After Mac and Stella arrives at the apartment, there's a close overview shot of the victim (Joel Ivey) showing his right arm over his chest with his thumb resting on his left arm. When the shot pans in on the arm, it is positioned alongside his body. When the camera pans out in a more distant shot, his arm is back over his chest but now his hand is in his armpit.
Towards the end of the episode, there is a shot of the apartment door which had been blasted with a shotgun through the peephole. The peephole is unscathed as compared to the gaping hole earlier in the episode directly after the crime.
Sheldon tells Danny that the umbrella was invented in China in "...during the Hui Dynasty, 326 AD." This is wrong on several counts. The dynasty in power in China at that time was the Jin (265-420). There is evidence that collapsible umbrellas date farther back, to the Han (206 BCE-220 CE) or Zhou (1046-256 BCE) Dynasties. There are also mentions of folding umbrellas or parasols in the Mesopotamian city of Nineveh (which was destroyed in 612 BCE), in Ancient Egypt and in the 5th century BCE Greece. So Sheldon was wrong about the Dynasty name, the time and may also have been wrong about the place.
While Lindsay is listening to the 911 call, there is a transcription of it running on her computer screen. The caller says, "...blast like a shotgun..." but the transcription says "... blast like a gunshot..."