A derivative is one of the key fundamental concepts of calculus.
The derivative’s most common use is to compute the slope at a specified
location on a curve, but has many other purposes too. When applied to 3
dimensions, a derivative of a surface is the same concept resulting in a vector
tangent to a surface at the specified location.
Basically a derivative transforms an equation into another (usually simpler)
form to isolate certain properties. The meaning of the properties vary
according to the context which the equation is defined. If the derivative is
applied recursively, the equation gets further isolated (simplified).....which
can be good or bad depending on your needs. You can always take a derivative
of an equation, but the result won't always be useful.
example:
equation of a parabola with root at coordinate (3,4) and opening upwards:
y = (x-3)2 + 4
When you specify a value of x, you get the value of Y at that X coordinate. In
this case, if x = 5, then Y = 8 meaning the curve passes through point (5,8).
that’s simple Algebra.
If you take the derivative of the equation, you get:
y' = 2x - 6
This equation describes the slope of the curve (eg; slope of tangent line) at a
specified X coordinate. If we want the slope at coordinate (5,8), then insert
X=5 into the equation to get a slope value of 4 which indicates the tangent
line rises 4 units in Y for every one unit it increases in X at coordinate
(5,8). Insert a different X value, and you'll likely get a different slope as
a result because the shape of the curve is continuously changing as you travel
along it.
Take the derivative of the slope equation and you'll get:
y'' = 2
This equation represents the concavity of the curve. The concavity defines
whether the curve is opening up or down at the specified X coordinate. Since
there is no X variable in the equation anymore, it reduces to a constant
indicating the curve is always opening upwards (because the constant is
positive, and the curve is a parabola) regardless of the X coordinate. When
used in conjunction with the slope equation, other information can be obtained
such as where the local minima and maxima exist, as well as where the critical
points occur (location where concavity flips between opening up vs. opening
down).
Take the derivative again and you’ll get:
y’’’ = 0
This isn’t useful. so we stop.
the single quote symbol is called ‘prime’ and indicates how many generations of
derivatives the original equation has gone through. y’’’ means the derivative
was applied 3 times in succession. There are other notations to indicate the
same thing. Prime is considered a shorthand notation and often frowned upon by
more serious mathematicians and scientists as it can be ambiguous in more
complicated contexts.
the (nearly) opposite of the derivative is the integral (infinite sum) and also
a key fundamental concept of calculus. Integrals are significantly more
difficult as they often involve fabricating variables/values out of thin air
and adjusting them to fit a specific scenario whose rules and boundaries aren’t
entirely known.
To complete that thought, here is how the integral would be applied to the
above equations. First, take the integral of the concavity equation y’’=2 to
get:
y’ = 2x + C
where ‘C’ is an arbitrary constant (offset) whose value is not yet known. We
must resolve X and/or Y before we can determine the value of C. But I’ll skip
that lesson for now and move on.
If we apply the integral to the integrated slope equation, we get:
y = x2 + C1x + C2
Which looks quite different from the original parabolic equation we started
with [y= (x-3)2 + 4]. The ‘C’ from the previous equation was renamed C1 and
the new ‘C’ introduced from integration was named ‘C2’. Some people prefer to
merge the two C’s, which is legal, but can send you down a more difficult path
to solution in some cases – it’s one of those things you learn from experience
rather than rote rules. Keep in mind, we have yet to determine ‘C’, but when
we do, the equations will be rearranged and proven equivalent even if the final
form is different. That’s the fun of calculus.
The above example is trivial, but real life can be quite messy, complicated,
and not always solvable. Take course(s) in calculus to learn more. the
concepts appear frequently in many areas of 3D animation including curves,
surfaces, rendering, simulations, and more.
A simple application of such information is computing paths of projectiles.
The original equation defines the path of the projectile over time. The slope
equation indicates the direction of travel at a given point on the path. The
concavity equation can tell you which direction the subject is rising (or
falling) and/or which way gravity and other forces are applied to the
projectile. Minima and maxima can tell you the maximum or minimum height the
projectile will reach. Derivatives and integrals can also tell you the
velocity and acceleration of the projectile at specific points in time. And so
on. A course in calculus will teach you the fundamentals of how to compute
derivatives and integrals in various contexts, and a few use cases. But
physics and the sciences will push you to use those tools in context of
something useful.
Matt
Date: Tue, 9 Aug 2016 21:26:49 +0200
From: Olivier Jeannel <[email protected]>
Subject: Re: Reminiscing
To: "softimage@listproc autodesk. com"
Would love some clearer info though. I don't know what's a derivative for a
vector and how to compute some. There must be tons of applications and uses
for such knowledge for sure.
Le 9 ao?t 2016 18:17, "Andy Nicholas" <[email protected]> a ?crit :
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