This may be dumb, but I’m wondering what are possible advantages and disadvantages of calculations in space as a sphere around a point. Instead of x,y,z being straight axes, you could imagine x and y as latitude and longitude on a sphere and z as distance from the centre. Moving in straight lines gets weird I think, but I don’t know enough about math to properly reduce the equations into their simplest forms. I’m wondering what it would mean to work in 3d space fully and only defined in this way though.
Not a dumb question. What you’ve described is well-known as spherical coordinates. They’re not used as often as the usual rectangular coordinates, but they do have their uses. Some things are easier to describe in one set of coordinates, some in another.
The change is easy enough if you’re familiar with it but for general day to day thinking Cartesian co-ordinates are simply three distance units (x, y, z). Spherical coordinates however require that you consider distance and 2 angles (r, θ, φ). Elegant but maybe not as intuitive.
I some ways, more intuitive. We experience the world in a kind of spherical coordinate system – when looking at an object in our field of vision we consider how far it is away from us (r) and where it is in our field of vision (or how we have to turn our head to be looking directly at it) which incorporates θ and φ. Traditional Cartesian coordinates are more abstract than that.
And just to be complete about it, θ is usually measured from 0 to 360 degrees or 0 to 2π radians rather than +/- or E/W longitude. φ is generally measured as 0 to 180 degrees or 0 to π radians from the “north” pole.
There are actually two different conventions about this. Mathematicians use the convention you describe; physicists swap the roles of θ and φ (i.e., φ is the azimuthal angle and θ is the polar angle.) But the systems are fundamentally the same.
All the Calculus textbooks I’ve ever used (admittedly, a small sample) have used the convention that OldGuy described, but have used the Greek letter ρ (rho) for the distance from the origin sqrt(x² + y² + z²), with r standing for the radial coordinate in polar and cylindrical coordinates sqrt(x² + y²). I hadn’t realized until this thread (and the research it led me to do) that there were so many different conventions.
Astronomers nearly always use spherical coordinates. The location of any star in the sky can be defined using the distance in light years, the Right Ascension (measured in hours, with twenty four hours in the full circle) and the Declination (measured in degrees above and below the celestial equator.
I very much prefer cartesian xyz coordinates myself when thinking about the universe, but these coordinates are subject to many inaccuracies. Right ascension and declination, on the other hand, can be measured very accurately.
Molecular structures (DNA, Polymers, etc.) are better described in spherical coordinates too. Physical phenomenon like momentum transfer , mass transfer (diffusion) , heat transfer etc. in Fluids are better visualized in spherical coordinates.
And just for grins, we also have cylindrical coordinates in which a point is located by a distance from the z-axis, an angle from the positive x-axis (so far, these two coordinates are just your normal polar coordinates), and a distance z parallel to the z-axis.
The reason students tend to prefer good old Cartesian coordinates, and shy away from the other kinds, is that all the other coordinate systems involve angles, which means ( <gasp!> <shudder!> ) using trigonometry <*the *horror!**> All those equations with sines and cosines and tangents seem to scare students away.
In fact, certain kinds of problems work out much easier in polar coordinates than in rectangular coordinates, if only students can get over their primal fear of equations with sines and cosines and tangents.
And, if you want to get really exotic, certain scattering problems in quantum mechanics with a Coulomb potential decompose really well in parabolic coordinates.
Oh, there are more exotic coordinate systems than that. There’s the Prolate Spheroidal Coordinate system, for instance
and Oblate Spheroidal Coordinates
and Ellipsoidal Coordinates
And others listed here (Orthogonal coordinates - Wikipedia ) and in books like Morse and Feshbach’s Methods of Theoretical Physics.
One generally chooses one’s coordinate system by the symmetries of the problems one is trying to solve, not from a sense of perversity. As such, I’ve used rectangular, cylindrical, and spherical coordinate systems (and flirted with non-orthogonal rectangular coordinates for problems with non-cubic crystal structures), but never had occasion to use any of the more exotic coordinates. They’re more of a curiosity to me. But many of the references will give the forms of vector operations like Div, Grad, and Curl on vectors as written in the basis systems for each coordinate system, and they’re interesting to look at.
I’m not sure that’s the only reason. It took me a while to really grok non-Cartesian coordinate systems in calculus.
A volume element in Cartesian space is exactly dxdydz. Easy to visualize since it’s just a box.
In cylindrical coordinates, it’s dzdrdθ*r. That’s already looking a bit weird since there’s a volume dependence on where you are in space. And you have to do a bit of geometry to prove to yourself that it’s the right answer.
In spherical coords, it’s drdθdφ*r[sup]2[/sup]*sin(φ). Now that’s really strange! The element is a weird little wedge of a shell. The formula isn’t really right anyway, but the errors go away as the parameters go to zero. So the student has to recognize this if they’re going to work out the formula from first principles.
I certainly agree that some stuff becomes much easier in the right coordinate system, but getting to that point can be a challenge.
A lot of things are easy to understand in math if you can visualize them. It helps to visualize things in the right (i.e., more intuitive) way. Picture a curve in a plane, and you are trying to find the area under it. Now picture a vertical line, moving from left to right leaving a bright area in its wake under the curve (think of filling an area on a chalkboard by dragging the side of the chalk on the board). Think of integration as toting up the area thus swept out, as opposed to thinking of a bunch of discrete narrow rectangles.
Now think, instead, of a circle where you are doing this. The math for toting up that area as it is horizontally swept out by a moving vertical line, is a bit messy. (Involves Pythagorean expressions with square roots of (x[sup]2[/sup] + y[sup]2[/sup]) sort of stuff.)
Now think, instead, of a point moving around the circle, and a radial line from the center to that moving point, sweeping out the area of the circle circularly. That’s a much more natural way to imagine producing a circular region. Think of the old-fashioned radar scopes you saw in old WWII movies, with the rotation radial line on the screen sweeping out the circular area, and lighting up the screen every time it swept over an enemy aircraft.
If that seems obvious and natural and straightforward (as it did to me), then ask yourself: What must the math be to describe that process? It must be natural and straightforward too (even if it does obviously require sines or cosines or polar coordinates or something). Then try to figure out the math to do it.
Exercise for the reader: If sweeping out a circle radially is such a “natural” way to do it, how about an ellipse? Picture an existing ellipse, with a point moving around the perimeter, and a radial line (following that point) sweeping out the area of the ellipse. Now, write the integral to do that in polar coordinates!
I really did that once – with a more complicated problem – on a FINAL EXAM. We were given the equation of an elliptical paraboloid, centered about the z-axis, opening downward with its vertex on the positive z-axis (getting the picture?), thus having an elliptical cross-section in the xy plane. Problem: Find the volume under the paraboloid above the xy plane.
It seemed so natural to do it with polar coordinates, but it took me a long time to figure out how to do it with an ellipse instead of a circle. But it seemed like it ought to be perfectly simple! Once I got it figured out, it was much simpler! (ETA: It helped that this was a take-home final exam!)