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### 17Calculus Subjects Listed Alphabetically

Single Variable Calculus

 Absolute Convergence Alternating Series Arc Length Area Under Curves Chain Rule Concavity Conics Conics in Polar Form Conditional Convergence Continuity & Discontinuities Convolution, Laplace Transforms Cosine/Sine Integration Critical Points Cylinder-Shell Method - Volume Integrals Definite Integrals Derivatives Differentials Direct Comparison Test Divergence (nth-Term) Test
 Ellipses (Rectangular Conics) Epsilon-Delta Limit Definition Exponential Derivatives Exponential Growth/Decay Finite Limits First Derivative First Derivative Test Formal Limit Definition Fourier Series Geometric Series Graphing Higher Order Derivatives Hyperbolas (Rectangular Conics) Hyperbolic Derivatives
 Implicit Differentiation Improper Integrals Indeterminate Forms Infinite Limits Infinite Series Infinite Series Table Infinite Series Study Techniques Infinite Series, Choosing a Test Infinite Series Exam Preparation Infinite Series Exam A Inflection Points Initial Value Problems, Laplace Transforms Integral Test Integrals Integration by Partial Fractions Integration By Parts Integration By Substitution Intermediate Value Theorem Interval of Convergence Inverse Function Derivatives Inverse Hyperbolic Derivatives Inverse Trig Derivatives
 Laplace Transforms L'Hôpital's Rule Limit Comparison Test Limits Linear Motion Logarithm Derivatives Logarithmic Differentiation Moments, Center of Mass Mean Value Theorem Normal Lines One-Sided Limits Optimization
 p-Series Parabolas (Rectangular Conics) Parabolas (Polar Conics) Parametric Equations Parametric Curves Parametric Surfaces Pinching Theorem Polar Coordinates Plane Regions, Describing Power Rule Power Series Product Rule
 Quotient Rule Radius of Convergence Ratio Test Related Rates Related Rates Areas Related Rates Distances Related Rates Volumes Remainder & Error Bounds Root Test Secant/Tangent Integration Second Derivative Second Derivative Test Shifting Theorems Sine/Cosine Integration Slope and Tangent Lines Square Wave Surface Area
 Tangent/Secant Integration Taylor/Maclaurin Series Telescoping Series Trig Derivatives Trig Integration Trig Limits Trig Substitution Unit Step Function Unit Impulse Function Volume Integrals Washer-Disc Method - Volume Integrals Work

Multi-Variable Calculus

 Acceleration Vector Arc Length (Vector Functions) Arc Length Function Arc Length Parameter Conservative Vector Fields Cross Product Curl Curvature Cylindrical Coordinates
 Directional Derivatives Divergence (Vector Fields) Divergence Theorem Dot Product Double Integrals - Area & Volume Double Integrals - Polar Coordinates Double Integrals - Rectangular Gradients Green's Theorem
 Lagrange Multipliers Line Integrals Partial Derivatives Partial Integrals Path Integrals Potential Functions Principal Unit Normal Vector
 Spherical Coordinates Stokes' Theorem Surface Integrals Tangent Planes Triple Integrals - Cylindrical Triple Integrals - Rectangular Triple Integrals - Spherical
 Unit Tangent Vector Unit Vectors Vector Fields Vectors Vector Functions Vector Functions Equations

Differential Equations

 Boundary Value Problems Bernoulli Equation Cauchy-Euler Equation Chebyshev's Equation Chemical Concentration Classify Differential Equations Differential Equations Euler's Method Exact Equations Existence and Uniqueness Exponential Growth/Decay
 First Order, Linear Fluids, Mixing Fourier Series Inhomogeneous ODE's Integrating Factors, Exact Integrating Factors, Linear Laplace Transforms, Solve Initial Value Problems Linear, First Order Linear, Second Order Linear Systems
 Partial Differential Equations Polynomial Coefficients Population Dynamics Projectile Motion Reduction of Order Resonance
 Second Order, Linear Separation of Variables Slope Fields Stability Substitution Undetermined Coefficients Variation of Parameters Vibration Wronskian

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17calculus > spherical coordinates

This page covers the basics of spherical coordinates and reference equations that use spherical coordinates for multi-variable calculus.

Spherical coordinates are very different from rectangular and cylindrical coordinates. There are still three coordinates, usually labeled $$(\rho, \theta, \phi)$$, but they are assigned so that spherical-type objects are easy to express. All distances and angles are measured based on a radial line drawn from the origin to the point. Table 1 describes how they are defined and Figure 1 shows this graphically.

Table 1 $$\rho$$ is the radial distance out from the origin to the point $$\theta$$ is the angle measured in the xy-plane from the positive x-axis to the shadow of the radial line. Note: This is the same angle as $$\theta$$ in cylindrical coordinates. $$\phi$$ is the angle measured from the positive z-axis to the radial line

Note - We have specifically chosen $$(\rho, \theta, \phi)$$ to describe spherical coordinates but not all books, instructors and videos use these same variables. Some use $$(r,\theta,\phi)$$ with $$\theta$$ and $$\phi$$ meaning different things depending on the context. (See the Wiktionary page on spherical coordinates for examples.) Check with your instructor and textbook to see which one they require.

$$(x,y,z) \to (\rho, \theta, \phi)$$

$$x=\rho\sin\phi\cos\theta$$

$$y=\rho\sin\phi\sin\theta$$

$$z=\rho\cos\phi$$

$$\rho^2=x^2+y^2+z^2$$

Table 2

The equations to convert from rectangular to spherical coordinates, in Table 2, are somewhat complicated. So take a few minutes to learn and memorize them (why memorize?). Look for similarities and differences between the equations and see if you can make some sense out of them.

Here is a quick video clip discussing these equations and showing a neat animation that will help you get a feel for spherical coordinates.

### MIP4U - Introduction to Spherical Coordinates [2mins-13secs]

video by MIP4U

Converting between Rectangular and Spherical Coordinates

Rectangular → Spherical

In order to convert from rectangular to spherical coordinates, the equations are Table 2 above are directly applied. The technique is not hard but it will help to watch a couple of quick videos showing how to do this with examples. The first video explains how to convert points from rectangular to spherical. The second video explains how to convert equations from rectangular to spherical. Both videos have plenty of examples. (If you need more examples, check the 17calculus spherical coordinates youtube playlist.)

### Krista King Math - Spherical Coordinates [3mins-53secs]

video by Krista King Math

### MIP4U - Converting Between Spherical and Rectangular Equations [8mins-15secs]

video by MIP4U

To be able to use spherical coordinates in triple integrals, you need to understand the unit vectors in spherical coordinates. Here is a video that will help.

### Michel vanBiezen - Spherical Unit Vector Conversions [9mins-45secs]

video by Michel vanBiezen

Before we go on, let's do a few practice problems.

Find an equation in spherical coordinates for the rectangular equation $$x^2 + y^2 - z^2 = 0$$.

Problem Statement

Find an equation in spherical coordinates for the rectangular equation $$x^2 + y^2 - z^2 = 0$$.

Solution

### 2209 solution video

video by MIP4U

Spherical → Rectangular

Sometimes it will be easier to evaluate an integral in rectangular coordinates rather than spherical coordinates. In this case, you may be given an equation in spherical coordinates and be asked to convert to rectangular coordinates.

To do this conversion, there is no set of equations with specific techniques. You just have to use trig identities and intuition to get an equation in rectangular coordinates. However, it sometimes helps to convert tangent, cotangent, secant and cosecant to sines and cosines and use $$\cos^2(t) + \sin^2(t) = 1$$. Try these practice problems to get the idea.

Instructions - Unless otherwise instructed, convert these equations in spherical coordinates to rectangular coordinates.

$$\rho = 2 \csc(\phi)$$

Problem Statement

$$\rho = 2 \csc(\phi)$$

$$x^2 + y^2 = 4$$

Problem Statement

$$\rho = 2 \csc(\phi)$$

Solution

### 2205 solution video

video by MIP4U

$$x^2 + y^2 = 4$$