| << *continued from page 02 | Secrets Revealed! |
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Week 2: During the second week of the course, three additional coordinate systems are incorporated into the framework of multivariable calculus: polar, spherical, and cylindrical. Combined with the Cartesian coordinates, these three additional systems equip you with the tools necessary to integrate functions over n-spaces not easily generalized in terms of the type I and type II regional dichotomy. Parameterizations of multivariable functions are also covered, with emphasis on implicit functions in z. Redefining implicit functions in terms of some parameter t enables the theoretical expansion of the fundamental theorem to include the alternate coordinate systems and their respective change of variable generalizations for triple integrals over unique and defined regions D. Also, during the exploration of parameterized functions, the mathematical description of lines, planes, and three-dimensional shapes is reviewed within the general context of vector-valued functions. The goal here is to master the use of scalar dot-products and vector cross-products in deducing various pertinent aspects of the continuous geometry in question. These analyses include determination of tangent lines and planes, normal vectors and planes, as well as level curves and surfaces. It is important to keep in mind that the advanced calculus concepts presented thus far are intended to serve as the tools with which you will explore the more challenging and scientifically relevant material of the course. In the sense that a majority of the principles covered within the past two weeks are logical, direct extensions of elementary calculus, the course so far has been review. At this point, it is difficult to see how a "sharper focus" and "deeper level of contemplation" on your part could even be possible, but yet that is exactly what the honorable Dr. Polinski has instructed the class to strive for during the coming weeks if adequate conceptual comprehension is to be achieved. It is ironic that more than half of the students originally enrolled in the class have already thrown in the towel and dropped the class. |
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| ( Week 2 notes ) | ||||
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| /return to *top of page ^^ |
| Written & Illustrated by Perishable for DLa #13 / Spring, 2002 | 050502-021105 |
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| Advanced Calculus Secrets : Revealed! ~ A Perishable Guide to Mastering Mathematics / ( page 03 ) |
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