Arc length function, Examples
[Pages:8]Math 20C Multivariable Calculus '
Lecture 8
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Slide 1
Arc length function, Examples
? Review: Arc length of a curve. ? Arc length function. ? Examples Sec. 13.4.
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Arc length of a curve
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The arc length of a curve in space is a number. It measures the extension of the curve.
Definition 1 The arc length of the curve associated to a vector valued function r(t), for t [a, b] is the number given by
b
ba = |r (t)| dt.
a
Suppose that the curve represents the path traveled by a particle in space. Then, the definition above says that the length of the curve is the integral of the speed, |v(t)|. So we say that the length of the curve is the distance traveled by the particle.
The formula above can be obtained as a limit procedure, adding up
the lengths of a polygonal line that approximates the original curve.
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Math 20C Multivariable Calculus
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Arc length of a curve
In components, one has,
r(t) = r (t) = |r (t)| =
ba =
x(t), y(t), z(t) ,
x (t), y (t), z (t) ,
[x (t)]2 + [y (t)]2 + [z (t)]2,
b
[x (t)]2 + [y (t)]2 + [z (t)]2dt.
a
The arc length of a general curve could be very hard to compute.
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Arc length function
Definition 2 Consider a vector valued function r(t). The arc length function (t) from t = t0 is given by
t
(t) = |r (u)|du.
0
Note: (t) is a scalar function. It satisfies (t0) = 0. Note: The function (t) represents the length up to t of the curve given by r(t).
Our main application: Reparametrization of a given vector valued function r(t) using the arc length function.
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Math 20C Multivariable Calculus
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Arc length function
Reparametrization of a curve using the arc length function: ? With r(t) compute (t), starting at some t = t0. ? Invert the function (t) to find the function t( ). Example: (t) = 3et/2, then t( ) = 2 ln( /3). ? Compute the composition r( ) = r(t( )). That is, replace t by t( ).
The function r( ) is the reparametrization of r(t) using the arc length as the new parameter.
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Slide 6
Examples, Sec. 13.4
? (Probl. 14 Sec. 13.4) Find the velocity, acceleration, and speed of the position function r(t) = t sin(t)i + t cos(t)j + t2k.
? (Probl. 16 Sec. 13.4) Find the velocity and position vectors given the acceleration and initial velocity and position:
a(t) = -10k, v(0) = i + j - k, r(0) = 2i + 3j.
? Problems with projectiles. Given the initial speed |v0| and the initial angle of the projectile with the horizontal, , describe the movement of the projectile.
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Math 20C Multivariable Calculus '
Lecture 9
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Slide 7
Scalar functions of 2, 3 variables
? Definition. ? Examples. ? Graph of the functions. ? Level curves and level surfaces.
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Scalar functions of 2 variables
Definition 3 A scalar function f of two variables (x, y) is a rule that assigns to each ordered pair (x, y) D IR2 a unique real number, denoted by f (x, y), that is,
f : D IR2 R IR.
Slide 8
Comparison: ? Vector valued functions, r : IR IR2 t x(t), y(t)
? Scalar function of two variables, f : IR2 IR
(x, y) f (x, y). &
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Graph and level curves
Definition 4 The graph of a function f (x, y) is the set of all points (x, y, z) in IR3 of the form (x, y, f (x, y)). Definition 5 The level curves of f (x, y) are the curves in in the domain of f , D IR2, solutions of the equation
f (x, y) = k,
for k R, a real constant in the range of f .
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Scalar functions of 3 variables
Slide 10
Definition 6 A scalar function f of three variables (x, y, z) is a rule that assigns to each ordered triple (x, y, z) D IR3 a unique real number, denoted by f (x, y, z), that is,
f : D IR3 R IR.
Note:
? In order to graph a function f (x, y, z) one needs four space dimensions. So, one cannot do such graph.
? The concept of a level curve can be generalized to functions of more than two variables. In this case they are called level surfaces.
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Lecture 10
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Limits and continuity
? Limit in 2, 3 space dimensions. ? Continuity. ? Examples.
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Limits
Idea: The function f (x, y) has the number L as limiting value at the point (x0, y0) roughly means that for all points (x, y) near (x0, y0) the value of f (x, y) differs little from L.
Definition 7 Consider the function f (x, y) and a point (x0, y0) IR2. Then,
lim f (x, y) = L
(x,y)(x0 ,y0 )
if for every number > 0 there exists another number > 0 such that |f (x, y) - L| < for every (x, y) D satisfying 0 < (x - x0)2 + (y - y0)2 < .
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Limits
In words: "f (x, y) has a limit L at (x0, y0) if the following holds: for all (x, y) D close enough in distance to (x0, y0) the values of f (x, y) approaches L." A tool to show that a limit does not exist is the following result.
Theorem 1 If f (x, y) L1 along a path C1 as (x, y) (x0, y0), and f (x, y) L2 along a path C2 as (x, y) (x0, y0), with L1 = L2, then
lim f (x, y) does not exist.
(x,y)(x0 ,y0 )
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Slide 14
Limits
Theorem 2 (Squeeze)
Assume f (x, y) g(x, y) h(x, y) for all (x, y) near (x0, y0); Assume
Then
lim f (x, y) = L = lim h(x, y),
(x,y)(x0 ,y0 )
(x,y)(x0 ,y0 )
lim g(x, y) = L.
(x,y)(x0 ,y0 )
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Math 20C Multivariable Calculus '
Lecture 10
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Slide 15
Continuity
Definition 8 A function f (x, y) is continuous at (x0, y0) if lim f (x, y) = f (x0, y0).
(x,y)(x0 ,y0 )
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Continuity
Examples of continuous functions: ? Polynomial functions are continuous in IR2, for example P2(x, y) = a0 + b1x + b2y + c1x2 + c2xy + c3y2.
? Rational functions are continuous on their domain,
f (x, y)
=
Pn(x, y) Qm(x, y)
,
for example,
f (x, y)
=
x2
+
3y - x2y2 x2 - y2
+
y4 ,
x = ?y.
? Composition of continuous functions are continuous, example
f (x, y) = cos(x2 + y2).
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