Multivariable Calculus

Coordinates in 3D

Understanding graphs and surfaces requires us to delve a little deeper into the place where they live: R3. \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 xyxy-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, z, and we have three coordinates (x,y,z) ( x,y,z) for every point in space R3. \mathbb{R}^3.

To get to the point (1,2,2) (1, 2 , 2) pictured below (red dot), we start at the origin where all axes meet, move 1 unit in the positive xx direction, 2 units in the positive yy direction, and then 2 units up in the positive zz 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 R3 \mathbb{R}^3 are just such coordinate systems, and both are built up from polar coordinates (r,θ) (r,\theta) in R2, \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) P =(x,y) by following a ray starting at the origin and making an angle of θ \theta with respect to the positive x x-axis for a distance of r. r.

Trigonometry then tells us that x=rcos(θ). x = r \cos(\theta ). What option best represents the relationship between y y and the polar coordinates (r,θ)? (r,\theta) ?


Coordinates in 3D

Polar coordinates and Cartesian coordinates are related through x=rcos(θ),y=rsin(θ). x = r \cos(\theta), \quad y = r \sin(\theta) . Suppose we're given the Cartesian coordinates for P P but want to know the polar form. Since sin2(θ)+cos2(θ)=1, \sin^2(\theta)+ \cos^2(\theta) =1 , x2+y2=[rcos(θ)]2+[rsin(θ)]2=r2[cos2(θ)+sin2(θ)]=r2=r. \begin{aligned} \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{aligned} What formula allows one to calculate θ \theta given the Cartesian coordinates (x,y)? (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)R3. P = (x,y,z) \in \mathbb{R}^3. The first two numbers represent a point in the plane, which we can describe using (r,θ). ( r, \theta). The cylindrical coordinates of P P are then (r,θ,z). (r,\theta,z).

Compute the cylindrical coordinates of (2,2,5). (-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 R3 \mathbb{R}^3 symmetric about the z z -axis, like describing CR, C_{R}, the cylinder of radius R R about the z z -axis, which is made up of all points of distance R R from this line.

Of the equations presented, which one best describes CR 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 zz-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)R3. P = (x,y,z) \in \mathbb{R}^3. We know x=rcos(θ),y=rsin(θ), x = r \cos(\theta),\quad y = r \sin(\theta) , where r r is the distance between the point (0,0) ( 0,0) and (x,y) (x,y) in the plane.

θ \theta is taken to be one of the new spherical coordinates; the other two are ρ \rho (\big(the distance between P P and (0,0,0)) (0,0,0)\big) and ϕ, \phi, the angle between the positive z z -axis and the ray joining P P with the origin.

Use the diagram to relate r r and ρ. \rho.


Coordinates in 3D

Since r=ρsin(ϕ), r = \rho \sin(\phi), x=ρsin(ϕ)cos(θ),y=ρsin(ϕ)sin(θ). x = \rho \sin(\phi) \cos(\theta), \quad y = \rho \sin(\phi) \sin(\theta). We need to find a formula for z z to complete the relationship between Cartesian and spherical coordinates.

Using the picture above, what is z z expressed in terms of (ρ,θ,ϕ)? ( \rho, \theta, \phi) ?


Coordinates in 3D

Finally, let's understand why the coordinates (ρ,θ,ϕ) ( \rho, \theta, \phi) given by x=ρsin(ϕ)cos(θ),y=ρsin(ϕ)sin(θ),z=ρcos(ϕ) 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 SR S_{R} of radius R R centered at the origin in spherical coordinates?


Coordinates in 3D

Visualizing mathematical objects like surfaces in R3 \mathbb{R}^3 can be very helpful in solving many multivariable calculus problems. We saw one example of this already when we visualized the graph of the depth function at the end of the optimization unit. At a glance we were able to see where the minimum and maximum values of the depth function occur.

The Cartesian, spherical, and cylindrical systems provide the means for visualizing a large variety of useful objects in multivariable calculus. The final unit of this introductory chapter shows one particular and very important example. There, we'll use the 3D coordinate system to understand what place integrals have in the world of multivariable calculus.


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