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I am slowly reading through Evan's PDE and am having trouble with some multivariate integral manipulations. My background in multivariate calculus is undergraduate at best, you could call it my Achilles heel. I thought I would start off with asking about the follow two items:

  1. What is $\frac{d}{dt} c\int_0^t u(x+(s-t)b,s)ds$, where $x,b\in \mathbb{R}^n$ and $s,t\in\mathbb{R}$? Using the multivariate chain rule on the integrand I get $b•Du - bu_t$, but this feels very wrong (it's in $n$ dimensions).
  2. Why is for $u(x,t)=\int_0^t \int_{\mathbb{R^n}} \Phi(y,s)f(x-y,t-s)dyds$ where $\Phi$ is smooth and $f$ is $C^2$ with compact support, $u_t(x,t)=\int_0^t \int_{\mathbb{R^n}} \Phi(y,s)f_t(x-y,t-s)dyds + \int_{\mathbb{R^n}} \Phi(y,t)f(x-y,0)dy$?
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Hint for (1): write $$t\longmapsto\int_0^t u(x+(s-t)b,s)ds$$ as a composition of $$\pmatrix{r\cr t}\longmapsto\int_0^r u(x+(s-t)b,s)ds$$ and $$ t\longmapsto\pmatrix{t\cr t}. $$

(I understant that the $c$ is a constant or a typo)

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  • $\begingroup$ That does not help me I'm afraid. Not sure how to work with your notation. I know this is basic stuff but I never learned these manipulations $\endgroup$ Commented Mar 12, 2018 at 13:42

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