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- https://math.libretexts.org/Bookshelves/Applied_Mathematics/Street-Fighting_Mathematics%3A_The_Art_of_Educated_Guessing_and_Opportunistic_Problem_Solving_(Mahajan)/06%3A_Analogy/6.02%3A_Topology-_How_many_regionsMust all the regions created by the lines be convex? (A region is convex if and only if a line segment connecting any two points inside the region lies entirely inside the region.) What about the thre...Must all the regions created by the lines be convex? (A region is convex if and only if a line segment connecting any two points inside the region lies entirely inside the region.) What about the three-dimensional regions created by placing planes in space? Hint: Explain first why the pattern generates the \(R_{2}\) row from the \(R_{1}\) row; then generalize the reason to explain the \(R_{3}\) row.
- https://math.libretexts.org/Bookshelves/Applied_Mathematics/Street-Fighting_Mathematics%3A_The_Art_of_Educated_Guessing_and_Opportunistic_Problem_Solving_(Mahajan)/05%3A_Taking_out_the_big_part/5.02%3A_Fractional_changes_and_low-enthropy_expressionsThe extent of the plausible alternatives measures the gap between our intuition and reality; the larger the gap, the harder the correct result must work to fill it, and the harder we must work to reme...The extent of the plausible alternatives measures the gap between our intuition and reality; the larger the gap, the harder the correct result must work to fill it, and the harder we must work to remember the correct result. Such gaps are the subject of statistical mechanics and information theory [20, 21], which define the gap as the logarithm of the number of plausible alternatives and call the logarithmic quantity the entropy.
- https://math.libretexts.org/Bookshelves/Applied_Mathematics/Street-Fighting_Mathematics%3A_The_Art_of_Educated_Guessing_and_Opportunistic_Problem_Solving_(Mahajan)/01%3A_Dimensions/1.04%3A_Summary_and_further_problemsIt is also a thermal phenomenon, so it depends on the thermal energy \(k_{B}T\), where \(T\) is the object’s temperature and \(k_{B}\) is Boltzmann’s constant. When the water depth is \(h = 5\)m, esti...It is also a thermal phenomenon, so it depends on the thermal energy \(k_{B}T\), where \(T\) is the object’s temperature and \(k_{B}\) is Boltzmann’s constant. When the water depth is \(h = 5\)m, estimate the rate at which the depth is increasing. Where M is the mass of the sun, m the mass of the planet, r is the vector from the sun to the planet, and \(\hat{r}\) is the unit vector in the r direction.
- https://math.libretexts.org/Bookshelves/Applied_Mathematics/Street-Fighting_Mathematics%3A_The_Art_of_Educated_Guessing_and_Opportunistic_Problem_Solving_(Mahajan)/05%3A_Taking_out_the_big_part
- https://math.libretexts.org/Bookshelves/Applied_Mathematics/Street-Fighting_Mathematics%3A_The_Art_of_Educated_Guessing_and_Opportunistic_Problem_Solving_(Mahajan)/01%3A_Dimensions/1.01%3A_Economics-_The_power_of_multinational_corporationsGDP, however, is a flow or rate: It has dimensions of money per time and typical units of dollars per year. (A dimension is general and independent of the system of measurement, whereas the unit is ho...GDP, however, is a flow or rate: It has dimensions of money per time and typical units of dollars per year. (A dimension is general and independent of the system of measurement, whereas the unit is how that dimension is measured in a particular system.) Comparing net worth to GDP compares a monetary amount to a monetary flow.
- https://math.libretexts.org/Bookshelves/Applied_Mathematics/Street-Fighting_Mathematics%3A_The_Art_of_Educated_Guessing_and_Opportunistic_Problem_Solving_(Mahajan)/zz%3A_Back_Matter/10%3A_Index
- https://math.libretexts.org/Bookshelves/Applied_Mathematics/Street-Fighting_Mathematics%3A_The_Art_of_Educated_Guessing_and_Opportunistic_Problem_Solving_(Mahajan)/01%3A_DimensionsOur first street-fighting tool is dimensional analysis or, when abbreviated, dimensions. To show its diversity of application, the tool is introduced with an economics example and sharpened on example...Our first street-fighting tool is dimensional analysis or, when abbreviated, dimensions. To show its diversity of application, the tool is introduced with an economics example and sharpened on examples from Newtonian mechanics and integral calculus.
- https://math.libretexts.org/Bookshelves/Applied_Mathematics/Street-Fighting_Mathematics%3A_The_Art_of_Educated_Guessing_and_Opportunistic_Problem_Solving_(Mahajan)/04%3A_Picture_Proofs
- https://math.libretexts.org/Bookshelves/Applied_Mathematics/Street-Fighting_Mathematics%3A_The_Art_of_Educated_Guessing_and_Opportunistic_Problem_Solving_(Mahajan)/02%3A_Easy_Cases/2.02%3A_Plane_Geometry_-_The_Area_of_an_EllipseThe candidate \(A = a^2 + b^2\) has correct dimensions (as do the remaining candidates), so the next tests are the easy cases of the radii \(a\) and \(b\). However, when \(a = 0\) the candidate \(A = ...The candidate \(A = a^2 + b^2\) has correct dimensions (as do the remaining candidates), so the next tests are the easy cases of the radii \(a\) and \(b\). However, when \(a = 0\) the candidate \(A = a^{2} + b^{2}\) reduces to \(A = b^{2}\) rather than to 0; so \(a^2 + b^2\) fails the \(a = 0\) test. Can you invent a second candidate for the area that has correct dimensions and passes the \(a = 0, b = 0, \text{ and } a = b\) tests?
- https://math.libretexts.org/Bookshelves/Applied_Mathematics/Street-Fighting_Mathematics%3A_The_Art_of_Educated_Guessing_and_Opportunistic_Problem_Solving_(Mahajan)/03%3A_Lumping
- https://math.libretexts.org/Bookshelves/Applied_Mathematics/Street-Fighting_Mathematics%3A_The_Art_of_Educated_Guessing_and_Opportunistic_Problem_Solving_(Mahajan)/02%3A_Easy_Cases/2.01%3A_Gaussian_Integral_RevisitedThis result refutes the option \(\sqrt{\pi α}\), which is infinite when \(α = \infty\); and it supports the option \(\sqrt{\pi/α}\), which is zero when \(α = \infty\). This result refutes the \(\sqrt...This result refutes the option \(\sqrt{\pi α}\), which is infinite when \(α = \infty\); and it supports the option \(\sqrt{\pi/α}\), which is zero when \(α = \infty\). This result refutes the \(\sqrt{\pi α}\) option, which is zero when \(α = 0\); and it supports the \(\sqrt{\pi/α}\) option, which is infinity when \(α = 0\). The table gives the area under the curve in the range \(x = −10 ... 10\), after dividing the curve into n line segments.
