can't power up arduino from dc jack
It would appear that the output of the power regulator pin has been vaporized.



From the Arduino R3 schematic we see this part as U1 which is an NCP1117ST50T3G power regulator.  Here is that data sheet for that part. From that data sheet we see the physical image of the part.



It appears that pin 2 has been vaporized.  If we look at the schematic of the Arduino R3 we see where pin 2 supplies the regulated power to the rest of the Arduino board and likely any connected electronics.  Note the red "X" indicating the vaporized pin 2.



The schematic and the data sheet both indicate the regulator's TAB is also an output.  If true it is difficult to explain why pin 2 vaporized.  Perhaps this particular version of the regulator does not use the TAB as an output.  Or perhaps the Arduino R3 PCB and schematic do not agree.

Added later...

There has been a comment that U1 (the NCP1117ST50T3G power regulator) may sometimes be manufactured with out the center pin.  Still, if you can power the Arduino R3 using the USB port and unable to power the Arduino R3 using power through the barrel jack, the likely suspects are U1 and D1.  However, to be sure, you would have to check the voltage before D1 and after U1.