That's pretty clear, with some exercices to complete in ice or vex :) Thank you, that's enormous !
On Wed, Aug 10, 2016 at 12:04 AM, Sebastien Sterling < sebastien.sterl...@gmail.com> wrote: > Dem knowledge bombs ! O-o > > On 9 August 2016 at 23:51, Matt Lind <speye...@hotmail.com> 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 <facialdel...@gmail.com> >> 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" <a...@andynicholas.com> a ?crit : >> >> >> ------ >> Softimage Mailing List. >> To unsubscribe, send a mail to softimage-requ...@listproc.autodesk.com >> with "unsubscribe" in the subject, and reply to confirm. >> > > > ------ > Softimage Mailing List. > To unsubscribe, send a mail to softimage-requ...@listproc.autodesk.com > with "unsubscribe" in the subject, and reply to confirm. >
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