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- Multivariable Calculus
- Three Dimensions

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Get Started Now- Intro Lesson: a9:09
- Intro Lesson: b3:50
- Intro Lesson: c7:29
- Lesson: 17:36
- Lesson: 27:48
- Lesson: 34:41
- Lesson: 44:38
- Lesson: 54:38
- Lesson: 63:56
- Lesson: 78:26

Instead of Cartesian coordinates, we use spherical coordinates for situations you will see in future sections.

For now, we will just learn how to convert from 3D Cartesians coordinates to 3D spherical coordinates, and vice versa. In other words,

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

Suppose we have the following graph:From the graph, we can obtain the following equations which will be useful for converting spherical to cartesian, or vice versa:

$r = \rho \sin\varphi$

$z = \rho \cos\varphi$

$z^2 + r^2 = \rho^2$

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

$y = \rho \sin \varphi \sin \theta$

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

- Introduction
**Spherical Coordinates Overview:**a)__Spherical Coordinates__- Instead of $(x,y,z)$, we have $(\rho,\theta,\varphi)$
- Graph of the coordinates in 3D

b)__Equations to Convert from Cartesian to Spherical__

- Finding the Equations
- Trig Ratios, Pythagoras
- Using Equations from last section to obtain more equations!

c)__Example of Converting Equations__- Cartesian to Spherical
- Spherical to Cartesian

- 1.
**Converting Cartesian Points into Spherical Coordinates**

Convert the following Cartesian Point into Spherical Coordinates:

$(2,5,1)$

- 2.Convert the following Cartesian Point into Spherical Coordinates:

$(-2,-1,-4)$

- 3.
**Converting Cartesian Equations into Spherical Coordinates**

Convert the following Cartesian Equation into Spherical Coordinates:$x^2 + y^2 - 2y = 3$

- 4.Convert the following Cartesian Equation into Spherical Coordinates:
$\frac{x}{z} = \frac{y}{x}$

- 5.
**Converting Spherical Equations into Cartesian Coordinates**

Convert the following Spherical Equation into Cartesian Coordinates:$\rho^2 = \sin\varphi \cos\theta$

- 6.Convert the following Spherical Equation into Cartesian Coordinates

$\csc \varphi = \rho \sin \theta + \rho \cos \theta$

- 7.
**Graphing the Spherical Equations by Changing into Cartesian Coordinates**

Graph the following Spherical Equation in the Cartesian Plane.$\rho^2 = 2\rho \sin\varphi \sin\theta + 4$