[skhunted]

Calculus 3 in the U.S. is multi-variable calculus. Sometimes and introduction to linear algebra is given in this course. Calculus 4 is differential equations. It’s almost never officially called Calculus 4. Sometimes an introduction to linear algebra is given in this course.

[ChainOfFools]

Okay that makes sense, that's how the classes went for me in the US, with Calc 3 being multi variable (and as you say, a brush with linear alg via vector-valued functions), and 2 sideways steps from there, one being a course on PDEs and the other on linear algebra, both called by those names rather than listed as a step in the calculus sequence itself.