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Conclusion

In this chapter, differentiation in several variables was organized around linear approximation.

Limits and continuity for maps $\R^n \to \R^m$ are defined in the same spirit as in one-variable calculus, and they can be checked componentwise.
Differentiability at $\va$ means that $f(\va+\vh)-f(\va)$ is approximated to first order by the linear map $Df(\va)\text{,}$ represented in coordinates by the Jacobian matrix.
Directional derivatives and partial derivatives describe first-order change, and the gradient packages this information into a single vector that points in the direction of greatest increase.
Second-order Taylor approximation and the Hessian provide tools for studying local maxima, local minima, saddle points, and constrained extrema through Lagrange multipliers.
When the derivative is invertible, the inverse and implicit function theorems show that equations can be solved locally in a stable, differentiable way.

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