#
Chapter 10
Analysis in ℝ^{
m
}

## 10.1 Derivatives

In this section, we review some of the key results in the differential calculus of one or more real variables. For the most part, proofs are not provided.

Let *U* be an open set in ℝ^{
m
}, let *F* : *U* → ℝ^{
n
} be a map, and let *p* be a point in *U*. We say that F is **differentiable at p
** if there is a linear map

*L*

_{ p }: ℝ

^{ m }→ ℝ

^{ n }such that

It can be shown that if such a map exists, it is unique. We call this map the **differential of F at p
** and henceforth denote it by

In the literature, the differential of *F* at *p* is also called the derivative of *F* at *p* or the total derivative of *F* at p, and is denoted variously by *dF*
_{
p
}
*,* d*F*
_{
p
}, *DF*(*p*), *D*
_{
p
}(*F*), or *F'*(*p*). We have chosen to include parentheses in the notation *d*
_{
p
}(*F*) to set the stage for viewing *d*
_{
p
} as a type of map.

It is usual to characterize *d*
_{
p
}(*F*) as being a “linear approximation” to *F* in the vicinity of *p*. For a more geometric interpretation, let us define the **graph of F
** by

We can think of *d*
_{
p
}(*F*)(ℝ^{
m
}), which is a vector space, as being the “tangent space” to graph(*F*) at *F*(*p*). This is a generalization of the tangent line and tangent plane familiar from the ...

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