From Section 6.1 Functions of Several Variables
A real-value function of f of x, y, z, … is a rule for manufacturing a new
number, written f(x, y, z …), from the values of a sequence of independent
variables (x, y, z, …) .
A real-value function is a rule. It’s a rule for coming up with/outputting a number. You follow the rules and you get a number. The output is based on the values of independent variables.
The graph of the function f
of two variables
consists of all points of the form
(x, y, f(x, y)) .
The graph of a function that has two variables consists of points coming from both variables and their output.
Hmm. Is the graph of a function that has one variable this then: (x, f(x)). I think I’ve seen it like that before. So graphs of a function consist of the points going into the function and the output.
In 3d space, there are 3
mutually perpendicular axes:
• x-axis (extend to the front)
• y-axis (extend to the right)
• z-axis (extend upwards)
Huh. That seems so odd to me. I’ve never exactly dealt with space before, but just going off of how an xy coordinate plane looks y just felt like the part that extends above.
So x-axis is in front and behind me
y-axis is to my right nad left
and z-axis is above and below
In 3d space,
there are 3 coordinate planes:
• xy-plane where z = 0
• xz-plane where y = 0
• yz-axis where x = 0
So a plane is when the graph looks flat? I guess? Also what does the coordinate have to do with anything here.
From Plane (mathematics) - Wikipedia :
In mathematics, a plane is a two-dimensional space or flat surface that extends indefinitely.
Ok. So I get what the plane here being referred to is. What’s the coordinate? From Gemini:
The term “coordinate plane” refers to a Cartesian plane, which is a two-dimensional plane defined by two perpendicular number lines called axes. It’s called a coordinate plane because it allows us to locate points by using coordinates, which are pairs of numbers that specify the point’s position relative to the axes. The plane is named in honor of René Descartes, who developed the system.
Hmm. It seems Gemini highlights the part it thinks I’m looking for. Neat.
Ok. That makes sense . So coordinate planes are planes that have coordinates. When you have three dimensional place you can form three different coordinate planes.
If we hold a variable constant,
the graph of the set of points is a plane.
Hmm. So the coordinate planes are when the variable being held constant is 0. At least thats what it seems like so far. Why is that? Hmm seeing this:
My thought process is that you could do a coordinate plane from there. Its just when doing coordinates I guess you assume its 0 otherwise you would list that third coordinate even if its a constant.
The graph of a function f of
two variables is the set of
points (x, y, f(x, y)) in three
dimensional space where we
restrict the values of (x, y) to
lie in the domain of f.
I forgot what a domain is. Let’s see. From Domain of a function - Wikipedia :
In mathematics, the domain of a function is the set of inputs accepted by the function.
Ok.