At German universities, math students typically take the courses “Analysis 1” and “Analysis 2” in their first year. Analysis 1 is proof-based real analysis rather than calculus, and analysis 2 is – well – “multivariable analysis”. The subtitle of my analysis 2 book was “Differential and integral calculus of several variables” (but, again, the book was rigorous and proof-based).
My impression is that multivariable analysis does not really exist as a subject in the US system. There seem to be the concepts of calculus and analysis, the former typically covering functions on $\mathbb{R}^n$ in an only non-rigorous way and the latter only as a special case of functions on more abstract spaces. In fact, the first Google hit for “multivariable analysis” is some statistical principle rather than mathematical analysis. Is my impression correct?
To add some more background information: Here is the table of contents of my German analysis 2 book. It covers “the $n$-dimensional space”, “vector-valued functions of one variable”, “differentiable functions”, “higher derivatives”, “applications of differentiation”, “integration theory”, and “measurability”. In which courses do students in the US learn about these topics?