2117: Differentiation and Integration
|Differentiation and Integration|
Title text: "Symbolic integration" is when you theatrically go through the motions of finding integrals, but the actual result you get doesn't matter because it's purely symbolic.
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This comic provides a flowchart purporting to show the process of differentiation, and another for integration.
However, Randall dramatically overstates this point here. After the first step of integration, Randall assumes that any integration can not be solved so simply, and then dives into a step named "????", suggesting that it is unknowable how to proceed. The rest of the flowchart is (we can assume deliberately) even harder to follow, and does not reach a conclusion. This is in contrast to the simple, straightforward flowchart for differentiation. The fact that the arrows in the bottom of the integration part leads to nowhere indicates that "Phone calls to mathematicians", "Oh no" and "Burn the evidence" are not final steps in the difficult journey. The flowchart could be extended by Randall to God-know-where extents.
It should be noted that Randall slightly undermines his point by providing four different methods, and an "etc", for attempting differentiation with no guidelines for selecting between them.
For any and , it follows that .
For any , it follows that .
For any and , it follows that if .
For any and , it follows that .
The "product rule" run backwards. Since , it follows that by integrating both sides you get , which is more commonly written as . By finding appropriate values for functions such that your problem is in the form , your problem may be simplified.
The "chain rule" run backwards. Since , it follows that . By finding appropriate values for functions such that your problem is in the form your problem may be simplified.
Cauchy's Integral formula is a result in complex analysis that relates the value of a contour integral in the complex plane to properties of the singularities in the interior of the contour. It is often used to compute integrals on the real line by extending the path of the integral from the real line into the complex plane to apply the formula, then proving that the integral from the parts of the contour not on the real line has value zero.
Partial fractions is a technique for breaking up a function that comprises one polynomial divided by another into a sum of functions comprising constants over the factors of the original denominator, which can easily be integrated into logarithms.
Mathematica is a modern technical computing system spanning most areas. One of its features is to compute mathematical functions. This step in the flowchart is install and use Mathematica to compute to do the integration for you.
Riemann Integration The Riemann integral is a definition of definite integration. Elementary textbooks on calculus sometimes present finding a definite integral as a process of approximating an area by strips of equal width and then taking the limit as the strips become narrower. Riemann integration removes the requirement that the strips have equal width, and so is a more flexible definition. However there are still many functions for which the Riemann integral doesn't converge, and consideration of these functions leads to the Lesbegue Integral. Riemann integration is not a method of calculus appropriate for finding the anti-derivative of an elementary function.
Stokes' Theorem ???
Risch algorithm The Risch algorithm is a notoriously complex procedure that, given a certain class of symbolic integrand, either finds a symbolic integral or proves that no elementary integral exists. (Technically it is only a semi-algorithm, and cannot produce an answer unless it can determine if a certain symbolic expression is equal to 0 or not.) Many computer algebra systems have chosen to implement only the simpler Risch-Norman algorithm, which does not come with the same guarantee. A series of extensions to the Risch algorithm extend the class of allowable functions to include (at least) the error function and the logarithmic integral.
Bessel function The Bessel function is a second-order differential equation used in solving problems in cylindrical and spherical coordinate systems.
Symbolic integration Mentioned in the title text. ???
Burn the evidence In a parody of detective novels, this may refer to the burning of your work to avoid the shame of being unable to solve the integration problem. Alternatively, this could be an ironic hint to the fact that in order to find the integral, it may even be necessary to break the law or upset higher powers, so that the negative consequences of a persecution can only be avoided by destroying the evidence.
|This transcript is incomplete. Please help editing it! Thanks.|
- [Two flow charts are shown.]
- [The first flow chart has four steps in simple order, one with multiple recommendations.]
- Try applying
- Chain Rule
- Power Rule
- Quotient Rule
- Product Rule
[The second flow chart begins like the first, then descends into chaos.]
- Try applying
- Integration by Parts
- Haha, Nope!
- [Chaos, Roughly from left to right, top to bottom, direction arrows not included.]
- Cauchy's Formula
- Partial Fractions
- Install Mathematica
- Riemann Integration
- Stokes' Theorem
- Risch Algorithm
- [Sad face.]
- What the heck is a Bessel Function??
- Phone calls to mathematicians
- Oh No
- Burn the Evidence
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