Geology: Calculus Video #6

3Blue1Brown Intro To Calculus Series Summary

Preface ✨

Hello everyone! My intentions for writing these articles are:

• Explain technical knowledge about geology in simple terms (to the public)
• Document my journey to getting an AD-T Geology degree in one year
• Store information and habits for my future self and others (in <7 minutes)

Coolio? Sweet. Enjoy the series :-)

notes as i watched the video 📹

p.s. — the video will be linked at the bottom of the article. cheers.

graphs and charts only have to worry about one dimensions.

but what about when we move it along an x-y axis?

for example, what if we try and move a specific point (x,y) → (3,4) on a circle…

and let’s say, we nudge it a bit up.

thx grant)

the extra nudge on the y-axis will become “dy”, and the extra nudge on the x-axis will become “dx”.

rise over run will be “dy/dx”…

but if only that could work…

but because we have two different variables, we cannot take the derivative of both at the same time… at least not like this.

in the equation “x² + y² = 5²”, x² & y² are not independent variables… they are part of an equation that gives us 5²…

and a derivative must be independent of any equation…

so no.

and math geeks call this an “implicit curve”,

a set (or specific numbers) for (x,y) coordinates that satisfy the properties* of the equation above; [x² + y² = 5²] , (*aka written in terms of the variables x & y.)

and carlos, how do u take a derivative with multiple variables like x & y?

1. take the derivative of everything, left & right side of the equation.

before: x² + y² = 5² ||| after: 2x * dx + 2y * dy = 0

remember: dx & dy are the nudges we made by going a step up on the circle (see photo above)

2. transform this equation into a dy/dx expression

step 1: 2x * dx + 2y * dy = 0 ||| step 2: 2y * dy = - 2x * dx

step 3: dy / dx = - (2x / 2y) ||| final: dy / dx = - (x/y)

remember, x = 3 , & y = 4, now plug it in…

dy / dx = - (3/4)

great.

but wtf does it mean to take a derivative of an expression with multiple variables (like x & y)?

in grant’s opinion, and mine too lmao, we gunna use related rates to explain y it works.

let’s use a ladder now, instead of a circle.

ladder is 5 meters held up against a wall.

x = 3 meters , y = 4 meters.

and let’s say a moron didn’t hold it up properly and ur dog is underneath the ladder.

the top of the ladder is dropping at 1 meter per second.

when it starts going down and murder ur pet…

in that initial moment of chaos…

what is the rate at which the bottom of the ladder is moving away from the wall?

guess what?

the distance from the bottom of the ladder (the part of the ladder touching the floor) to the wall is 100% determined by the distance between the top of the ladder (part that is touching the wall) and the floor.*

*(see the photo above)

now let’s put some labels on them.

because we’re looking for time, we’re going to write the variables with a dependence on time, or (t) for short.

welp, u see that equation?…

x(t)² + y(t)² = 5²

anywhere that it has (t), in it… it means this:

it is a function of time*, and it will be labeled via seconds.

*it depends on how many seconds have occured, and that’s what will go in for (t).

but because we have nudged the ladder down from 0 seconds to 1 second,

there will be a decrease in the height of ladder to the ground (y-axis), and an increase of the bottom-ladder’s distance from the wall (x-axis).

so what will happen?

here is how the equation will look because of the nudge in time (dt)

and when the entire derivative has been taken, we get this:

because the time in (t) is a variable, we took the derivative of a variable, only to get it’s extra nudge in time according to what axis it was.

so what do we plug in for those numbers?

x = 3, y = 4… when the time is equal to zero.

and then dx/dt & dy/dt are the numbers when (t) = 1.

so let’s find out.

let’s do the math.

6 * dx/dt + 8 * (-1 m/s) = 0

6 * dx/dt + (-8 m/s) = 0

the -1 came from how fast the ladder was falling, which was 1 m/s, but falling means going down, negative.

and yes, -1 is the change on the y-axis, the change in height of the top of the ladder to the ground.

and now, to find out what dx/dt is, separate the constants from the term.

6 * dx/dt = 8 m/s

dx/dt = 8/6 m/s

dx/dt = 4/3 meters per second (m/s)

so the change in the x-axis means that the part of the ladder that’s touching the ground already (farthest from the wall)…

is getting farther away from the wall by 4/3 meters per second (or 1.333 m/s).

in many problems, related rates are tied to a common variable (especially time or distance).

ok, now going back to our problem with the circle.

as i mentioned up above:

in many problems, related rates are tied to a common variable (especially time or distance).

only this time, in the circle problem, there is no common variable we can use.

there is only coordinates on a graph.

so what do we do when we face situations like these?

in the beginning of this article, we had this:

but now, let’s distill it further.

the S in this plot has a meaning.

x²+y² has been given an abbreviation, known as “S” in this example.

depending on what “x” & “y” is…

wherever u see a point “S”, the numbers will be plugged in / replaced for “x” & “y”.

now let’s focus on one of these “S” points.

let’s choose the “S” point on the top-right corner.

when we make a tiny nudge to that original point, (altering the original location of “x” & “y”)…

we now must find out: in what direction did the change occur?

did the change go inside of the circle, or outside?

and that change will also alter the positioning of “S”, which will be known as “dS” (derivative of S).

in a picture, it would look something like this:

add the original + altered positioning of the (x,y) coordinates

the starting point for the upper right hand corner point was (3,4).

x = 3, y = 4

now, let’s say that “dx” is -0.02 & “dy” is -0.01

plugging in those numbers, we end up with…

no numbers: x² + y² + 2x*dx + 2y*dy

with numbers: (3² + 4²) + 2*(3)(−0.02) + 2*(4)(−0.01)

before the change, the value was 25.

now, it’s 24.8, damn.

that’s what this derivative expression 2x⋅dx + 2y⋅dy means. It tells us how much the value x² + y² changes,

as determined by the point (x,y) where you started,

and the tiny step (dx, dy) that we took.

- grant sanderson

and the smaller the “dx” & “dy” value become, the closer we get to knowing what the change in value (the slope of the curve) was according to that nudge.

btw, slope of the curve is the rise over run of two specific points, but the closer they are, the more accurate the rise over run is at one specific point.

for nerds, it means, the ratio of the change in the value of the variable on the vertical axis to the change in the value of the variable on the horizontal axis.

CJ

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