This doesn't quite answer the question but multivariable calculus is baby differential topology and there are a few topological theorems that economists quote frequently.

The most common one is probably Brouwer fixed point theorem, which is required for the existence of Nash equilibria of a pretty wide class of games. Game theory is the main ingredient of microeconomics. (Actually the relevant general theorem is Kakutani's fixed point theorem for compact- and convex-valued correspondences but that is a stretch for most econ folks.)

Also, the Hairy Ball theorem has an economic interpretation. In a market, for each corresponding price there is an excessive demand. If there are $n$ goods, then price and demand are vectors in $\mathbb{R}^n$. When excessive demand is zero, economics says the market clears and is in equilibrium. So Hair Ball theorem says that the market clears for some price.

If you cover Lagrange multipliers, economics students taking intermediate microeconomics and macroeconomics see that everyday. The econ view Lagrange multiplier is the "shadow value of money", meaning that if the budget constraint is relaxed by $\epsilon$ at the optimal bundle (in the direction of the gradient), consumer utility increases by $\epsilon \cdot \lambda$. The equation

$$\nabla u = \lambda \cdot \nabla g$$

describes the consumer substituting between goods in his basket as he compares marginal utilities (entries in $\nabla u$) and marginal cost (entries in $\nabla g$).

As for the gradient and Maximal Likelihood estimation procedure, at the very basic level it's just a calculation. Suppose you have observations $\{ x_i \}$ drawn independently from the probability space $(\mathbb{R}, f(x, \theta)dx)$, where $\theta$ lies in some compact parameter space $\subset \mathbb{R}^n$. To estimate $\theta$, you maximize the likelihood function

$$
L(\theta) = \Pi_i f(x_i, \theta).
$$

The gradient of $\log L$ is called the score function. On a deeper level, although I am sure how much of it you can mention to your undergrads, MLE is the beginning point of information geometry, where statistics and differential geometry interact.

If you must, provide links from each Instance to the others. $\endgroup$