is the legal principle that if the boundary between two political entities is stated to be a waterway, without further description (e.g., a median line, right bank, eastern shore, low tide line, etc.), the boundary follows the thalweg of that watercourse. In particular, the boundary follows the center of the principal navigable channel of the waterway (which is presumably the deepest part). If there are multiple navigable channels in a river, the one principally used for downstream travel (likely having the strongest current) is used. This definition has been used in specific descriptions as well. The Treaty of Versailles, for example, specifies that "In the case of boundaries which are defined by a [navigable] waterway" the boundary is to follow "the median line of the principal channel of navigation." The precise drawing of river boundaries has been important on countless occasions. Notable examples include the Shatt al-Arab between Iraq and Iran, the Danube in central Europe (Croatia–Serbia border dispute), the Kasikili/Sedudu Island dispute between Namibia and Botswana (settled by the International Court of Justice in 1999), and the 2004 dispute settlement under the UN Law of the Sea concerning the offshore boundary between Guyana and Suriname, in which the thalweg of the Courantyne River played a role in the ruling. The doctrine is also applied to sub-national boundaries, such as those between American states. In rivers with occasional flooding and especially ice cover which breaks in the spring, the thalweg can change. This can mean that the border might move if it is defined as the thalweg. The Finland–Sweden border has been moved several times due to this. Mathematics In the field of topology, the thalweg can be described as a parametrised curve. If the elevation of the valley floor at some position with coordinates is represented by a surface , then the negative gradient vector field equips each position with a vector pointing downhill in the direction of steepest slope (or by analogy, the direction a ball would roll if released from this position in the valley). Any parametrised curve that is everywhere directed along the gradient vector field eventually joins the thalweg (or using the same analogy, you can release a ball from any position in the valley, and it will eventually meet and then roll along the thalweg, assuming a smooth rolling surface so the ball doesn't get stuck, and also assuming sufficient rolling resistance or wind