I am reading about the one dimensional heat equation and boundary conditions. This is what the author writes about the left hand end of the rod:
Can someone explain to me why the slope must be positive at $x=0$? If the rod is hotter shouldn't it loose temperature, i.e. negative slope?
The slope in the $t$ direction is negative because the temp is dropping over time. But in the $x$ direction, the change in heat is positive, because things are cooler to the left.