Basic Notation Q's: x, y, z, u, f(x)

[The_ArtofScience]

I don't know if this is a terribly ignorant question but I was wondering what exactly is y(0)=1? Is that like f(0)=1 where the x is 0 and the y is 1?

I'm also curious about the expression u=f(x, y, z). What is that exactly? How is it related to the x, y, z coordinates?

 

[HallsofIvy]

Since you haven't said what you mean by "f" or "y", it's hard to be precise. IF you have some function f(x) (or g(t), h(u), etc.) that you want to graph on an xy-coordinate system, since it is standard to use the x-axis to represent the independent variable (x or t or u or whatever) and the y-axis to represent the dependent variable (f(x) or g(t) or h(u) or whatever), yes it is standard to y= f(x) or y(x) in place of f(x). If f(0)= 1 and you have assigned x to the independent variable and y to the dependent variable, perhaps for graphing purposes, then you would have y(0)= 1. On a graph, that would be a point exactly one unit above the origin on the y-axis.

u= f(x,y,z) is a function that assigns a value to each possible combination of three independent values. Although, like all of mathematics, it can be used in many different ways, it is a common application to think of x, y, z as coordinates that designate points in space. One specific application would be to think of u as air temperature. A coordinate system would assign three numbers to each point in space. u(x,y,z) would be the temperature at each point.

 

[The_ArtofScience]

Thank you, Ivy. I really liked your explanation.

I'm still curious though, if f(x,y) represents z, then what would f(x,y,z) be? Its hard to imagine another coordinate in the xyz coordinate plane, and if there is another coordinate, I might be overlooking it.

 

[CompuChip]

f(x, y, z) is a function that gives a value for each point in space, rather than each point in a plane. The example that HallsOfIvy gave, is the function that assigns to every point in the room the air temperature at that point.

The problem with functions of three and more variables is that we cannot draw them because we don't have enough dimensions. A function f(x) of one variable, can be drawn in two dimensions (set up an x-axis, and for each point x draw the corresponding value f(x) directly above it on the y-axis). Functions of two variables can still be drawn: above each point in the plane, draw the function value on the z-axis, like https://www.physicsforums.com/latex_images/96/968788-0.png . But to draw a function of three variables, you would have to have four perpendicular axes. Unfortunately we only live in a three-dimensional space, so it's very hard imagining this (and impossible doing it intuitively, let along projecting it on a two-dimensional piece of paper). Best thing you can do is try to think of the temperature analogy. You can then extend this also to a function of four variables, f(x, y, z, t) which would give the air temperature in the room at a given point (x, y, z) and a time t.
Another way to look at it: consider a function like a slot machine. It has some number of input slots, which we label by arbitrary letters (x, y, z, ...) -- if you put numerical values on these slots the function machine spits out a value. For example, the function $f(x) = x^2$ has one input slot x: if I insert x = 3 it spits out 9 and if I put in x = 1/2 it spits out 1/4. The advantage of a function of one (two) variables is that it can be graphed, by putting all possible input values on the x-axis (x and y axes) and the corresponding output on the y (z) axis, which gives you an insight in how the number is produced (e.g. if I slightly increase x, what will happen to f(x)), an advantage you don't have in more dimensions (though, in more advanced mathematics there are ways to describe the behavior of arbitrary functions which are almost as good as drawing graphs).

 

[HallsofIvy]

Thank you, Ivy. I really liked your explanation.

I'm still curious though, if f(x,y) represents z, then what would f(x,y,z) be? Its hard to imagine another coordinate in the xyz coordinate plane, and if there is another coordinate, I might be overlooking it.

I didn't say "f(x,y) represents z". f(x,y) is a function of two variables and may have nothing to do with 'z'. If could, for example, represent the temperature at some point (x,y) on a heated plate. It depends upon the application of the function.

Of course, you can't have both "z= f(x,y)" and "f(x,y,z)"- but that has nothing to do with z. You are using "f" to mean two different things; it can't be both a function of two variables and a function of three variables.