Download Multivariate Calculus and Geometry (3rd Edition) (Springer by Seán Dineen PDF

By Seán Dineen

Multivariate calculus should be understood top by way of combining geometric perception, intuitive arguments, specific motives and mathematical reasoning. This textbook has effectively this programme. It also presents a fantastic description of the elemental techniques, through known examples, that are then established in technically tough situations.

In this new version the introductory bankruptcy and of the chapters at the geometry of surfaces were revised. a few workouts were changed and others supplied with extended solutions.

Familiarity with partial derivatives and a path in linear algebra are crucial must haves for readers of this ebook. Multivariate Calculus and Geometry is aimed basically at better point undergraduates within the mathematical sciences. The inclusion of many sensible examples concerning difficulties of a number of variables will attract arithmetic, technological know-how and engineering scholars.

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Extra resources for Multivariate Calculus and Geometry (3rd Edition) (Springer Undergraduate Mathematics Series)

Example text

20 2 Level Sets and Tangent Spaces normal space P tangent space Fig. 2 In Rn there are various ways of presenting lines, planes, etc. The normal form consists of a description as the set of points satisfying a set of equations while the parametric form is in terms of independent variables and this, as we shall see in Chaps. 7 and 10, is almost a parametrization of the space. 4 Let S denote the set of all points in R3 which satisfy the equation x 2 + 2y 2 − 5z 2 = 1. We wish to find the tangent space and the normal line at the point (2, −1, 1) on S.

2 If ∇g(x1 , . . , xn ) = λ∇ f (x1 , . . , xn ) then x1 · · · xn /xi = λ and xi = x1 · · · xn /λ for all i. This shows x1 = x2 = · · · = xn . Since x1 + x2 + · · · + xn = 1 we have xi = 1/n for all i and g(1/n, 1/n, . . , 1/n) = n −n . As g(x1 , . . , xn ) = 0 whenever one of the xi ’s is equal to zero it follows that the maximum of g, on the set f (x1 , . . , xn ) = 1 and xi ≥ 0 all i, is (1/n)n . n If xi , i = 1, . . , n, are arbitrary positive numbers let yi = xi j=1 x j for each i. We have n n i=1 and, by the first part, yi = x1 · · · xn n xj n xi i=1 n j=1 x j = 1, = y1 · · · yn ≤ 1 n n .

The above method does not identify absolute or global maxima and minima. 3. We now describe a useful method which can be applied to certain functions on convex open sets. A subset U ⊂ Rn is convex if the straight line joining any two points in U is contained in U. The interior of a circle, sphere, box, polygon, the first quadrant or octant, and the upper half-plane are typical examples of convex open sets. The exterior of a circle or polygon is not convex. Suppose f : U (open, convex) −→ R has continuous first and second order partial derivatives at all points.

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