Dem knowledge bombs ! O-o On 9 August 2016 at 23:51, Matt Lind <[email protected]> wrote:
> 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 : > > > ------ > Softimage Mailing List. > To unsubscribe, send a mail to [email protected] > with "unsubscribe" in the subject, and reply to confirm. >
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