Multivariable Calculus — Vectors & Gradients

Functions of two variables

continues from lesson 1 — values defined earlier in the course stay live here

ELI5: a function like f(x, y) = x² + y² is a landscape — give it a spot (x, y) and it returns a height. (Its graph is a bowl.) We can't draw the 3-D surface here, but we can take a slice — fix y and watch the height change with x.

f2\left(x, y\right) = x^{2} + y^{2}

a bowl-shaped landscape

height = \mathrm{f2}\left(3, 4\right)

25

the height above (3, 4): 25

f2(x, 1)

a slice at y = 1 — height vs x along one line across the bowl

Real-world hook: two-variable functions are terrain elevation (height from latitude & longitude), profit (from price & quantity), and a model's error (from each of its settings).