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Calculus III Hughs-Hallett
Math 232 A,B
Br. Joel Baumeyer
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Multivariable Calculus
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A function in Three Dimensional
Space (3-D), z = f(x,y):
is a function with two independent
variables; it is still a rule that
assigns for each of the two
independent variables, x and y,
one and only one value to the
dependent variable z.
e.g. h(x,t) = 5 + cos(0.5x - t) pg 4
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3-D Graphing Basics
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x coordinate measures the distance from the
yz-plane whose name is: x = 0
y coordinate measures the distance from the
xz-plane whose name is: y = 0
z coordinate measures the distance from the
xy-plane whose name is: z = 0
distance:
d 
( x  a)2  ( y  b)2  ( z  c) 2
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Graphs of Functions in 2
Variables
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The graph of a function of two
variables, f, is the set of all points
(x,y,z) such that z = f(x,y).
The domain of such a function is
is a subset of points in the real
Euclidean plane. In general, the
graph of a function of two variables
is a surface in 3-space.
In particular, the graph of a linear
function in 3-space is a plane:
ax + by + cz = d.
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Section of a Graph of a
Function in 3-D
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For a function f(x,y), the function
we get by holding x fixed and
letting y vary is called a section of
f with x fixed. The graph of the
section of f(x,y) with x = c is the
curve, or cross-section, we get by
intersecting the graph of f with the
plane x = c. We define a section
of f with y fixed similarly.
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Contour Lines (or Level Curves)
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Contour lines, or level curves, are
obtained from a surface by slicing
it by horizontal planes.
If z = f(x,y) then for some value c,
where z = c, c = f(x,y) is a level
curve for the function.
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Equation of a Plane in 3-D
If a plane has slope m in the x
direction, slope n in the y direction, and
passes through the point: ( x 0 , y 0 , z 0 )
then its equation is:
z  f ( x0 , y 0 ) )  z 0  m ( x  x0 )  n ( y  y 0 )
 if we write: c  z 0  mx 0  ny 0
then: f(x,y) = c + mx + ny
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Level Surfaces
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A function of three variable f(x,y,z),
is represented by a family of
surfaces of the form:
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f(x,y,z) = c,
each of which is called a level
surface.
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3-D  4-D
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A single surface representing a
two-variable function: z = f(x,y)
can always be thought of a one
member of the family of level
surfaces representing a three
variable function:
G(x,y,z) = f(x,y) - z.
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The graph of z = f(x,y) is the level
surface of G = 0.
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Limits and Continuity
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The function f as a limit at the point
f ( x, y)  L
(a,b), written: ( x, ylim
)( a ,b )
if the difference |f(x,y) - L| is as small
as we wish whenever the distance
from the pint (x,y) to the point (a,b) is
sufficiently small, but not zero.
A function f is continuous at a point
if
lim f ( x, y)  f (a, b).
( x , y )( a ,b )
A function is continuous if it is
continuous at each point of its
domain
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