# 6: Vector-Valued Functions of Several Variables

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IN THIS CHAPTER we study the differential calculus of vector-valued functions of several variables.

• SECTION 6.1 reviews matrices, determinants, and linear transformations, which are integral parts of the differential calculus as presented here.
• SECTION 6.2 defines continuity and differentiability of vector-valued functions of several variables. The differential of a vector-valued function $$\mathbf{F}$$ is defined as a certain linear transformation. The matrix of this linear transformation is called the differential matrix of $$\mathbf{F}$$, denoted by $$\mathbf{F}'$$. The chain rule is extended to compositions of differentiable vector-valued functions.
• SECTION 6.3 presents a complete proof of the inverse function theorem.
• SECTION 6.4. uses the inverse function theorem to prove the implicit function theorem.

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