1 Dimensions
(Relevant Section of the Textbook: 12.1 Three Dimensional Coordinate Systems, and 10.3 Polar Coordinates)
To do calculus in higher dimensions, we need first a precise mathematical language that will allow us to describe these spaces. And that language begins with a foundational, but straightforward definition: the \(n\)-tuple.
Definition 1.1 (n-Tuple) An \(n\) tuple of real numbers is an ordered list of real numbers. For example, a 2-tuple like \((3,7)\) is often called an ordered pair. Tuples are sometimes written horizontally and sometimes vertically, depending on convenience. Various styles of brackets are used on tuples, depending on the author and usage. Below are examples of a 3-tuple, a 4-tuple, and a 7-tuple in several styles: \[\langle 0,-12,0.3\rangle\hspace{1cm}\pmat{3\\ -7\\ \pi \\4}\hspace{1cm}[1,2,3,4,5,6,7]\]
Just as numbers represent a location on the line, tuples can be used to represent locations in space. When we use them as such, we call the entire \(n\)-tuple a point, and we call each of the entries a coordinate.
Definition 1.2 (Point) A point is an \(n\)-tuple when it is being used to represent a location in space.
You are already familiar with this from single variable calculus, where we use orderd pairs \((x,y)\) to represent points of the 2-dimensional plane \(\RR^2\). By extension, we can use three-tuples \((x,y,z)\) to represent points in the physical space around us. But what about even bigger tuples, like the \(7\)-tuple \([1,2,3,4,5,6,7]\)? What kind of space does this represent a point in? This is a point in a seven dimensional space!
Definition 1.3 The dimension of a space is the number of coordinates needed to describe a point in it.
A plane is 2-dimensional, but so is the surface of a sphere: if your friend called you and gave you two numbers - their latitude and longitude - you could precisely locate them on the Earth’s surface.
\[(37.7749^\circ N, 122.4194^\circ W)\]
The space around us is three dimensional because if I wanted to direct you to my apartment I would need to give you not only the two street intersections (two numbers, specifying a point on the earth’s surface) but also the floor I live on (the height above the surface).
\[(1^{st}\mathrm{ Street}, 3^{rd}\mathrm{ Ave}, 4^{th}\mathrm{ Floor})\]
But the space-time we live in is four-dimensional because if we wanted to meet for lunch I would need to give you four numbers, my position in space and also when to meet, so that we do not miss each other, you thinking luch is at 11 and I thinking noon.
\[(1^{st}\mathrm{ Street}, 3^{rd}\mathrm{ Ave}, 4^{th}\mathrm{ Floor}, 12\mathrm{pm})\]
Thus, there are direct phyiscal reasons to consider calculus in two, three and four dimensions. And, our best physical theories of the world at human scales (classical mechanics) are written in this language. Understanding weather, planetary motion, fluid flow, and black holes requires a solid grounding in multivariable calculus. But, since the real world is only four dimensional, does that mean there is no need for the calculus of 7, or 13, or 132,234,453 dimensional space?
1.1 Higher Dimensions and Configurations
All the spaces we have talked about so far represent physical space, but mathematics is allows us to be much more general than this. Imagine you are designing a tin can, and you want to start by thinking of the space of possibilities: what are all the possible shapes of a cylindrical can? As such a can is fully determined by its radius and its height, we can think of these as being two coordinates, and expressing a particular can by an ordered pair \((r,h)\). Thus, the space of possible cans is 2-dimensional!
What about the space of \(L\)-spaced desks? What is the dimension of this space? This space has 5-dimensions: the length and width of each of the two sides of the desk, and also its height.
\[(\ell_1,\ell_2,w_1,w_2,h)\]
But where really mind-blowing numbers of dimensions begin to arise is in the study of data. Imagine you are modeling the conditions in the San Francisco bay, and you take a measurement of the sea height for every square kilometer. The bay has an area of 4000 square kilometers, so this means your datapoint has \(4,000\) numbers in it! Your approximation to the bay’s surface is a point in four thousand dimensional space!
Or, consider an image taken by a digital camera: for simplicity assume the image is in black and white, and 10 megapixels. This means each pixel is determined by a single number (how light or dark the pixel is), and the image has 10 million pixels, so it’s encoded by ten million numbers! That means even simple images are worked with mathematically as points in a space with millions of dimensions.
1.2 Cartesian Coordinates
What do the actual numbers in the \(n\)-tuple mean? In the examples above we have been implicitly using \(xyz\) or length-width-height coordinates: the first number tells you the distance left/right, the second back/forth and the third up/down. These are called Cartesian Coordinates after the mathematician-philosopher Rene Descartes. While this sort of thinking comes from physical space, it is useful to help give a concrete picture even to non-physical spaces such as the space of soup cans, or the space of images.
Definition 1.4 (Cartesian Coordinates) Cartesian coordinates starts by choosing \(n\) perpendicular lines in \(n\) dimensional space: for example the \(x\) and \(y\) axes in the plane, or the \(x,y,z\) axes in \(\RR^3\). A point in space is given coordinates \((a,b,c)\) if it lies at distance \(a\) along the first axis, \(b\) along the second, and \(c\) along the third.
Here’s an animation showing a point in 3D space, and its components along the \(x,y\) and \(z\) axes.
Perhaps the most famous theorem of geometry is the Pythagorean Theorem, which tells us how to compute distance in the cartesian coordinates on the plane:
Theorem 1.1 The distance of the point \((a,b)\) from the point \((0,0)\) in the plane is \[\mathrm{dist}=\sqrt{a^2+b^2}\]
Once we know this theorem is true in \(\RR^2\) (thanks, Pythagoras!) we can use this proof to extend it to higher dimensions!
Theorem 1.2 In \(\RR^3\), the distance of \((x,y,z)\) from the origin is \[\mathrm{dist}=\sqrt{x^2+y^2+z^2}\] In general, if \((x_1,\ldots, x_n)\) is a point in \(\RR^n\), its distance from the origin is \[\mathrm{dist}=\sqrt{\sum_{i=1}^n x_i^2}\]
Given this description of distance, we can give a precise description of circles and spheres: a circle is the set of points which are all a fixed distance (the radius) from a fixed point (the center). The unit circle is the set of points distance 1 from the origin. Similarly, the unit sphere is the set of points distance 1 from the origin in 3-dimensional space.
In more generality, we can define a hypersphere in any higher dimension the same way: by taking the set of points which are a fixed distance from the origin in that space!
Definition 1.5 The unit circle (sometimes called the unit 1-sphere) is the set of points in \(\RR^2\) given by \[x^2+y^2=1\] The unit sphere (sometimes called the unit 2-sphere) is the set of points in \(\RR^3\) satisfying \[x^2+y^2+z^2=1\] The unit 3-sphere is the set of points in four dimensional space satisfying \[x^2+y^2+z^2+w^2=1\] And so on…
Exercise 1.1 Why is the circle called a 1-sphere, and the sphere in $3D space called the 2-sphere?
We can also use cartesian coordiantes to describe other simple shapes, like lines. In \(\RR^2\) with cartesian coordinates, the \(x\)-axis is given by the set of points \(\{(x,0)\}\): that is, every point on the \(x\)-axis has \(y\)-coordinate equal to zero.
Definition 1.6 (Coordinate Axes and Planes) The \(x\) axis in the plane is given by the equation \(y=0\), and the \(y\) axis by the equation \(x=0\). Similarly in \(\RR^3\), the \(xy\) plane is given by the equation \(z=0\), the \(yz\) plane by the equation \(x=0\), and the \(xz\) plane by the equation \(y=0\).
A point can be projected onto a coordinate axis or plane by setting that coordinate to zero. The resulting point is the closest point on that line or plane to the original point. This makes it a relatively straightforward calculation to find the distance from a point to a coordinate axis/plane!
Here’s some practice problems:
1.3 Other Coordinate Systems
While the majority of our class will occur in Cartesian coordinates as they are the first coordinate system everyone must master, we will at times consider a couple of other interpretations of \(n\)-tuples, which make describing certain systems with circular or spherical symmetry easier. The first of these may already be familiar from earlier calculus classes: polar coordinates on the plane.
Definition 1.7 (Polar Coordinates) A point \((r,\theta)\) in polar coordinates on the plane, for \(r>0\) and \(\theta\in [0,2\pi)\) represents the point which lies at a distance \(r\) from the origin, and makes an angle of \(\theta\) with the positive \(x\)-axis.
From this description we can convert a point in polar coordinates to cartesian coordinates \((x,y)\) using trigonometry:
Definition 1.8 The conversion from polar coordinates \((r,\theta)\) to cartesial coordiantes \((x,y)\) is given by \[\pmat{x\\ y}=\pmat{ r\cos\theta\\ r\sin\theta}\]
Using polar coordinates simplifies many things when circles are involved: for example, the equation of the unit circle \(x^2+y^2=1\) becomes the much simpler equation \(r=1\) in polar coordinates!
Using polar coordinates for \(x,y\) in the 3-dimensional space \((x,y,z)\) gives a coordinate system called cylindrical coordinates.
Definition 1.9 (Cylindrical Coordinates) A 3-tuple \((r,\theta,z)\) represents a point in \(\RR^3\) using cylindrical coordiantes where the position in the \(xy\) plane is given by the polar coordinates \((r,\theta)\) and the height above the \(xy\) plane is given by \(z\). The conversion to carteisan coordinates is \[\pmat{x\\ y\\ z}=\pmat{ r\cos\theta\\ r\sin\theta \\ z}\]
Finally we will come across a third coordinate system in this class: spherical coordinates. This represents 3-dimensional space starting with a collection of concentric spheres. Don’t worry too much about this now, we will come back to it in some weeks! I’ve only placed it here for your future reference.
Definition 1.10 (Spherical Coordinates) A 3-tuple \((\rho,\theta,\phi)\) represents a point in \(\RR^3\) using spherical coordiantes where \(\rho\) is the distance from the origin, \(\theta\) is the angle with the positive \(x\) axis (as in polar coordiantes) and \(\phi\) is the angle with the positive \(z\)-axis. The conversion to cartesian coordinates is given by \[\pmat{x\\ y\\ z}=\pmat{ \rho \cos\theta\sin\phi\\ \rho\sin\theta\sin\phi\\ \rho\cos\phi}\]