# As Z Notes

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The syllabus includes the main theoretical concepts which are fundamental to the subject, some current applications of physics, and a strong emphasis on advanced practical skills. The emphasis throughout is on the understanding of concepts and the application of physics ideas in novel contexts as well as on the acquisition of knowledge.

If you don’t want to mess around here between notes, slides, e-books etc and just want to have past papers of Cambridge International AS and A Level Physics (9702). Due to the nature of the mathematics on this site it is best views in landscape mode.

If your device is not in landscape mode many of the equations will run off the side of your device (should be able to scroll to see them) and some menu items will be cut off due to the narrow screen width. In the last two sections of this chapter we’ll be looking at some alternate coordinate systems for three-dimensional space.

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This one is fairly simple as it is nothing more than an extension of polar coordinates into three dimensions. The third equation is just an acknowledgement that the $$z$$-coordinate of a point in Cartesian and polar coordinates is the same.

Since we are now in three dimensions and there is no $$z$$ in equation this means it is allowed to vary freely. B $${r^2} + {z ^2} = 100$$ Show Solutions equation will be easy to identify once we convert back to Cartesian coordinates.

C $$z = r$$ Show SolutionAgain, this one won’t be too bad if we convert back to Cartesian. For reasons that will be apparent eventually, we’ll first square both sides, then convert.

Z lecture notes: Introduction Apply logic and simple mathematics to computing. Gain understanding through analysis rather than experiments (testing).

Can be applied to behavior (specification), structure (design, “refinement''), implementation (verification). Discover errors by analysis (earlier) instead of by testing or customer experiences (later).

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Document development thoroughly to permit more objective technical evaluation and review (even automated analysis, machine checking). Infallible (but some products can be checked more conclusively).

Formal semantics (unlike most programming languages): you can calculate whether two formulas (specifications, programs) mean the same thing by formula manipulation. Notational conventions for logic and simple mathematics.

Discrete mathematics with declarations and structure (“paragraphs''). Literature, reference manual, draft ANSI/ISO standard.

More effort in early project stages. Show Mobile NoticeS how All Notes Hide All Notes You appear to be on a device with a “narrow” screen width (i.e. you are probably on a mobile phone).

Due to the nature of the mathematics on this site it is best views in landscape mode. If your device is not in landscape mode many of the equations will run off the side of your device (should be able to scroll to see them) and some menu items will be cut off due to the narrow screen width.

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\ Let’s start simple by integrating over the box, \ Note that we integrated with respect to $$x$$ first, then $$y$$, and finally $$z$$ here, but in fact there is no reason to the integrals in this order.

Let’s do a quick example of this type of triple integral. \ \times \left \times \left\] Show Solutions to make the point that order doesn’t matter let’s use a different order from that listed above.

\begin{align*}\ii int\limits_{B}{{8xyz\, dV}} & = \int_{{\,1}}^{{\,2}}{{\int_{{\,2}}^{{\,3}}{{\int_{{\,0}}^{1}{{8xyz\, dz}}\, dx}}\, dy}}\\ & = \int_{{\,1}}^{{\,2}}{{\int_{{\,2}}^{{\,3}}{{\left. {2{x^2}y} \right|_2^3\, dy}}\\ & = \int_{{\,1}}^{{\,2}}{{10y\, dy}} = 15\end{align*} Before moving on to more general regions let’s get a nice geometric interpretation about the triple integral out-of-the-way so we can use it in some examples to follow.

\\, dA}}\] where the double integral can be evaluated in any of the methods that we saw in the previous couple of sections. Example 2 Evaluate $$\display style \ii int\limits_{E}{{2x\, dV}}$$ where $$E$$ is the region under the plane $$2x + 3y + z = 6$$ that lies in the first octane.

Show Solution should first define octane. We can get a visualization of the region by pretending to look straight down on the object from above.

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Show Solution In this case we’ve been given $$D$$ and so we won’t have to really work to find that. Here is a sketch of the region $$D$$ as well as a quick sketch of the plane and the curves defining $$D$$ projected out past the plane, so we can get an idea of what the region we’re dealing with looks like.

Example 4 Evaluate $$\display style \ii int\limits_{E}{{\sort {3{x^2} + 3{z ^2}} \, dV}}$$ where $$E$$ is the solid bounded by $$y = 2{x^2} + 2{z ^2}$$ and the plane $$y = 8$$. The region $$D$$ in the $$oz$$-plane can be found by “standing” in front of this solid, and we can see that $$D$$ will be a disk in the $$oz$$-plane.

This disk will come from the front of the solid, and we can determine the equation of the disk by setting the elliptic paraboloid and the plane equal. \ This region, as well as the integral, both seems to suggest that we should use something like polar coordinates.

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