Differentiation Under the Integral Sign

By H. Flanders. 1973

How researchers see this paper

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Sensing and coverage for a network of heterogeneous robots (2008, background): “(7) Proof: By applying the differentiation under the integral sign (see [9]) we can write ∂H ∂xi = ∫”
Reparameterization Gradient for Non-differentiable Models (2018, methodology): “The proof of the theorem uses an existing but less known theorem about interchanging integration and differentiation under moving domain [3], together with the divergence theorem.”
Home 3D body scans from noisy image and range data (2011, background): “The derivative of this integral is easily computed in terms of the derivative of the path of integration and the derivative of the integrand [11].”
Monte Carlo Gradient Estimation in Machine Learning (2019, background): “…a process of limits, the validity of the interchange will relate to the conditions for which it is possible to exchange limits and integrals, in such cases most often relying on the use of the dominated convergence theorem or the Leibniz integral rule (Flanders, 1973; Grimmett and Stirzaker, 2001).”
Optimizing the CVaR via Sampling (2014, background): “Such an extension is given by (Flanders 1973), which we now state.”
Awareness coverage control over large scale domains with intermittent communications (2008, methodology): “We first consider the following Lemma (see [26] for a detailed exposition), which will be used shortly.”