if geology, then calculus “if you understand geology, then you understand calculus”
TRANSCRIPT
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
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
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 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 ?
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
f(x)
f'(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.
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
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
tangent
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) 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.
The Limit Concept and the Invention of Calculus
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 ?
The Limit Concept and the Invention of Calculus
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
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
The Differential Coefficient
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 ?
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