Differential calculus in several variables asks how a function changes when its input changes in many possible directions. The central idea is that near a point \(\va\text{,}\) a differentiable map is approximated by a linear map \(Df(\va)\text{.}\)
We begin by visualizing scalar-valued functions through graphs and level sets. We then study limits and continuity, define differentiability, and examine directional derivatives, partial derivatives, and the gradient. We use these tools to build Taylor approximations, analyze extrema, and prove the inverse and implicit function theorems.
