Miscellaneous Notes on Mathematics
1. Here is a quick reference to the elements of symbolic or propositional logic, including straightforward descriptions of the basic operations on propositions together with their truth tables:
elements_of_logic-new1.pdf | |
File Size: | 32 kb |
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2. The problem of computing the convolution of a rectangular pulse with itself arose in recent discussions with a student. The following file contains a derivation of this convolution for a general rectangular function of width (b - a) and height (or amplitude) A, but after re-reading it a few months later I see that it can be simplified considerably...a project for later.
rectangular_pulse_convolution-update.pdf | |
File Size: | 91 kb |
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3. An instructor in one of the trigonometry courses often asks students for a derivation of an expression for the radius of a circle inscribed in a triangle, in terms of the lengths a, b, and c of the sides of the triangle. The result requires Heron's formula for the area of the triangle in terms of the lengths of its sides and its semi-perimeter, s = (a + b + c)/2, which is just half its perimeter (a + b + c). A derivation of the required result, together with a derivation of Heron's formula, is given here, just in case you are stuck somewhere:
radius_circle_in_triangle_revision2.pdf | |
File Size: | 62 kb |
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4. Euler's formula (named for the great mathematician Leonhard Euler, his last name being pronounced "Oyler") is extremely useful for the derivation of various trigonometric identities. It is this simple but beautiful formula connecting the complex exponential function with the cosine and sine functions:
$$e^{i\theta} = \cos{\theta} + i \sin{\theta}$$
The following note gives a heuristic (which means not mathematically rigorous to the satisfaction of pure mathematicians) derivation of the formula, requiring concepts from calculus, together with several applications:
$$e^{i\theta} = \cos{\theta} + i \sin{\theta}$$
The following note gives a heuristic (which means not mathematically rigorous to the satisfaction of pure mathematicians) derivation of the formula, requiring concepts from calculus, together with several applications:
note_on_eulers_formula_rev1.pdf | |
File Size: | 89 kb |
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5. A favorite topic in pre-calculus courses is an introduction to arithmetic and geometric sequences and series, as well as the binomial theorem for the series expansion of the nth power of the general binomial expression (ax + by). Some details are given in the following document:
help_topic-arithmetic_geometric_binomial_series.pdf | |
File Size: | 131 kb |
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