20e-lecture1

# 20e-lecture1 - Lecture 1 Monday March 30th [email protected]

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Unformatted text preview: Lecture 1 : Monday March 30th [email protected] Key concepts : Open ball, open set, continuous surface Know whether a given set is open 1.1 Functions of several variables Recall that R n denotes the set of all n-tuples, or n-dimensional vectors , of real numbers. For n = 1 we have the usual real line, and for n = 2 we have the plane. In this course, a function f : R n → R m assigns to each vector in R n a vector in R m . This is called a function of n variables , since the domain consists of n-dimensional vectors. So, for example, the function f : R 2 → R 3 defined by f ( x,y ) = ( x,x + y,x/y ) is a function of two variables. Sometimes we will refer to a function f : R n → R m for m > 1 as a vector-valued function or a vector field , especially later in the course. In certain instances, the domain of a function of several variables is restricted to be a subset of R n . For example, the domain of the function above is { ( x,y ) ∈ R 2 : y 6 = 0 } ....
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20e-lecture1 - Lecture 1 Monday March 30th [email protected]

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