Lecture 1: Points in Space
1.1 Two and three dimensional Cartesian space Fermat (1601 - 1665) and Descartes (1596 - 1650) had the idea of identifying points in the plane with a unique ordered pair of real numbers. That is, given a plane, we rst identify t
Lecture 11: Limits and Continuity
11.1 Limits
Denition Suppose f : Rn R. We say the limit of f (x) as x approaches a is L,
denoted lim f (x) = L, if for every > 0 there exists > 0 such that
xa
|f (x) L| <
whenever
0 < |x a| < .
Proposition
If lim f (x) =
Lecture 10: Functions from Rn to R
10.1 Graphs
Denition
The graph of a function f : Rn R is the set
cfw_(x1 , x2 , . . . , xn , xn+1 ) : xn+1 = f (x1 , x2 , . . . , xn ).
Note that if f : Rn R, then the graph of f is in Rn+1 . Hence we may visualize the
g
Lecture 8: Arc Length and Curvature
8.1 Arc Length
Suppose f : R Rn parametrizes the curve C in Rn as t goes from a to b. If we think of
an object moving along C so that its position at time t is f (t), then the speed of the object
at time t is |f (t)|. L
Lecture 14: The Chain Rule
14.1 The gradient
Denition
We call
Suppose f : Rn R and all rst-order partial derivatives of f exist at a.
f (a) =
f (a),
f (a), . . . ,
f (a)
x1
x2
xn
the gradient of f at a.
Example
If f (x, y, z) = xyz 10x2 , then
f (x, y, z)
Lecture 13: Derivatives
13.1 Partial derivatives do not imply dierentiability
Recall that for functions f : R R, dierentiability is a stronger condition than continuity.
That is, if f is dierentiable at a, then f is continuous at a. Geometrically, you can
Lecture 12: Directional and Partial Derivatives
12.1 Directional derivatives
Denition Suppose f : Rn R, x is a point in the domain of f , and u is a unit vector
in Rn . Providing the limit exists, we call
f (x + hu) f (x)
h0
h
Du f (x) = lim
the direction
Lecture 9: Motion Along a Curve
9.1 Components of velocity and acceleration
Suppose f (t) is the position of a particle moving along a curve C in Rn , where f : R Rn ,
and let v(t) = f (t) and a(t) = f (t) be the velocity and acceleration of the particle
Lecture 7: Derivatives of Functions from R to Rn
7.1 Derivatives
Recall that the derivative of a function f : R R at a point t is
f (t + h) f (t)
,
h0
h
f (t) = lim
provided the limit exists.
The derivative of a function f : R Rn at a point t is
Denition
Lecture 4: The Cross Product
4.1 The cross product
There is no general way to dene multiplication for vectors in Rn , with the product also
being a vector of the same dimension, which is useful for our current purposes. However,
in the special case of R3
Lecture 6: Functions from R to Rn
6.1 Some terminology
Given a function f : Rn R, the set of all points x in Rn satisfying x = f (t) for some t
in the domain of f is called a curve. Note that f (t) is a vector in Rn , so we may dene
functions fk : R R, k
Lecture 3: The Dot Product
3.1 The angle between vectors
Suppose x = (x1 , x2 ) and y = (y1 , y2 ) are two vectors in R2 , neither of which is the zero
vector 0. Let and be the angles between x and y and the positive horizontal axis,
respectively, measure
Lecture 5: Lines and Planes
5.1 Lines in Rn
Denition
Given vectors p and v in Rn , with v = 0, the set of all vectors x satisfying
x = p + tv,
< t < , is called a line.
If p = (p1 , p2 , . . . , pn ), v = (v1 , v2 , . . . , vn ), and x = (x1 , x2 , . . .
Lecture 2: Vectors
2.1 Points and vectors
When we think of a point x = (x1 , x2 , . . . , xn ) in Rn as representing both a distance and
a direction from the origin 0, we may call x a vector. Geometrically, it is often convenient
to picture x as an arrow.
Lecture 15: The Gradient
15.1 Directional derivatives revisited
Recall: If f : Rn R and u is a unit vector, then the directional derivative of f at a in
the direction of u is
f (a + hu) f (a)
Du f (a) = lim
,
h0
h
provided the limit exists. Now if we dene