Seán Dineen - Multivariate Calculus and Geometry

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Springer-Verlag London 2014

Sen Dineen Multivariate Calculus and Geometry Springer Undergraduate Mathematics Series 10.1007/978-1-4471-6419-7_1

1. Introduction to Differentiable Functions

Sen Dineen 1

(1)

School of Mathematical Sciences, University College Dublin, Dublin, Ireland

Sen Dineen

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Summary We introduce differentiable functions, directional and partial derivatives, graphs and level sets of functions of several variables .

In this concise chapter we introduce continuous and differentiable functions between arbitrary finite dimensional spaces. We pay particular attention to notation, as appropriate notation is often the difference between simple and complicated presentations of several-variable calculus. Once this is in place many of our calculations follow the same lines as in the one dimensional calculus. We do not include proofs but, for readers familiar with analysis, we provide suggestions that lead to proofs along the lines that apply in the one variable calculus.

The following extremely simple example illustrates the type of calculation we will be executing frequently and the reader should practice similar examples until they become routine and the intermediate step is unnecessary.

Example 1.1

Let

The partial derivative of with respect to , or , is obtained by treating and as constants and differentiating with respect in the usual one variable way. Thus if and then and

Similarly if and then and

and, if and , then and

We now recall concepts and notation from linear algebra. First we define the distance between vectors in . This will enable us to define convergent sequences, open and closed sets, continuous and differentiable functions, and state the fundamental existence theorem for maxima and minima.

If let and call the length (or norm ) of . If and are vectors in then is the distance between and . The inner product (or dot product or scalar product ) of and , or , is defined as

We have and two vectors and are perpendicular if and only if their inner product is zero.

For , let , where 1 lies in the th position. The set is a basis , the standard unit vector basis , for . If

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