9
$\begingroup$

Let $f$ be a Riemann Integrable function over $\mathbb{R}^2$. When can we do this?

$$\frac{\partial}{\partial\theta}\int_{a}^{b}f(x,\theta)dx=\int_{a}^{b}\frac{\partial}{\partial\theta}f(x,\theta)dx$$

(Here, $a$ and $b$ are not a function of $\theta$.)

In the problem, which I am solving recently, are like this:

$f_{\theta}(x)$, here $\theta$ is constant and $\theta\in\mathbb{R}$ (usually). For example $f_{\theta}(x)=x^2\theta$. So, I am blindly interchanging integration and differentiation because of continuity over $\theta$. But I want to know little bit more.

Also, what happens if $a$ and $b$ are function of $\theta$? Thanks.

$\endgroup$
  • $\begingroup$ see liebniz integral rule and the dominated convergence thoerem $\endgroup$ – qbert Nov 21 '17 at 3:57
  • $\begingroup$ Your example function is separable and so you just pull the theta out and takes its derivative. If the limits are a function of theta, then the chain rule is required. In probably most cases that one comes across in calculus courses, you can interchange derivative and integral. $\endgroup$ – jdods Nov 21 '17 at 4:01
9
$\begingroup$

You may interchange integration and differentiation precisely when Leibniz says you may. In your notation, for Riemann integrals: when $f$ and $\frac{\partial f(x,t)}{\partial x}$ are continuous in $x$ and $t$ (both) in an open neighborhood of $\{x\} \times [a,b]$.

There is a similar statement for Lebesgue integrals.

$\endgroup$
  • $\begingroup$ Thanks, Now I understand, this is exactly what I need to know! $\endgroup$ – user467365 Nov 21 '17 at 4:36
  • $\begingroup$ How is it true for weak derivatives? For instance, to show the conservation of energy in heat equation, we do it again, but I don't understand why it is still true. $\endgroup$ – Fardad Pouran Apr 18 '18 at 9:35
  • $\begingroup$ @FardadPouran : It sounds like you have a different question, so perhaps should ask a new Question. With essentially no context in your comment, my half-blind guess is that you use the definition of weak derivative to shift the derivative to the other factor under the integral. (This is somewhat analogous to integration by parts.) $\endgroup$ – Eric Towers Apr 18 '18 at 14:14
  • $\begingroup$ I meant the case like $f\in C([0,T],H)$ for some Hilbert space $H$ and integrating the $H$-valued functions. Hopefully, I could find a chapter in Evans book related to this topic (Space involving times Section 5.9.2) $\endgroup$ – Fardad Pouran Apr 19 '18 at 18:30

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.