Student Lectures - All lectures - Multidimensional Analysis & Geometry: introduction to the derivative in higher dimensions, lecture 1

Oxford Mathematics 2nd Year Student Lecture - Kevin McGerty The classical definition of the derivative is that it is the limiting ratio of the change in the function y relative to the change in the input variable x, or dy/dx. Here x is a real number and y a real-valued function of x. But suppose now that we wish to consider a function y of the position of a particle in 3-dimensional space . Then the classical 'dy/dx' definition breaks down, as we cannot divide by a vector representing a small change of position. Going back to the intuitive notions that motivate the definition of the derivative, however, such as the velocity of a moving particle or the slope of a graph, one is led to the crucial insight that the derivative at a point p say, is not a scalar but rather a linear map which is, in a sense we will make precise, the 'best linear approximation' to our function near p. Once this idea is rigorously defined, many of the basic properties of the derivative that are familiar from the single-variable case extend readily to higher dimensions. You can watch many other student lectures via our main Student Lectures playlist (also check out specific student lectures playlists): https://www.youtube.com/playlist?list=PL4d5ZtfQonW0A4VHeiY0gSkX1QEraaacE All first and second year lectures are followed by tutorials where students meet their tutor in pairs to go through the lecture and associated problem sheet and to talk and think more about the maths. Third and fourth year lectures are followed by classes.