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I am looking for some differential equation lectures from a theoretical perspective, not a strictly computational one. I found the MIT 18.03 lectures which (as the professor says towards the end of the first lectures) "is not going to be a course for those theoretically inclined." I would prefer something more geared towards the theory of differential equations. Book and other resource recommendation are also welcome, but I would prefer lectures.

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I know you're looking for lectures (unfortunately I also failed to find any) but as a compromise I have been really impressed with VI Arnold's Ordinary Differential Equations. It's very geometrical, and rigorous in the enlightening sense (rather than the paranoid-the-sky-will-fall-on-your-head sense.)

To give one very simple example - in the first chapter he gives a rigorous discussion of coordinate transforms. In most calculation-based courses, to transform to polar coordinates, you just algebraically replace $x$ by $ r\cos\theta $ and you blindly apply the chain rule to get

$$ \dot{x} =\dot{r}\cos \theta - r\dot{\theta }\sin \theta$$

and similarly for $\dot{y}.$ But Arnold instead discusses the geometric meaning of such a transform - as a transform of the underlying vector field, and shows that the substitution above should really be thought of as multiplying the local velocity field by the Jacobian (which approximates differentiable transformations arbitrarilly well in an arbitrarilly small region.)

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  • $\begingroup$ +2, if that were possible. +1 for the great recommendation, which I will definitely look into, and +1 for "rigorous in ... the paranoid-the-sky-will-fall-on-your-head sense." $\endgroup$ – Juan Sebastian Lozano Jun 16 '15 at 3:05

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