6. Law of Conservation of Energy in Higher Dimensions

The paragraph begins with an explanation of how the change in a function, denoted as Δf, can be approximated by the derivative of the function at a point times the distance moved, Δx. It emphasizes that this is an approximation and corrections are present, symbolized by dots, and proportional to higher powers of Δx. The discussion then transitions into applying the Work Energy Theorem and the Law of Conservation of Energy in two dimensions, aiming to derive a relation like K_1 + U_1 = K_2 + U_2, assuming no friction. The potential energy U_2 is introduced as a function of two variables, x and y, and the importance of understanding multivariable functions is highlighted. The visualization of such functions is compared to a canopy over the xy-plane, with the function value representing the height of the canopy at a given point. The concept of partial derivatives, denoted by df/dx and df/dy, is introduced as a means to measure how the function changes with respect to x while keeping y constant, and vice versa. The paragraph concludes with a note on the equality of cross derivatives, a property of 'reasonable' functions, and the importance of understanding why this property holds.

This paragraph delves into practical exercises involving partial derivatives. A function f = x^(3)y^(2) + y is given, and the task is to find its partial derivatives with respect to x and y, treating the other variable as constant. The process of finding higher-order derivatives, such as d^(2)f/dx^(2) and d^(2)f/dy^(2), is also demonstrated. The concept of cross derivatives, d^(2)f/dxdy and d^(2)f/dydx, is introduced, and it is shown that for 'reasonable' functions, these cross derivatives are equal. The reason behind this equality is explained through the concept of incremental changes in function values as one moves from one point to another in the xy-plane. The paragraph concludes with a reminder of the general formula for the change in a function due to changes in x and y, which is a straightforward extension of the one-dimensional case to two dimensions.

The paragraph explores the concept of work done by a force in two dimensions and its dependency on the path taken. It starts by discussing the need to introduce an intermediate point when calculating the change in a function and the importance of being cautious about where the derivatives are taken. The paragraph then presents a scenario where the work done by a force depends on the path taken from one point to another, contrasting it with the one-dimensional case where the work done is path-independent. The concept of a conservative force is introduced, and it is shown that not all forces are conservative, as evidenced by a force generated by the class. The paragraph concludes with a demonstration that for a randomly chosen force, the work done is indeed path-dependent, which implies that potential energy cannot be defined for such forces in the same way it is in one dimension.

This paragraph focuses on defining conservative forces and their associated potential energy in two dimensions. It explains that for a force to be conservative, the work done in moving from one point to another should be independent of the path taken. The paragraph introduces a method to generate conservative forces by starting with a potential energy function U and then deriving the force components F_x and F_y from it. The condition for a force to be conservative is given by the equality of cross derivatives, d^(2)U/dxdy and d^(2)U/dydx. The paragraph also provides a test to check if a given force is conservative by comparing the y-derivative of F_x with the x-derivative of F_y. The concept is illustrated with the example of gravity near the Earth's surface, which is a conservative force, and the potential energy function mgy is derived from it. The paragraph concludes with an example of a roller coaster moving under the influence of gravity and the normal force, which does not affect the total mechanical energy due to its perpendicularity to the motion.

The final paragraph uses the example of a roller coaster to illustrate the concept of energy conservation. It describes the roller coaster's path as a function of x, with the height at every x representing the potential energy U. The total energy E of the roller coaster is introduced as a constant sum of kinetic and potential energy. The behavior of the roller coaster is analyzed as it moves along the track, gaining and losing kinetic and potential energy while maintaining a constant total energy. The possibility of the roller coaster stopping and turning around at a point where the potential energy equals the total energy is discussed. The paragraph also touches on the concept of quantum tunneling as an alternative process that could occur in a quantum mechanical context. Finally, the role of the normal force exerted by the track is considered, explaining that it does not contribute to the work done or change in total energy because it is perpendicular to the displacement. The paragraph concludes by reinforcing the importance of understanding the conditions under which the Law of Conservation of Energy applies.