vector functions 13. vector functions the functions that we have been using so far have been...
TRANSCRIPT
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VECTOR FUNCTIONSVECTOR FUNCTIONS
13
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VECTOR FUNCTIONS
The functions that we have been
using so far have been real-valued
functions.
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VECTOR FUNCTIONS
We now study functions whose values
are vectors—because such functions are
needed to describe curves and surfaces
in space.
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VECTOR FUNCTIONS
We will also use vector-valued functions
to describe the motion of objects through
space.
In particular, we will use them to derive Kepler’s laws of planetary motion.
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13.1Vector Functions
and Space Curves
VECTOR FUNCTIONS
In this section, we will learn about:
Vector functions and drawing
their corresponding space curves.
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FUNCTION
In general, a function is a rule that
assigns to each element in the domain
an element in the range.
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VECTOR FUNCTION
A vector-valued function, or vector
function, is simply a function whose:
Domain is a set of real numbers.
Range is a set of vectors.
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VECTOR FUNCTIONS
We are most interested in vector functions r
whose values are three-dimensional (3-D)
vectors.
This means that, for every number t in the domain of r, there is a unique vector in V3 denoted by r(t).
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COMPONENT FUNCTIONS
If f(t), g(t), and h(t) are the components of
the vector r(t), then f, g, and h are real-valued
functions called the component functions of r.
We can write:
r(t) = ‹f(t), g(t), h(t)› = f(t) i + g(t) j + h(t) k
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VECTOR FUNCTIONS
We usually use the letter t to denote
the independent variable because
it represents time in most applications
of vector functions.
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VECTOR FUNCTIONS
If
then the component functions are:
Example 1
3( ) , ln(3 ),t t t t r
3( ) ( ) ln(3 ) ( )f t t g t t h t t
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VECTOR FUNCTIONS
By our usual convention, the domain of r
consists of all values of t for which the
expression for r(t) is defined.
The expressions t3, ln(3 – t), and are all defined when 3 – t > 0 and t ≥ 0.
Therefore, the domain of r is the interval [0, 3).
Example 1
t
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LIMIT OF A VECTOR
The limit of a vector function r is defined
by taking the limits of its component
functions as follows.
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LIMIT OF A VECTOR
If r(t) = ‹f(t), g(t), h(t)›, then
provided the limits of the component functions
exist.
Definition 1
lim ( ) lim ( ), lim ( ), lim ( )t a t a t a t a
t f t g t h t
r
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LIMIT OF A VECTOR
If , this definition is equivalent to
saying that the length and direction of the
vector r(t) approach the length and direction
of the vector L.
lim ( )t a
t L
r
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LIMIT OF A VECTOR
Equivalently, we could have used
an ε-δ definition.
See Exercise 45.
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LIMIT OF A VECTOR
Limits of vector functions obey
the same rules as limits of real-valued
functions.
See Exercise 43.
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LIMIT OF A VECTOR
Find ,
where
Example 2
lim ( )t o
tr
3 sin( ) (1 ) t tt t te
t r i j k
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LIMIT OF A VECTOR
According to Definition 1, the limit of r is
the vector whose components are the limits
of the component functions of r:
(Equation 2 in Section 3.3)
Example 2
3 1
0 0 0 0
sinlim ( ) lim(1 ) lim limt t t t
tt t te
t
r i j k
i k
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CONTINUOUS VECTOR FUNCTION
A vector function r is continuous at a
if:
In view of Definition 1, we see that r is continuous at a if and only if its component functions f, g, and h are continuous at a.
lim ( ) ( )t a
t a
r r
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CONTINUOUS VECTOR FUNCTIONS
There is a close connection
between continuous vector functions
and space curves.
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CONTINUOUS VECTOR FUNCTIONS
Suppose that f, g, and h are
continuous real-valued functions
on an interval I.
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SPACE CURVE
Then, the set C of all points (x, y ,z) in space,
where
x = f(t) y = g(t) z = h(t)
and t varies throughout the interval I
is called a space curve.
Equations 2
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PARAMETRIC EQUATIONS
Equations 2 are called parametric
equations of C.
Also, t is called a parameter.
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SPACE CURVES
We can think of C as being traced out by
a moving particle whose position at time t
is:
(f(t), g(t), h(t))
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SPACE CURVES
If we now consider the vector function
r(t) = ‹f(t), g(t), h(t)›, then r(t) is the position
vector of the point P(f(t), g(t), h(t)) on C.
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SPACE CURVES
Thus, any continuous vector function r
defines a space curve C that is traced out
by the tip of the moving vector r(t).
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SPACE CURVES
Describe the curve defined by the vector
function
r(t) = ‹1 + t, 2 + 5t, –1 + 6t›
Example 3
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SPACE CURVES
The corresponding parametric equations
are:
x = 1 + t y = 2 + 5t z = –1 + 6t
We recognize these from Equations 2 of Section 12.5 as parametric equations of a line passing through the point (1, 2 , –1) and parallel to the vector ‹1, 5, 6›.
Example 3
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SPACE CURVES
Alternatively, we could observe that
the function can be written as r = r0 + tv,
where r0 = ‹1, 2 , –1› and v = ‹1, 5, 6›.
This is the vector equation of a line as given by Equation 1 of Section 12.5
Example 3
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PLANE CURVES
Plane curves can also be
represented in vector notation.
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PLANE CURVES
For instance, the curve given by
the parametric equations
x = t2 – 2t and y = t + 1
could also be described by the vector equation
r(t) = ‹t2 – 2t, t + 1› = (t2 – 2t) i + (t + 1) j
where i = ‹1, 0› and j = ‹0, 1›
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SPACE CURVES
Sketch the curve whose vector equation
is:
r(t) = cos t i + sin t j + t k
Example 4
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SPACE CURVES
The parametric equations for this curve
are:
x = cos t y = sin t z = t
Example 4
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SPACE CURVES
Since x2 + y2 = cos2t + sin2t = 1,
the curve must lie on the circular
cylinder
x2 + y2 = 1
Example 4
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SPACE CURVES
The point (x, y, z) lies directly above
the point (x, y, 0).
This other point moves counterclockwise
around the circle x2 + y2 = 1 in the xy-plane.
See Example 2 in Section 10.1
Example 4
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HELIX
Since z = t, the curve
spirals upward around
the cylinder as t
increases.
The curve is called a helix.
Example 4
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HELICES
The corkscrew shape of the helix
in Example 4 is familiar from
its occurrence in coiled springs.
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HELICES
It also occurs in the model of DNA
(deoxyribonucleic acid, the genetic
material of living cells).
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HELICES
In 1953, James Watson
and Francis Crick
showed that the structure
of the DNA molecule is
that of two linked,
parallel helixes that are
intertwined.
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SPACE CURVES
In Examples 3 and 4, we were given
vector equations of curves and asked
for a geometric description or sketch.
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SPACE CURVES
In the next two examples, we are given
a geometric description of a curve and are
asked to find parametric equations for
the curve.
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Find a vector equation and parametric
equations for the line segment that joins
the point P(1, 3, –2) to the point Q(2, –1, 3).
Example 5SPACE CURVES
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SPACE CURVES
In Section 12.5, we found a vector equation
for the line segment that joins the tip of
the vector r0 to the tip of the vector r1:
r(t) = (1 – t) r0 + t r1 0 ≤ t ≤ 1
See Equation 4 of Section 12.5
Example 5
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SPACE CURVES
Here, we take
r0 = ‹1, 3 , –2› and r1 = ‹2 , –1, 3›
to obtain a vector equation of the line
segment from P to Q:
or
Example 5
( ) (1 ) 1,3, 2 2, 1,3 0 1
( ) 1 ,3 4 , 2 5 0 1
t t t t
t t t t t
r
r
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SPACE CURVES
The corresponding parametric equations
are:
x = 1 + t
y = 3 – 4t
z = – 2 + 5t
where 0 ≤ t ≤ 1
Example 5
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SPACE CURVES
Find a vector function that represents
the curve of intersection of the cylinder
x2 + y2 = 1 and the plane y + z = 2.
Example 6
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SPACE CURVES
This figure shows how
the plane and
the cylinder intersect.
Example 6
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SPACE CURVES
This figure shows the
curve of
intersection C, which is
an ellipse.
Example 6
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SPACE CURVES
The projection of C onto the xy-plane is
the circle x2 + y2 = 1, z = 0.
So, we know from Example 2 in Section 10.1 that we can write:
x = cos t y = sin t
where 0 ≤ t ≤ 2π
Example 6
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SPACE CURVES
From the equation of the plane,
we have:
z = 2 – y = 2 – sin t
So, we can write parametric equations for C as:
x = cos t y = sin t z = 2 – sin t
where 0 ≤ t ≤ 2π
Example 6
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PARAMETRIZATION
The corresponding vector equation is:
r(t) = cos t i + sin t j + (2 – sin t) k
where 0 ≤ t ≤ 2π
This equation is called a parametrization of the curve C.
Example 6
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SPACE CURVES
The arrows indicate the
direction
in which C is traced as
the parameter t
increases.
Example 6
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USING COMPUTERS TO DRAW SPACE CURVES
Space curves are inherently
more difficult to draw by hand than
plane curves.
For an accurate representation, we need to use technology.
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USING COMPUTERS TO DRAW SPACE CURVESThis figure shows a
computer-generated
graph of the curve with
the following parametric
equations:
x = (4 + sin 20t) cos t
y = (4 + sin 20t) sin t
z = cos 20 t
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TOROIDAL SPIRAL
It’s called a toroidal
spiral because
it lies on a torus.
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TREFOIL KNOT
Another interesting
curve, the trefoil knot,
is graphed here.
It has the equations:
x = (2 + cos 1.5 t) cos t
y = (2 + cos 1.5 t) sin t
z = sin 1.5 t
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SPACE CURVES BY COMPUTERS
It wouldn’t be easy to
plot either of these
curves by hand.
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Even when a computer is used to draw
a space curve, optical illusions make it difficult
to get a good impression of what the curve
really looks like.
SPACE CURVES BY COMPUTERS
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SPACE CURVES BY COMPUTERS
This is especially
true in this figure.
See Exercise 44.
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The next example shows
how to cope with this problem.
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Use a computer to draw the curve with
vector equation
r(t) = ‹t, t2, t3›
This curve is called a twisted cubic.
TWISTED CUBIC Example 7
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We start by using the computer to plot
the curve with parametric equations
x = t, y = t2, z = t3
for -2 ≤ t ≤ 2
SPACE CURVES BY COMPUTERS Example 7
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The result is shown here.
However, it’s hard to see the true nature of the curve from this graph alone.
Example 7
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Most 3-D computer graphing programs
allow the user to enclose a curve or surface
in a box instead of displaying the coordinate
axes.
SPACE CURVES BY COMPUTERS Example 7
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When we look at
the same curve in
a box, we have
a much clearer picture
of the curve.
Example 7
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We can see that:
It climbs from a lower corner of the box to the upper corner nearest us.
It twists as it climbs.
Example 7
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We get an even better idea of the curve
when we view it from different vantage
points.
SPACE CURVES BY COMPUTERS Example 7
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This figure shows the
result of rotating the
box to give another
viewpoint.
Example 7
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These figures show
the views we get when
we look directly at a
face of the box.
Example 7
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In particular, this figure
shows the view from
directly above the box.
It is the projection of the curve on the xy-plane, namely, the parabola y = x2.
Example 7
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This figure shows the
projection on
the xz-plane, the cubic
curve z = x3.
It’s now obvious why the given curve is called a twisted cubic.
Example 7
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Another method of visualizing
a space curve is to draw it on
a surface.
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For instance, the twisted cubic in
Example 7 lies on the parabolic cylinder
y = x2.
Eliminate the parameter from the first two parametric equations, x = t and y = t2.
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This figure shows both the cylinder
and the twisted cubic.
We see that the curve moves upward from the origin along the surface of the cylinder.
SPACE CURVES BY COMPUTERS
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We also used this method in Example 4
to visualize the helix lying on the circular
cylinder.
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A third method for visualizing
the twisted cubic is to realize that
it also lies on the cylinder z = x3.
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So, it can be viewed as the curve
of intersection of the cylinders
y = x2 and z = x3
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We have seen that an interesting
space curve, the helix, occurs in
the model of DNA.
SPACE CURVES BY COMPUTERS
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Another notable example of a space curve
in science is the trajectory of a positively
charged particle in orthogonally oriented
electric and magnetic fields E and B.
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Depending on the initial velocity given
the particle at the origin, the path of
the particle is either of two curves, as
follows.
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It can be a space curve
whose projection on the
horizontal plane is the
cycloid we studied in
Section 10.1
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It can be a curve whose
projection is
the trochoid investigated
in Exercise 40
in Section 10.1
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Some computer algebra systems provide
us with a clearer picture of a space curve
by enclosing it in a tube.
Such a plot enables us to see whether one part of a curve passes in front of or behind another part of the curve.
SPACE CURVES BY COMPUTERS
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For example, the new figure shows
the curve of the previous figure as rendered
by the tubeplot command in Maple.
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For further details concerning the physics
involved and animations of the trajectories
of the particles, see the following websites:
www.phy.ntnu.edu.tw/java/emField/emField.html
www.physics.ucla.edu/plasma-exp/Beam/
SPACE CURVES BY COMPUTERS