunately, our set of tools for this task is nearly identical to our set of tools for converting between Cartesian and polar coordinates in two dimensions, because the cylindrical coordinate system extends the polar coordinates system into three dimensions in the same way the Cartesian coordinate system does: via the introduction of a z-axis. One can readily verify that r2=x2+y2,r=+x2+y2,x=rcosθ,y=rsinθ,tanθ=yx,θ=tan−1(yx),andz=z is the complete set of relationships needed to convert any point in three-dimensional, Cartesian coordinates to cylindrical coordinates and vice versa. It is often the case that a problem in calculus or differential equations that models physical phenomena with inherent rotational symmetry about an axis, such as heat distribution in a metal rod, is better understood and solved in cylindrical coordinates. We can now turn our attention to the third and final coordinates system of our discussion. To unlock this lesson you must be a Study.com member [Create an account](/content/academy/plans.html) **Spherical Coordinates** The system of **spherical coordinates** constitutes an alternative approach to extending the polar coordinate system to three dimensions, in contrast to the cylindrical coordinate system. Unlike the cylindrical coordinate system, which takes as its third coordinate a distance z, the spherical coordinate system takes as its third coordinate a second angle, typically denoted as ϕ. Whereas θ measures the angle of counterclockwise rotation from the positive x-axis in the plane and satisfies the inequality 0≤θ<2π, the angle ϕ measures rotation from the positive z-axis in three-dimensional space and satisfies the inequality 0≤ϕ<π. Additionally, in spherical coordinates, the coordinate r, which denotes distance in the xy-plane, is replaced by a coordinate typically denoted as ρ. This coordinate measures the Euclidean distance between the origin and the point being expressed in spherical coordinates. Unlike r, we require ρ to be nonnegative. In symbols, then, 0≤ρ. By definition of the Euclidean distance metric, it immediately follows that ρ=+x2+y2+z2, when viewed as a conversion from Cartesian coordinates. From the perspective of cylindrical coordinates, we have ρ=+r2+z2, since r2=x2+y2. Thus, a point in R3, as expressed in spherical coordinates, is written (ρ,θ,ϕ) and is thought of as existing on a sphere that's centered at the origin. | | | --- | | | The spherical coordinate system expresses points in 3D space as points on a sphere that Here we note that notational discrepancies may arise, as other sources use (ρ,ϕ,θ) to denote a point in spherical coordinates. It's always a good idea to carefully check the way an author introduces notation, regardless of one's familiarity with the topic. Spherical coordinates are particularly useful for solving triple integrals over solids such as spheres, domes and cones. Furthermore, whereas Cartesian coordinates in two dimensions are preferable for identifying locations on a flat map of city streets, spherical coordinates are preferable for identifying locations on the earth. In that case, θ corresponds to the meridian, ϕ corresponds to latitude, and ρ is the point's elevation above the surface. ### Spherical Coordinate Conversion Deriving the complete set of equations needed to readily convert Cartesian and cylindrical coordinates into spherical coordinates and vice versa requires slightly more work and imagination than conversion between Cartesian and cylindrical coordinates. First, we'll consider the point (x,y,z), as written in Cartesian coordinates, with the goal of converting it into spherical coordinates (ρ,θ,ϕ). We've already seen that ρ is just the Euclidean distance metric in R3: ρ=+x2+y2+z2, but what about the angles θ and ϕ? | | | --- | | | | | | --- | | | The angle θ is the same coordinate as the angle θ used in polar and cylindrical coordinates, so the familiar relationships x=rcos⁡θ and y=rsin⁡θ still hold. However, since r=ρsin⁡ϕ by trigonometric observation, we can go a bit further by saying x=rcos⁡θ=ρsin⁡ϕcosθ and y=rsin⁡θ=ρsin⁡ϕsin⁡θ. Finally, our right triangle tells us z=ρcos⁡ϕ, which implies ϕ=cos−1⁡zx2+y2+z2=cos−1⁡zr2+z2=cos−1⁡zρ. Now we have all the necessary tools to convert, not only between spherical and Cartesian coordinates, but between spherical and cylindrical coordinates as well. In summary, the complete set of equations used to convert to and from spherical coordinates is the following: r=ρsin⁡ϕ,x=rcos⁡θ=ρsin⁡ϕcos⁡θ,y=rsin⁡θ=ρsin⁡ϕsin⁡θ,z=ρcos⁡ϕ,ρ=+x2+y2+z2=+r2+z2,andϕ=cos−1⁡zρ. Now that we have all the notation and equations necessary to describe points in R3, we'll practice converting between coordinate systems. To unlock this lesson you must be a Study.com member [Create an account](/content/academy/plans.html) **Examples of Cylindrical and Spherical Coordinate Conversion** ### Example 1: Cartesian to Cylindrical Consider the point (x,y,z)=(−5,12,2), as written in Cartesian coordinates. We wish to write this same point in cylindrical coordinates, which are of the form (r,θ,z). By the equation r=+x2+y2, it follows that r=−52+122=25+144=169=13. Next, we wish to compute the angle of rotation, called θ. Since x=−5 is negative and y=12 is positive, our point lives in Quadrant II as projected into the xy-plane. We wish to choose θ such that π2<θ<π. Well, tan⁡θ=−125⟹θ=tan−1−125=π−tan−1⁡125≈1.97. Therefore, our angle of rotation θ is approximately 1.97 radians. Finally, the trivial conversion z=z gives us the following (approximate) equivalence: (−5,12,2)≡(13,1.97,2). ### Example 2: Cartesian to Spherical Consider the sphere of radius 1 that's centered at (0,0,1) in Cartesian coordinates and given by x2+y2+(z−1)2=1. If the spherical coordinate system is of any use at all, we'd expect a much simpler equation to describe the same object in spherical coordinates. Let's start by converting x,y, and z to ρsin⁡ϕcos⁡θ,ρsin⁡ϕsin⁡θ,andρcos⁡ϕ, respectively yielding ρ2sin2⁡ϕcos2⁡θ+ρ2sin2⁡ϕsin2⁡θ+(ρcos⁡ϕ−1)2=1. We can factor out a ρ2sin2⁡ϕ from the first two terms, expand (ρcos⁡ϕ−1)2, and write ρ2sin2⁡ϕ(cos2⁡θ+sin2⁡θ)+ρ2cos2⁡ϕ−2ρcos⁡ϕ+1=1. Since cos2⁡θ+sin2⁡θ=1 by the Pythagorean identity, we can subtract 1 from both sides of the equation and write ρ2sin2⁡ϕ+ρ2cos2⁡ϕ−2ρcos⁡ϕ=0. Again, factoring out ρ2 from the first two terms leaves the Pythagorean identity, sin2⁡ϕ+cos2⁡ϕ=1, so after adding 2ρcos⁡ϕ to both sides of the equation we're left with ρ2=2ρcos⁡ϕ. Since ρ≠0, we can divide both sides of the equation by ρ. That gives us ρ=2cos⁡ϕ, a far more elegant equation than the original x2+y2+(z−1)2=1. ### Example 3: Spherical to Cartesian Consider the point (16,π3,π6) as written in spherical coordinates. We wish to write this same point in R3 as (x,y,z), using Cartesian coordinates. The conversions are relatively straightforward, so long as we remember that sin⁡π6=12,sin⁡π3=32,cos⁡π6=32,andcos⁡π3=12. By our conversion formulas, x=ρsin⁡ϕcos⁡θ=16sin⁡π6cos⁡π3=16(12)(12)=4,y=ρsin⁡ϕsin⁡θ=16sin⁡π6sin⁡π3=16(12)(32)=43, and z=ρcos⁡ϕ=16cos⁡π6=16(32)=83. Thus, (8,π3,π6)≡(4,43,83), as desired. ### Example 4: Spherical to Cylindrical Consider the point (2,3π8,π12) as written in spherical coordinates. We wish to express the same point in R3, using cylindrical coordinates. Since θ=θ in both coordinate systems, we only need to use the equations r=ρsin⁡ϕ and z=ρcos⁡ϕ to complete our conversion. First, r=ρsin⁡ϕ=2sin⁡π12=2(3−122)=32−12. Next, z=ρcos⁡ϕ=2cos⁡π12=2(3+122)=32+12. Now we have the equivalence (2,3π8,π12)≡(32−12,3π8,32+12), exactly as desired. ### Example 5: Integration in Cylindrical Coordinates Coordinate systems other than the Cartesian coordinate system shine in the setting of multivariable calculus. In particular, certain triple integrals, which are used to integrate functions over three-dimensional regions, are easier to solve in cylindrical or spherical coordinates. Consider the following general form of a triple integral of a function f over a three-dimensional region D in cylindrical coordinates: ∭Df(r,θ,z)dzrdrdθ. It's important to note that the volume differential in cylindrical coordinates is dV=dzrdrdθ, as opposed to the volume differential dV=dxdydz in Cartesian coordinates. We integrate the innermost integral with respect to z, then integrate the middle integral with respect to r. Finally, integrate the outermost integral with respect to θ. Consider the triple integral ∫02π∫01∫r2−r2dzrdrdθ in cylindrical coordinates. We first compute ∫r2−r21dz=2−r2−r and rewrite our triple integral as the double integral ∫02π∫01r(2−r2−r)drdθ. Next, we compute ∫01r(2−r2−r)dr=23(2−1) and rewrite our double integral as the integral ∫02π23(2−1)dθ=4π3(2−1). We're done. Without digging into the details, it suffices to say that integrating over the same region in Cartesian coordinates would've been a headache, but still possible. ### Example 6: Integration in Spherical Coordinates Consider the following general form of a triple integral of a function f over a three-dimensional region D in spherical coordinates: ∭Df(ρ,θ,ϕ)ρ2sin⁡ϕdρdθdϕ. Again, it's important to note that the volume differential in spherical coordinates is dV=dρ⋅ρsin⁡ϕdθ⋅ρdϕ=ρ2sin⁡ϕdρdθdϕ, as opposed to the volume differential dV=dxdydz in Cartesian coordinates or dV=dzrdrdθ in cylindrical coordinates. We integrate the innermost integral with respect to ρ, then integrate the middle integral with respect to θ. Finally, integrate the outermost integral with respect to ϕ. Consider the triple integral ∫0π∫0π∫02sin⁡ϕρ2sin⁡ϕdρdθdϕ in spherical coordinates. We first compute ∫02sin⁡ϕρ2sin⁡ϕdρ=8sin4⁡ϕ3 and rewrite our triple integral as the double integral ∫0π∫0π8sin4⁡ϕ3dθdϕ. Next, we compute ∫0π8sin4⁡ϕ3dθ=8πsin4⁡ϕ3 and rewrite our double integral as the integral ∫0π8πsin4⁡ϕ3dϕ=π2. That's an elegant solution to our original problem. Again, we emphasize that if this triple integral had been computed using Cartesian coordinates, it would've taken significantly longer as compared to using spherical coordinates. In addition, it wouldn't have conveyed the essence of the surface over which we integrated. To unlock this lesson you must be a Study.com member [Create an account](/content/academy/plans.html) **Lesson Summary** There are several **coordinate systems** that systematically organize two- and three-dimensional space by using different pieces of numerical information called **coordinates**. In particular, we examined **polar coordinates** as an alternative to Cartesian (rectangular) coordinates in two dimensions, as well as polar coordinates' three-dimensional counterparts: **cylindrical coordinates** and **spherical coordinates**. Then we leveraged our knowledge of trigonometry to derive equations that can be used to translate points between those coordinate systems. These systems can be used to integrate over three-dimensional surfaces that would be far more difficult two work with as Cartesian coordinates. To unlock this lesson you must be a Study.com member [Create an account](/content/academy/plans.html) ## Video Transcript ## Mapping 3D Space When you look into the night sky, you can see about five thousand of the more than one hundred billion stars in our Milky Way Galaxy. Humans have charted the stars as long as they have been mapping large-scale features on Earth's surface. So why are maps of Earth's features so widely available, while maps of the locations of nearby stars in our galaxy are hard to come by? There are likely a few reasons. One big reason is that humans use land and sea maps regularly for navigation, but star maps? Not so much. Another reason is that accurately and meaningfully representing locations of features and relative distances in a three-dimensional space is not easy. The map itself needs to be 3-D if you don't want to lose information by suppressing one of the dimensions. Maps of Earth typically suppress elevation, which is why Earth's surface can be represented on a 2-D map. Let's see how spherical coordinates provide a natural way of representing the locations of stars in our local region of the galaxy. ## Polar Coordinate Systems The idea behind cylindrical and spherical coordinates is to use angles instead of Cartesian coordinates to specify points in three dimensions. Sometimes, employing angles can make mathematical representations of functions simpler. **Polar coordinates** represent points in the coordinate plane, not with the usual Cartesian ordered pair _(x, y)_, but with two different coordinates _(r, phi)_ that are functionally related to _(x, y)_. Specifically, for a given point _P_, _r_ is the absolute distance from the origin to _P_. The angle _phi_ is the angular position of _P_, with angle measured from the positive _x_-axis. Cylindrical and spherical coordinate systems are extensions of 2-D polar coordinates into a 3-D space. ## Cylindrical Coordinates **Cylindrical coordinates** are most similar to 2-D polar coordinates. Let's consider a point _P_ that has coordinates _(x, y, z)_ in a 3-D Cartesian coordinate system. The same point can be represented in cylindrical coordinates _(r, phi, z)_ where _r_ and _phi_ are the 2-D polar coordinates of _P_'s image in the _xy_ plane ( _z = 0_), and _z_ is exactly the same as _P_'s Cartesian z-coordinate. Here is the relationship between a point's Cartesian and cylindrical coordinates on a graph: | | | --- | | | Fig. 1: Cylindrical coordinates r, phi and z The coordinate transformations to go from Cartesian _x_ and _y_ coordinates to cylindrical _r_ and _phi_ coordinates are as follows: | | | --- | | | Eqs. 1: Cartesian to Cylindrical Coordinate Transformation These are the same as the transformation to 2-D polar coordinates. There are a few features of this transformation to notice. First, the coordinate _r_ under this transformation is always a positive number. _r_ is interpreted as the smallest distance from _P_ to the _z_ axis. Also, _phi_, expressed in radians, will always be between _-pi_ and _pi_. The _z_ coordinate keeps the same value as you transform from one system to the other. The inverse transformation from _(r, theta, z)_ to _(x, y, z)_ may also be familiar from 2-D polar coordinates as well. | | | --- | | | Eqs. 2: Cylindrical to Cartesian Coordinate Transformation Let's consider the following question as an example of applying the coordinate transformation: what are the Cartesian coordinates _(x, y, z)_ of the point P specified by cylindrical coordinates _(2, -pi/6, 1)_? In moving from cylindrical to Cartesian coordinates, the _z_-coordinate does not change. _z_ is conventionally the third value in the ordered triplet, therefore, _z = 1_ in both cylindrical and Cartesian coordinates. Now, to find _x_ and _y_, we should plug in values _r = 2_ and _phi = -pi/6_ into the transformation equations. | | | --- | | | Therefore, the Cartesian coordinates of this point are _(sqrt(3), -1, 1)_. ## Spherical Coordinates Cylindrical coordinates are not the only way to specify a point in a 3-D space using an angle. **Spherical coordinates** are another generalization of 2-D polar coordinates. However, in this coordinate system, there are two angles, _theta_ and _phi_. Let's consider a point _P_ that is specified by coordinates _(x, y, z)_ in a 3-D Cartesian coordinate system. The same point can be represented in spherical coordinates as _(r, theta, phi,)_ where _r_, _theta_, and _phi_ are functionally related to _x_, _y_, and _z_, as we will see. In spherical coordinates, _r_ is the distance from the origin to point _P_ along the line connecting them. The first angle, _theta_, is often called the **polar angle** because it runs between the 'poles' of the coordinate system, the negative and positive _z_-axes. _Theta_ takes the value 0 along the positive _z_-axis and _pi_ along the negative _z_-axis. The second angle, _phi_, is called the **azimuth angle**, and it is identical to the angle _phi_ in cylindrical coordinates. | | | --- | | | Eqs. 3: Cartesian to Spherical Coordinate Transformation The coordinate transformations take a point _P_ in Cartesian coordinates to its corresponding spherical coordinates. There are a few features to note in this transformation. The radial coordinate _r_ under this transformation is always a positive number that is exactly equal to the Euclidean distance from _P_ to the origin. Also note that the two angles take different ranges of values. Polar angle _theta_, expressed in radians, must always be between 0 and _pi_, but azimuth angle _phi_ can point in any direction in the _xy_ plane, so it takes values from _-pi_ to _pi_. Based on the geometry, the inverse transformation takes spherical coordinates back to Cartesian coordinates. | | | --- | | | Eqs. 4: Spherical to Cartesian Coordinate Transformation Let's consider an example: what are the spherical coordinates _(r, theta, phi)_ of the point P specified by Cartesian coordinates _(3, -sqrt(3), -2)_? In moving from Cartesian to spherical coordinates, we should use these calculations: | | | --- | | | ## Application: Celestial Maps Star maps are a common application of spherical coordinates. As observed on Earth, the stars appear to us on the inside of a sphere centered on us. Although early astronomers and philosopher believed the stars were all equidistant from us, we know now that the exact distance to each star we see is different. In terms of spherical coordinates, the relative positions of the stars in the sky can be specified by two numbers _(theta, phi)_. The actual 3-D distance can be incorporated by given an _r_ coordinate for each star. Celestial maps for stargazers tend to use a complementary angle to polar angle _theta_, which is called the **altitude**, to locate a star. | | | --- | | | Fig. 3: A schematic of coordinates used in star maps. ## Lesson Summary Polar coordinate systems use angles as coordinates of points. Cylindrical and spherical coordinate systems are generalizations of 2-D polar coordinates into three dimensions. **Polar coordinates** represent points in the coordinate plane, not with the usual Cartesian ordered pair ( _x_, _y_), but with two different coordinates ( _r_, _phi_). **Cylindrical coordinates** are most similar to 2-D polar coordinates. They use ( _r_, _phi_, _z_) where _r_ and _phi_ are the 2-D polar coordinates of _P_'s image in the _x_- _y_ plane and _z_ is exactly the same as _P_'s Cartesian _z_ coordinate. In **spherical coordinates**, another angle, the polar angle _theta_, is also defined to specify a point in 3-D. These coordinate systems are used by astronomers and engineers to simplify mathematical models of systems of interest. #### Register to view this lesson Are you a student or a teacher? I am a student I am a teacher #### Unlock your education See for yourself why 30 million people use Study.com Become a Study.com member and start learning now. [Become a member](/content/academy/lesson/cylindrical-spherical-coordinates-definition-equations-examples.html#partialRegFormModal/index.html) Already a member? [Log in](/content/academy/login.html) Go back ### Resources created by teachers for teachers Over 30,000 video lessons & teaching resources—all in one place. 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