Does heat flow always in the direction of maximum decrease? I was wondering if at a given gradient temperature heat could flow in any direction satisfying moving from a higher to a lower temperature. Or it can only flow in the direction of gradient? Consider a point $(x_0,y_0,z_0)$ where $T_0=T(x_0,y_0,z_0)$ and $T_1$, $T_2$ are the temperatures of two points very close to the $(x_0,y_0,z_0)$. It is true that $T_0>T_1>T_2$. Would some amount of heat flow to the point with $T_1$ and another amount to the point with $T_2$ or all the heat would flow to the point with the maximum temperature difference? What if $T_1=T_2$?
 A: It is useful to think of level surfaces of the same temperature, around the hottest region. Surfaces far from there will be colder than the closer ones. 
The heat flow follows the gradient, that is normal to that surfaces at each point.
In your example, if $T_1 = T_2$ points $1$ and $2$ are at the same level surface, and there is no flow between that points.
A: One can also think about it in the following way. At point $0$, the temperature is higher than at point $1$ and $2$. What it means is that the molecules of the medium at point $0$ are "jiggling" more rapidly than the molecules at points $1$ and $2$. These rapidly moving molecules at point $0$ will exchange momentum with neighboring molecules, which are "jiggling" little slower, making them "jiggle" little faster. This "jiggling" will "propagate" towards point $1$ as well as towards point $2$, the difference is that "the rate of propagation" of this "jiggling" $(\propto \nabla T)$ towards point $2$ will be larger than it is towards point $1$ because $T_0>T_1>T_2$. The "propagation of jiggling" is called heat transfer and "the rate of propagation" is related to the heat flux $\mathbf{q}$. Of course to calculate "total amount of heat transfer", one has to consider where the points $1$ and $2$ lie relative to point $0$, are they on the same line? Is point $1$ closer to point $0$ than the point $2$? How much time has elapsed? Are the points maintained at constant temperature? The situation may become even more complicated if the medium is a fluid! Hope this gives the idea and probably now you can guess what happens if $T_1=T_2$.
