Math 128, Modern Geometry
D. Joyce, Clark University
19 Oct 2005
Due Today. From Chapter 7: 1, 6, 9, 10.
Due Monday. From Chapter 8: 1, 2, 8.
Properties of distance. The important thing
about this distance is that all the transformations
of the hyperbolic
Math 128, Modern Geometry
D. Joyce, Clark University
25 Aug 2005
Theres more than one kind of geometry
You have probably only seen one kind of plane geometry so far, classical Euclidean geometry, but you may have seen it from two dierent points of view. O
Summary of basic probability theory, part 3
D. Joyce, Clark University
Math 218, Mathematical Statistics, Jan 2008
Joint distributions. When studing two related
real random variables X and Y , it is not enough
just to know the distributions of each. Rathe
Summary of basic probability theory, part 1
D. Joyce, Clark University
Math 218, Mathematical Statistics, Jan 2008
9. If event E is a subset of event F , then P (E)
P (F ).
10. Statement 7 above is called the principle of
inclusion and exclusion. It gene
Common probability distributions
D. Joyce, Clark University
Aug 2006
1
Introduction.
I summarize here some of the more common distributions used in probability and statistics.
Some are more important than others, and not all of them are used in all elds.
A short introduction to Bayesian statistics, part V
D. Joyce, Apr 2009
Normal distributions with unknown variances. Lets now consider normal distributions
with both unknown mean and unknown variance
2 . Well discover a way to nd a natural family of
conju
A short introduction to Bayesian statistics, part IV
D. Joyce, Apr 2009
6
Normal distributions.
Thus, the family of all normal distributions is a
conjugate family for .
Example. Lets take an example. Suppose
were monitoring the production line of cans of
A short introduction to Bayesian statistics, part III
D. Joyce, Apr 2009
5
The Poisson process
Gamma(, r). There are a couple dierent ways
that gamma distributions are parametrizedeither
in terms of and r as done here, or in terms of
and . The connection
A short introduction to Bayesian statistics, part II
D. Joyce, Apr 2009
Bayes pool table example. The process we
just completed is what Thomas Bayes (17021761)
did. He illustrated the problem with balls on a
table. Ill paraphrase his illustration using th
A short introduction to Bayesian statistics, part I
D. Joyce, Apr 2009
2nk
2nk + 2k
P (B|X) = 1 P (A|X)
2k
= nk
2
+ 2k
Ill try to make this introduction to Bayesian
statistics clear and short. First well look as a
specic example, then the general setting,
Example 1 (A radial vector eld). Well look as
several vector elds in the plane, and draw them
by drawing a few vectors F(x) with their tails attached to the points x. For instance, the vector
eld F(x) = (1, 1) is a constant vector eld with all
Intro to ve
speed, x , to be the length of velocity, so speed is
a nonnegative scalar.
So, at a time t, an object has a position x = x(t);
a velocity x = x (t), which is sometimes denoted
v = v(t); an acceleration x = x (t), which is
sometimes denoted a = a(t); and a
is a good approximation of f near a in the sense
that
f (x) h(x)
= 0.
lim
xa
xa
Note that when m = 1, this denition says the
derivative Df of the scalar-valued function f is the
1 n row-matrix
Total derivatives
Math 131 Multivariate Calculus
D Joyce, Spri
A Lorenz attractor. Math 131 Multivariate Calculus
D Joyce, Spring 2014
A Lorenz attractor. Sometimes surprising things can happen. Lets start with this vector
eld in space.
8
F(x, y, z) = (10(y x), 28x y xz, 3 z + xy)
It seems pretty innocuous since its
Note how y and z dont change in this limit.
This kind of derivative is called a partial derivative since only one of the variables changes. Furthermore, the concept of limit used here is just the
Partial derivatives
scalar limit you used in calculus, not
Whereas a line is parametrized by one variable
weve usually used ta surface is parametrized by
two variableswell usually use s and t.
It helps to have a standard notation. Well use S
to denote the surface in R3 that were parametrizSurfaces
ing. Well use X
Surface integrals
Math 131 Multivariate Calculus
D Joyce, Spring 2014
The area dierential of a surface, and a double integral for the area of the surface.
Recall that were using X(s, t) to describe a paramaterization of a surface S in 3-space. Also
we hav
Theorem 1. When the rst and second partials of
a function of two variables x and y are all continuous functions, the order that the partials are taken
doesnt matter, that is, fyx = fxy , or in partial notation
2f
2f
=
.
x y
y x
Higher-order partial deri
About some topological concepts. In the
text theres a discussion of various topological
conceptsopen, closed, boundary, neighborhood,
and accumulation point. These are particularly important when the concepts of limit and continuity
are extended to more g
the standard (x, y, z ) coordinate three-space. Notations for vectors. Geometric interpretation as vectors as displacements as well as vectors as points.
Vector addition, the zero vector 0, vector subtracThings you need to know about
tion, scalar multipli
and its derivative is the acceleration,
a(t) = v = x = (x , y , z ).
Now, Kepler determined that the acceleration of
the planet was toward the sun. Newton explained
that in terms of gravitational force by saying, rst,
that the gravitation of the sun is an
Now lets add limits of integration. If the limits
of integration for u are a and b, then the limits of
integration for x will be x(a) and x(b).
x(b)
Jacobians
Math 131 Multivariate Calculus
b
f (x) dx =
x(a)
f (x(u)
a
dx
du.
du
D Joyce, Spring 2014
We wan
system mentioned above has these three equations
fx (x, y ) = 2y = gx (x, y ) = 2x
fy (x, y ) = 2x + 2y = gy (x, y ) = 2y
g (x) = x2 + y 2 = c = 1
Lagrange multipliers
Math 131 Multivariate Calculus
in the three unknowns x, y , and . The rst two
equations
Gradients
Math 131 Multivariate Calculus
D Joyce, Spring 2014
f
Last time. Introduced partial derivatives like
x
of scalar-valued functions Rn R, also called
scalar elds on Rn .
Total derivatives. Weve seen what partial
derivatives of scalar-valued functi
Onto functions (surjective), one-to-one functions (injective), one-to-one correspondences
(bijective), range. These terms mean the same
thing in multivariate calculus as they in calculus of
a single variable. Be sure youre familiar with the
following conc
D
M
dA as follows.
y
M
dA
y
D
b
(x)
M
dy dx
y
a
(x)
Proof of Greens theorem
Math 131 Multivariate Calculus
=
D Joyce, Spring 2014
(x)
b
M (x, y)
=
a
Summary of the discussion so far.
b
M (x, (x) M (x, (x) dx
=
M
N
x
y
M dx + N dy =
D
D
dx
y=(x)
a
b
dA.
b
Here, the symbol is just a variant of thats often
used for line integrals when the line is a closed curve
or a nite union of closed curves.
Greens theorem
Math 131 Multivariate Calculus
Example 1. Let F = (M, N ) = (2y, x) and D is
the semicircular region
The surface x2 + y 2 + 1 = z is a paraboloid opening
upward (positive z being upward) with vertex on
the z -axis at z = 1. Above that surface and below
the plane z = 5 lies the 3-dimenional region D. The
top surface of D is a circle of radius 2. Let F be
If the second derivative f (a) was 0, then you had
to try some other way to determine whether it
was a max or min or neither. We need some sort
of second-derivative test for the multidimensional
case.
Maxima and minima of scalar elds
Math 131 Multivariate