Review Vectors & Mapping

How Do We Combine Vectors ?

Vectors can be added together (they are commutative)

Just start the tail of one next to the head of another.

Vectors are denoted with a bar over the letter or a bold letter

ab

c

a + b = c

Components of a Vector

A single vector can be broken down into two orthogonal vectors in a Cartesian reference frame

X (i)

Y (j)

r

r = 250 i + 145 j

Here, the term 250 i is a vector with magnitude 250 pointing in the i direction

250

145

Components of a Vector

Since directions i and j are at right angles to each other we can write this in another way using the Pythagorean Theorem

This will give us the length of the vector r, length is noted by vertical bars

X (i)

Y (j)

r

250

145

r = sqrt ( 2502 + 1452 )

Can We Obtain the Direction of the Vector ?

We can use rules for right triangles to obtain the angle

250

145

= arctan ( y / x) = arctan ( 145 / 250 ) = 30o

Convert from Polar to Cartesian Coordinates

E

N

r

E30oS means 30o south of east

Vector convention: Angles are measured CCW from X axis (or east on a map)

The angle shown here is: - 30o

Vector Dot Product:Multiplying 2 vectors

v1

To multiply 2 vectors:

v2

v1 *

v2 =

v1 *

v2

cos ()

Where

v1

is length of vector,

v1 .

You can rearrange to solve for .

If Geology, Then Calculus

“If you understand Geology,then you understand Calculus”

If Geology, Then Calculus I will make the assertion that

“If you understand Geology, then you understand Calculus”

Can we prove it ?

The “if-then” statement is a conditional. In Critical Thinking, the laws of logic state these as truth values (either true or false – no “maybes”): p = “you understand geology” (antecedent) q = “you understand calculus” (consequent)

(p q) ~ (p ^ ~ q) (1) impossible to have p and not q

(p q) ~ (q ~ p) (2) if q if false, then p is false

If Geology, Then Calculus

The “if-then” statement is a conditional. In Critical Thinking, the laws of logic state these as truth values (either true or false – no “maybes”): p = “you understand geology” (antecedent) q = “you understand calculus” (consequent)

(p q) ~ (p ^ ~ q) (1) impossible to have p and not q

(p q) ~ (q ~ p) (2) if q if false, then p is false

It is not possible for you to understand geology and not understand calculus If you do not understand calculus,

you do not understand geology

If Geology, Then Calculus

Understanding geology is a sufficient condition for understanding calculus Understanding calculus is a necessary condition for

understanding geology

The assertions may be described as “sufficient” and “necessary” p is a sufficient condition of q q is a necessary condition of p (unavoidable)

p = “you understand geology” q = “you understand calculus”

If Geology, Then Calculus

You may know about calculus more than you think

Don't let the skills of differentiating and integrating get in

the that way of concepts.

There are some things about geology that guarantee an instinctive understanding of calculus.

If Geology, Then Calculus

We are NOT arguing the reverse, that “If you know calculus, then you understand geology”

Lord Kelvin (1824 – 1907) a mathematical physicist clearly understood calculus. He proved from first principles of heat conduction that the Earth could not be as old as the Uniformitarians claimed. His proof showed that the Earth was between 20 and 40 million years old. He scoffed at Earth scientists who suggested that the theory of uniformatarianism indicated a much older earth. Thus, calculus is clearly not sufficient to understand geology.

Lord Kelvin

Proof: If Geology, Then Calculus

Steps to solving problems: 1. Understand problem 2. Devise a plan 3. Carry out the plan 4. Look back

Step 1: Understand the problem – Draw a picture

People who don't know calculus

People who know calculus

People who understand geology

Rectangle represents all peopledivided into 2 parts

People who understand geology (circle) also understand calculus.

A person cannot both understand geology and not understand calculus.

Proof: If Geology, Then Calculus

Step 2: Make a plan

Where “r” is an intermediate step. Conclusion is true only if the premises are true Must use a sound argument What if r = “ you understand snakes” This argument may be valid but it is not very sound

because neither of the premises are true.

Premise 1: p r

Premise 2: r q

Conclusion: p q

Proof: If Geology, Then Calculus

Step 2: Make a plan

Is there a substitution for “r” that makes both P1 and P2 true ?

Propose a solution:

r = “you understand rates and maps”

This argument is valid and sound.

Premise 1: p r

Premise 2: r q

Conclusion: p q

Proof: If Geology, Then Calculus

P1: If you understand geology, you understand rates and maps

P2: If you understand rates and maps, you understand calculus

C1: If you understand geology, you understand calculus

Proof: If Geology, Then CalculusStep 3: Carry out the plan - Understanding calculus

What does it mean to “Understand calculus” ?

a) Do you know what a derivative is ? b) Do you know what an integral is ? c) Do you know that finding a derivative and finding an integral are inverse processes ? (Fundamental Theorem of Calculus)

Proof: If Geology, Then Calculus

A geologist may not know all these terms, but a geologist probably knows these things intuitively – from geological experience because a geologist understands:

a) topographic slopes

b) volumes as portrayed on topographic maps

c) uniformitarianism and sediment loading (e.g. Colorado river beds in Grand Canyon)

Hillside Topography

Understanding Calculus

Let's go for a hike!

Hillside Topography

Understanding Calculus

a

e

c

d

b

The grade of the topography can be broken up todescribe which part of the hike is more difficult than other sections.

Which section is the easiest ?

Which is the most difficult ?

Hillside Topography:

Describing Slopes

a

e

c

d

b

Steepness of the Slope:

0

10

20

Hillside Topography:

Describing Slopes

a

e

c

d

b

Steepness of the Slope:

0

10

20

Describing Slopes

Steepness of the Slope:

0

10

20

How can we describe a slope mathematically on a graph ?

Slope = riserun

How is the slope determined on a hillside ?

rise is difference in elevation between 2 pointsrun is horizontal distance between these points

Let's try it! (measure with brunton...)

Describing Slopes

The Slope or multiplied by 100 is the percent grade.

Scenic highways with 6% grade or higher have warning signs.

riserun

Hillside Topography:

Understanding Calculus

a

e

c

d

b

Steepness of the Slope:

0

10

20

We can think ofthe hillside as a continuous function,f(x) where elevation changes for step (x) along the path.

We can also think of the slopes as another function, the rate of change in elevation along the path. This function, f'(x), is called a derivative. x

If Geology, Then Calculus

Geologists know and feel what a derivative is.

A derivative is the slope function

As geologists walk around the topography, they experience the slope function under their feet!

Fermat's Ratio – Measuring the Slope

Measurement of a hillside slope is same approach used by Pierre de Fermat (1601 – 1665) to calculate the slope of the tangent to a curve.

ratioFermat

= f (x+a) - f (x)a

Where “a” is a little bit added on to x.

What is a in our example slope on the hillside ?

The two points of measurement are x and (x+a) .

The elevation change is the rise or the numerator of Fermat's ratio

= riserun

Fermat's Ratio – Measuring the Slope

h (elevation)

distance

Ah(x)

Read h(x) as “elevation “h at x” at point A

Point B is a little further away, a distance x + a

h(x+a)B

xx

x x + a

Fermat's Ratio – Measuring the Slope

The tangent is a straight line draw from A to the x axis

We measure “s” as the horizontal distance from here to x.

The tangent extended upward intersects vertical line for B This point is called B'. The elevation here is b.

h (elevation)

distance

Ah(x)

h(x+a)B

xxx x + as

B'

b

Fermat's Ratio – Measuring the Slope

Use similar triangles to get:

h (elevation)

distance

Ah(x)

h(x+a)B

xxx x + as

B'

b

s + a = bs h(x)

Fermat's Ratio – Measuring the Slope

Fermat recognized b is nearly the same as h(x+a)

h (elevation)

distance

Ah(x)

h(x+a)B

xxx x + as

B'

b

s + a = h(x+a)s h(x)

rearranging

h(x+a) - h(x) = h(x)a s

Fermat's Ratio – Measuring the Slope

Where h(x)/s is just the slope or rise/run.

Assume an example where h(x) = mx2 + c

h(x+a) - h(x) = h(x)a s

[ m(x+a)2 + C ] - [ mx2 + C ] a

Simplifying this gives: ratioFermat

= 2mx This is also known as the derivative, f'(x)