Single variable calculus is typically (and reasonably) taught over a whole year, with the first semester being devoted to "differential" calculus, and the second semester being devoted to "integral" calculus.
In my own experience, "multivariable" calculus is taught in one semester. That is a course with vector calculus, partial derivatives, gradient and the chain rule. Then onto line integrals, multiple integrals, and Green's Gauss' and Stokes' theorems. That's a lot for one semester.
Would it make sense to break "multivariable up into a two semester course, differential and integral, as with single variable calculus?
That would be a "differential" course with vector calculus, partial derivatives, gradient and the chain rule, Taylor series, constrained optimization, matrices, mappings, and determinants, plus the Jacobian in the first semester. And an "integral" course with improper integration, power series, change of variables, line integrals, multiple integrals, Fourier integrals and Green's, Gauss' and Stokes' theorems in the second semester.