The purpose of this lecture is to provide a “crash course” on calculus with multiple variables. We will (briefly) cover:

• Differentiability with multiple variables
• Partial derivatives, directional derivatives and the gradient
• Second-order derivatives and the Hessian
• Multi-variable Taylor theorem

Our general approach will be based on reducing the multi-variable case to the single-variable case, and using classical results we know from basic calculus.

Differentiation of Real-valued Functions

Recap: Differentiability in single variable

<aside> 💡 Definition: differentiable, single variable For $f : (a,b) \to \R$ and $x \in (a,b)$, if

$$

\lim_{h\to 0} \frac{f(x+h) - f(x)}{h} \quad\text{exists}, $$

then:

• We say that $f$ is differentiable at $x$, and that the above limit is the derivative of $f$ at $x$.
• We write $f'(x)$ to denote the derivative of $f$ at $x$. </aside>

We would like to generalize this definition to multivariate functions $f : \R^d \to \R$.

Preliminary: Limits of multivariate functions

<aside> 💡 Definition: limit, multivariate case Assume that $S \subseteq \R^d$ and that $f : S \to \R$ is a function. The statement $\lim_{x \to a} f(x) = L$ is defined to mean:

$$ \begin{align*} &\forall \epsilon >0, \; \exists \delta>0 \;\;\text{s.t.:}\;\; \\ &x \in S \text{\;\;and\;\;} 0 < \|x - a\| < \delta \;\implies\; |f(x) - L| < \epsilon. \end{align*} $$

</aside>

• In words: as we approach closer to $x$, in terms of the Euclidean norm and no matter "from which direction", the function values approach $L$.
• This is identical to the familiar definition from single-variable calculus, with the Euclidean norm $\|x-a\|$ replacing an absolute value $|x-a|$.