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Concept# Multivariable calculus

Summary

Multivariable calculus (also known as multivariate calculus) is the extension of calculus in one variable to calculus with functions of several variables: the differentiation and integration of functions involving multiple variables (multivariate), rather than just one.
Multivariable calculus may be thought of as an elementary part of advanced calculus. For advanced calculus, see calculus on Euclidean space. The special case of calculus in three dimensional space is often called vector calculus.
Typical operations
Limits and continuity
A study of limits and continuity in multivariable calculus yields many counterintuitive results not demonstrated by single-variable functions. For example, there are scalar functions of two variables with points in their domain which give different limits when approached along different paths. E.g., the function.
:f(x,y) = \frac{x^2y}{x^4+y^2}
approaches zero whenever the point (0,0)

Official source

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It is well known and readily seen that the maximum of n independent and uniformly on [0, 1] distributed random variables, suitably standardised, converges in total variation distance, as n increases, to the standard negative exponential distribution. We extend this result to higher dimensions by considering copulas. We show that the strong convergence result holds for copulas that are in a differential neighbourhood of a multivariate generalised Pareto copula. Sklar's theorem then implies convergence in variational distance of the maximum of n independent and identically distributed random vectors with arbitrary common distribution function and (under conditions on the marginals) of its appropriately normalised version. We illustrate how these convergence results can be exploited to establish the almost-sure consistency of some estimation procedures for max-stable models, using sample maxima.

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Background: Metabolic side effects induced by psychotropic drugs represent a major health issue in psychiatry. CREB-regulated transcription coactivator 1 (CRTC1) gene plays a major role in the regulation of energy homeostasis and epigenetic mechanisms may explain its association with obesity features previously described in psychiatric patients. This prospective study included 78 patients receiving psychotropic drugs that induce metabolic disturbances, with weight and other metabolic parameters monitored regularly. Methylation levels in 76 CRTC1 probes were assessed before and after 1 month of psychotropic treatment in blood samples. Results: Significant methylation changes were observed in three CRTC1 CpG sites (i.e., cg07015183, cg12034943, and cg 17006757) in patients with early and important weight gain (i.e., equal or higher than 5% after 1 month; FDR p value = 0.02). Multivariable models showed that methylation decrease in cg12034943 was more important in patients with early weight gain (>= 5%) than in those who did not gain weight (p = 0.01). Further analyses combining genetic and methylation data showed that cg12034943 was significantly associated with early weight gain in patients carrying the G allele of rs4808844A>G (p = 0.03), a SNP associated with this methylation site (p = 0.03). Conclusions: These findings give new insights on psychotropic-induced weight gain and underline the need of future larger prospective epigenetic studies to better understand the complex pathways involved in psychotropic-induced metabolic side effects.

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