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Category Archives: Notes
Thinking Infinitesimally – Multivariate Calculus (II)
Chain Rule for Multivariate Calculus We continue our discussion of multivariate calculus. The first item here is the analogue of Chain Rule for the multivariate case. Suppose we have parameters f, u, v, x, y, z. Suppose {u, v} are independent parameters (in particular, … Continue reading
Thinking Infinitesimally – Multivariate Calculus (I)
[ Background required: some understanding of single-variable calculus, including differentiation and integration. ] The object of this series of articles is to provide a rather different point-of-view to multivariate calculus, compared to the conventional approach in calculus texts. The typical … Continue reading
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Tagged calculus, differentiation, intermediate, multivariate, partial differentiation
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Why Do We Need Eigenvalues and Eigenvectors?
[ Prerequisites : basic linear algebra, matrices and determinants. ] Eigenvalues and eigenvectors are often confusing to students the first time they encounter them. This article attempts to demystify the concepts by giving some motivations and applications. It’s okay if … Continue reading
Kinetic Theory, Entropy and Information Theory
This is really a continuation from the series “Thermodynamics for Mathematicians”. Our discussion then wasn’t quite complete without some justifications of the facts we used from kinetic theory of gases, in particular, we will figure out the constant c in the … Continue reading
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Tagged energy, entropy, heat, information theory, non-imo, physics, statistical physics, thermodynamics
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Thermodynamics for Mathematicians (IV)
Note: for this installation, we now require multi-variate calculus, specifically partial differentiation. 10. More State Parameters For this section, let’s consider the case of a homogeneous gas, whose state is parametrised by (P, V, N), where N is kept constant for now. … Continue reading
Thermodynamics for Mathematicians (III)
Note: calculus, specifically integration, is essential in this article, though students can possibly just substitute it with summation to achieve the same result. In the previous installation, we defined a temperature scale by looking at the efficiency of a heat … Continue reading
Thermodynamics for Mathematicians (II)
We continue our discussion of thermodynamics. Warning: some of the examples will involve calculus and a bit of (ordinary) differential equations. 3. Work Done by/on the System A system can do work, e. g. when a gas expands and pushes against … Continue reading
Thermodynamics for Mathematicians (I)
“Every mathematician knows it is impossible to understand an elementary course in thermodynamics.” – V. I. Arnold. The following is the culmination of at least three attempts to understand thermodynamics at various times during the last two years. Why’s it … Continue reading
Mathematics and Such 