if geology, then calculus “if you understand geology, then you understand calculus”

If Geology, Then Calculus

“If you understand Geology,then you understand 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

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


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.


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 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

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

Describing Slopes


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


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

f(x)

f'(x)


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.

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

ratioFermat

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

= riserun


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


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

tangent


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 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) Assuming that a is very small relative to x.

h (elevation)

distance

Ah(x)

h(x+a)B

x x + as

The Limit Concept and the Invention of Calculus

Sir Isaac Newton 1643 - 1727

Sir Isaac Newton (1643-1716) and Gottfried W. Leibniz (1646-1716) realized that the tangent needs to be described in terms of infinitely small quantities.


In the example of our hillside: We estimated the slope (rise/run) using a brunton

How well does this represent the details of the

topography ? How could we improve this estimate ?


Details are better estimated by using smaller intervals.

The slope at a point (the derivative) is the limit of a ratio.

This concept was formalized by Augustin-Louis Cauchy (1789-1857) and took on the rigorous footing we know today:

f (x + x) – f (x) x

f ' (x) = lim x 0

Try with a plumb bob

The Two Dreadful Symbols

In a book titled Calculus Made Easy, Silvanus Thompson (1851) writes:

“The preliminary terror which chokes off (students) from attempting to learn how to calculate, can be abolished.. by stating.. the meaning in common-sense terms – of the two principle symbols... These dreadful symbols are:

1) d which merely means 'a little bit of ' Thus dx means 'a little bit of x'. Mathematicians think of these 'little bits'

as infinitely small

The second dreadful symbol is yet to come...

The Differential Coefficient

If you have two quantities that depend on each other.

A change in one (x) will bring about a change in the other (y)

If we alter x just a little, say by dx then we cause x to become x + dx

Because x has been changed, y will change by some amount (not necessarily small) to y + dy

In calculus, we want to know this ratio, dy/dx.

This assumes, however, that these quantities are related.

y

x

dx

dy


The differential coefficient, dy/dx is shown by a little triangle along any curve.

Geologist think of this triangle as rise/run on a hill side.

Thus, if you understand geology, you understand slopes and rise/run, which means that you understand df / dx.

y

x

dxdy


You also understand the slope at a point, expressed as df / dx or f '(x)

This slope can only be approximated accurately when looking at y over small changes in x, that is when x becomes infinitely small.

Only when x is small does the ratio df / dx truly represent the actual slope.

f(x)

Areas and Volumes

Let's say you are a millionaire (from gold discoveries in the California hills) and have just bought San Nicolas Island.

You want to know just how big your island is, What is it's area, it's volume ? How can you calculate this ?

Areas and Volumes

One idea is that you could make a 3D model of the island and slice it up into a few layers (like a wedding cake).

Determine the area and thickness of each layer.

Then add up the volumes of each slice.

V = Ai h

i =1

n

How good of an approximation is this ?

Could is be improved ?

Areas and Volumes

How would you determine the area of one slice of this island model ?

How accurate is your area estimate ?

Could you improve this area estimate ?

What if cake is heart shaped ?

Areas and Volumes

Could we cut it up into small square pieces ? What size is best ? .....does size matter ? How well can you represent the edges with squares ?

Areas and Volumes

Could we cut the cake into smaller square pieces ?

A = xy

The Second Dreadful Symbol

In a book titled Calculus Made Easy, Silvanus Thompson (1851) writes:

“ which is merely a long S may be called ' the sum of '. Thus S dx means the sum of all the bits of x.

Now any fool can see that if you add up all the little bits of dx, you get the whole x. The word 'integral' simply means ' the whole '. “

-Thompson

Integral: Summing up Small Pieces of a Whole

Think of the duration of time for one hour

We can break it up into 60 minutes

Or into 3600 seconds

If you add up 3600 seconds, you get the full hour - this is integrating!

“When you see an expression with this terrifying symbol, you will now know that it is merely giving you instructions to total up all the little bits of what follows.” - Thompson

Back to our Cake!

To determine how much cake we have (volume) We can just intergrate (or add up the volume of) each slice.

Vcake

= A dh

Let Them Eat Cake!

To determine the area of the heart shaped cake We can just intergrate (or sum up) each small square.

Acake

= dx dy

if geology, then calculus “if you understand geology, then you understand calculus”

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