Department

MathematicsCourse Code

MAT237Y1Professor

John BlandStudy Guide

FinalThis

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2.1.

dot product and cross product.

Important points include:

Definitions of the dot product and cross product.

Cauchy’s Inequality and the Triangle Inequality.

The formula

a b

=

a b

cos

θ

.

if

u

is a unit vector and

v

is any vector, then (

u v

)

u

is the projection of

v

onto

the line generated by

u

.

This means that is we write

v

=

v

1

+

v

2

, where

v

1

is parallel to

u

and

v

2

is

orthogonal to

u

, then

v

1

=

(

u v

)

u

.

Equivalently, if we form a right triangle whose hypotenuse is

v

and whose base

lies along the line generated by

u

then the base is given by (

u

·

v

)

u

.

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The “meaning” of the cross product:

a

×

b

is the vector that

–

is orthogonal to both

a

and

b

, with direction determined by the “right-hand

rule”, and

–

has length

|a| |b|

sin

θ

= area of the paralellogram generated by

a

and

b

.

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algebraic properties of the cross product. For example

–

a × b

=

−b × a

–

(

c

1

a

1

+

c

2

a

2)

b

=

c

1

a

1

b

+

c

2

a

2

b

–

etc.

The definition of the dot product, and all its properties mentioned above, should be familiar

from linear algebra.

Students should remember all of the above facts and should be able to use them to solve

problems.

Only pages 1-2 are available for preview. Some parts have been intentionally blurred.

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⊂

2.2.

open and closed sets. Students should know

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the definition of a

sphere

and an (open)

ball

in

R

n

.

These are just matters of terminology and notation. We will normally say

“ball”

instead of “(open) ball”.

the definition of an

open

subset of

R

n

.

the definition of a

closed

subset of

R

n

.

Given a set

S

R

n

, you should know the definitions of its

–

interior Sint

–

boundary

∂S

,

–

closure

S

¯

.

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the definition of a

bounded

subset of

R

n

.

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A set is open if and only if its complement is closed.

You should also be able to use the definitions to solve problems at the level of the practice

problems. For example, you should be able to recognize open and closed sets and detemine

the interior/boundary/closure of a set.

Of the above concepts, the most important are probably

open

and

closed

. The textbook

gives two equivalent characterizations of open sets: a set

S

is

open

if it doesn’t contain any

of its boundary points, or if every point of

S

is an interior point. We can also write

S

⊂

R

n

is

open

if, for every

x

∈

S

, there exists

ε >

0 such that

B

(

ε,

x

)

⊂

S

.

This is just another way of saying that every point of

S

is an interior point.

We will later show that many sets can be instantly recognized as open or closed; see

Section 2.4.2 below.

Note that, contrary to what the names suggest, it is possible for a set to be both open

and closed at the same time.

2.2.1.

A remark.

Note that in his proof of Proposition 1.4, Folland asserts that

(1)

∂S

=

∂

(

S

c

)

.

This follows directly by combining

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the definition of a boundary point, and

the fact that (

S

c

)

c

=

S

If you are not convinced that (

S

c

)

c

=

S

or that (1) follows directly from combining from

the above considerations, then you should check these points in detail.

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