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Functions of several variables
Now that we have a basic understand of Rn and its geometry
webegin studying functions whose domain (and range) are subsets
ofRn.Real valued functions of a real variable: e.g. f : R→ R,f (x)
= x2 + 1.
www.math.uiuc.edu/∼clein/classes/2014/fall/241.html
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Functions of several variables
For a real valued functions of two real variables we input
TWOnumbers and the output is a single number:e.g. h : R2 → R, h(x ,
y) = x2y − ey .
www.math.uiuc.edu/∼clein/classes/2014/fall/241.html
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Functions of several variables
R2–valued functions of four real variables:e.g. g : R4 → R2,
g(x1, x2, x3, x4) = (x1 cos(x2), x4 log(x23 + 1))
www.math.uiuc.edu/∼clein/classes/2014/fall/241.html
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Functions of several variables
Real world examples:
• T : R2 → R, T (x , y) = temperature atlatitude x , longitude y
.
• W : R2 → R2 wind speed and direction:W(x , y) = wind vector at
latitude x , longitude y .
www.math.uiuc.edu/∼clein/classes/2014/fall/241.html
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Graphs
Question: How do we “picture” functions of one variable,f : R→
R?Answer: Graphs.
{(x , f (x)) ∈ R2 | x ∈ R}.
f(x) (x,f(x))
x
Example: f (x) = x2 + 1.
www.math.uiuc.edu/∼clein/classes/2014/fall/241.html
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Graphs
Question: How do we “picture” functions of two variables,f : R2
→ R?
Answer: Graphs again!
{(x , y , f (x , y)) ∈ R3 | (x , y) ∈ R2}.
2
(1,2,5)
(1,2)
5
1
Example: f (x , y) = x2 + y2.
Can construct graph by intersecting with planes...
www.math.uiuc.edu/∼clein/classes/2014/fall/241.html
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Slicing with planes
To picture the graph, we can intersect with planes of our
choice.
• y = constant... f (x , y) becomes essentially a function of
theremaining variable x .
• x = constant... f (x , y) becomes essentially a function of
theremaining variable y .
• z = constant... level curves.
www.math.uiuc.edu/∼clein/classes/2014/fall/241.html
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Level curves
For a function f (x , y), the level curves are the family of
curves inthe xy–plane, {(x , y) | f (x , y) = c}c∈R.
Familiar example: h(x , y) = height above sea level at (x ,
y)
www.math.uiuc.edu/∼clein/classes/2014/fall/241.html
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Level curves
Example: f (x , y) = y2 − x2.
Sketch the level curves, then sketch the graph.
www.math.uiuc.edu/∼clein/classes/2014/fall/241.html
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Quadric surfaces
Similar techniques ⇒ visualize solutions to quadratic
equations.
www.math.uiuc.edu/∼clein/classes/2014/fall/241.html
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Higher dimension
Same idea for functions of more variables (limited
visualization).
Example: f (x , y , z) = x2 + y2 + z2.Level surfaces: {(x , y ,
z) | f (x , y , z) = constant }.
www.math.uiuc.edu/∼clein/classes/2014/fall/241.html
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Higher dimensions still...
Lots of pretty pictures from functions f : R3 → R3.Hopf
fibration is such an example when image lies in the
unitsphere...
www.math.uiuc.edu/∼clein/classes/2014/fall/241.html
