Multivariate Calculus and Geometry (Springer Undergraduate Mathematics Series)
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This book covers multivariate calculus with a combination of geometric insight, intuitive arguments, detailed explanations and mathematical reasoning. It features many practical examples involving problems of several variables.
to use coordinates we usually let if and, when , we let . It is helpful to think of as an interval of time and as the position of a particle at time as it travels along the route from to . We call thevelocity and the speed at time . Since distance speed time the formula where is the length of , is not surprising. We shall, however, pause to prove this formula in order to show the usefulness of vector notation. Since is differentiable we have for all and in where as for any fixed . Hence for
our first two coordinates so that the tangent plane to the surface at corresponds to the -plane and then take the -direction as one of the normal directions (Fig. 16.4b). Fig. 16.5. We define a function on the tangent plane near by letting denote the distance squared from the tangent plane in the direction to the surface (Fig. 16.5). This means that the surface near is the graph of and that , defined on the tangent space near , has a local minimum at . One can carry out a similar analysis at
and . Alternatively, if and then results in Chap. 8 show it suffices to prove is constant. , , and 7.11 implies is unit speed and . Hence (since and is positive) and . . Hence . 7.12 . Chapter 8 8.1 By inspection one can take in (a) and (b) and in (c). 8.2 , , , , . Note By inspection so lies in the plane . By Proposition 8.1, and . To know which sign to take it was necessary to do the above calculations. 8.3 , , , , , . 8.4 , , , , , , , , , , . 8.5 , . 8.6 The normal plane
vision of mathematics. This possibility arises from the extensive nature of the subject. Multivariate calculus links together in a non-trivial way, perhaps for the first time in a student’s experience, four important subject areas: analysis , linear algebra , geometry and differential calculus . Important features of the subject are reflected in the variety of alternative titles we could have chosen, e.g. “Advanced Calculus”, “Vector Calculus”, “Multivariate Calculus”, “Vector Geometry”,
level set where . To apply the methods of the previous chapter we suppose that has full rank on or equivalently that are linearly independent vectors for all in . Suppose has a local maximum on at the point . Since we only need to examine near we may suppose that is the graph of a function of variables. By rearranging the variables, if necessary, we can assume that , open in , that (the first coordinates of ) lies in and . Let Since has a local maximum on at the function (3.2) has a local