Partial Derivatives and vectors are essential to two important
concepts in multivariable calculus-*surfaces in 3 dimensional space *and
*transformations *from one coordinate system into another.
Thus, in this chapter, we combine the ideas of the first two chapters in
order to study and explore surfaces and
transformations.

In so doing, we will introduce several
concepts which will be important in the chapters on integration which are
to follow. We will also discover some very interesting and elegant
results about surfaces. In particular, we will introduce the
concept of *intrinsic* properties of a surface, which will include
some of the concepts that have greatly influenced mathematics over the
past 100 years.

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