Multivariable Calculus
Coordinates in 3D
Understanding graphs and surfaces requires us to delve a little deeper into the place where they live: \( \mathbb{R}^3.\)
This unit covers the essentials of three-dimensional coordinate systems. We'll need some right triangle trigonometry in order to construct the polar, cylindrical, and spherical coordinate systems.
We'll start with the 3D Cartesian system, which extends the familiar 2D \(xy\)-coordinate system into the full three dimensions of our day-to-day experience by adding a new axis. (Check out the animation below!)
Coordinates in 3D
The new axis is labeled \( z, \) and we have three coordinates \( ( x,y,z)\) for every point in space \( \mathbb{R}^3.\)
To get to the point \( (1, 2 , 2) \) pictured below (red dot), we start at the origin where all axes meet, move 1 unit in the positive \(x\) direction, 2 units in the positive \(y\) direction, and then 2 units up in the positive \(z\) direction.
What are the coordinates of the green triangle?
Coordinates in 3D
Now, what are the coordinates of the blue rectangle?
Coordinates in 3D
Not all creatures prefer Cartesian coordinates. Bees do a waggle-dance to communicate distance and direction information, in effect using a coordinate system based on circles instead of perpendicular lines. If we're to be at least as clever as bees, we should develop alternative coordinate systems, too!
Cylindrical and spherical coordinate systems for \( \mathbb{R}^3 \) are just such coordinate systems, and both are built up from polar coordinates \( (r,\theta) \) in \( \mathbb{R}^2, \) the system of choice for talkative bees. The rest of the unit will develop all three coordinate systems from scratch.
Coordinates in 3D
Let's start with polar coordinates.
We arrive at the planar point \( P =(x,y)\) by following a ray starting at the origin and making an angle of \( \theta \) with respect to the positive \( x\)-axis for a distance of \( r.\)
Trigonometry then tells us that \( x = r \cos(\theta ).\) What option best represents the relationship between \( y \) and the polar coordinates \( (r,\theta) ?\)
Coordinates in 3D
Polar coordinates and Cartesian coordinates are related through
\[ x = r \cos(\theta), \quad y = r \sin(\theta) .\]
Suppose we're given the Cartesian coordinates for \( P\) but want to know the polar form. Since \( \sin^2(\theta)+ \cos^2(\theta) =1 ,\)
\[ \begin{align}
\sqrt{x^2+y^2}
&= \sqrt{ \big[ r \cos(\theta) \big]^2 + \big[r \sin(\theta)\big]^2} \\
& = \sqrt{ r^2 \big[ \cos^2(\theta)+ \sin^2(\theta)\big]} \\
&= \sqrt{r^2 } \\
&= r.
\end{align}\]
What formula allows one to calculate \( \theta\) given the Cartesian coordinates \( (x,y)?\)
Coordinates in 3D
For 3D problems, we can build a hybrid of the 2D polar system and Cartesian coordinates called polar cylindrical (or just cylindrical) coordinates.
Suppose \( P = (x,y,z) \in \mathbb{R}^3.\) The first two numbers represent a point in the plane, which we can describe using \( ( r, \theta).\) The cylindrical coordinates of \( P \) are then \( (r,\theta,z). \)
Compute the cylindrical coordinates of \( (-2,-2, 5).\) The visualization provided below may be of some assistance. Adjust the controls until the point sits at the top of the vertical line....
Coordinates in 3D
Cylindrical coordinates are ideally suited for problems in \( \mathbb{R}^3 \) symmetric about the \( z \)-axis, like describing \( C_{R},\) the cylinder of radius \( R \) about the \( z \)-axis, which is made up of all points of distance \( R\) from this line.
Of the equations presented, which one best describes \( C_{R} \) in cylindrical coordinates?
Note: We use dashed lines to indicate portions of the picture that continue out to infinity. The cylinder above continues parallel to the \(z\)-axis in both the positive and negative directions.
Coordinates in 3D
When a problem has complete symmetry around the origin, spherical coordinates are usually better than cylindrical coordinates.
Suppose \( P = (x,y,z) \in \mathbb{R}^3. \) We know \[ x = r \cos(\theta),\quad y = r \sin(\theta) ,\] where \( r \) is the distance between the point \( ( 0,0) \) and \( (x,y) \) in the plane.
\( \theta \) is taken to be one of the new spherical coordinates; the other two are \( \rho\) \(\big(\)the distance between \( P \) and \( (0,0,0)\big) \) and \( \phi, \) the angle between the positive \( z \)-axis and the ray joining \( P \) with the origin.
Use the diagram above to relate \( r \) and \( \rho. \)
Coordinates in 3D
Since \( r = \rho \sin(\phi), \) \[ x = \rho \sin(\phi) \cos(\theta), \quad y = \rho \sin(\phi) \sin(\theta). \] We need to find a formula for \( z \) to complete the relationship between Cartesian and spherical coordinates.
Using the picture above, what is \( z \) expressed in terms of \( ( \rho, \theta, \phi) ?\)
Coordinates in 3D
Finally, let's understand why the coordinates \( ( \rho, \theta, \phi) \) given by \[ x = \rho \sin(\phi) \cos(\theta),\quad y = \rho \sin(\phi) \sin(\theta),\quad z = \rho \cos( \phi) \] are called spherical.
Of the options presented, which one correctly describes the sphere \( S_{R} \) of radius \( R \) centered at the origin in spherical coordinates?
Coordinates in 3D
We wouldn't get very far in multivariable calculus without the ability to visualize mathematical objects like surfaces in \( \mathbb{R}^3.\) The Cartesian, spherical, and cylindrical systems provide means for doing so.
We saw one example of this already when we visualized the graph of the depth function in the optimization unit. At a glance we were able to see where the extreme values of the depth function occur.
To close out our introductory chapter, we'll use the 3D coordinate system to understand what place integrals have in the world of multivariable calculus.