In the US it's typically covered at the end on calc II
A typical course looks like:
The 3-D Coordinate System
Equations of Lines
Equations of Planes
Functions of Several Variables
Calculus with Vector Functions
Tangent, Normal and Binormal Vectors
Arc Length with Vector Functions
Interpretations of Partial Derivatives
Higher Order Partial Derivatives
Tangent Planes and Linear Approximations
Gradient Vector, Tangent Planes and Normal Lines
Relative Minimums and Maximums
Absolute Minimums and Maximums
Double Integrals over General Regions
Double Integrals in Polar Coordinates
Triple Integrals in Cylindrical Coordinates
Triple Integrals in Spherical Coordinates
Change of Variables
Line Integrals of Vector Fields
Fundamental Theorem for Line Integrals
Conservative Vector Fields
Curl and Divergence
Surface Integrals of Vector Fields
there are two main ideas:
1. in differentiation of two variable functions f, the main point is the gradient vector gradf, which lies in the domain of f, and points in the direction of greatest increase of f, hence at any point p, is perpendicular to the "level set" passing through p, i.e. the set where f has the same value as at p. the components of gradf are "partial" derivatives (fx,fy) computed by considering f as a function first only of x, then only of y.
2. in integrating functions of two variables, the main idea is that of volumes by slicing, or cross sections, which means you can reduce a double integral to the successive computation of two single integrals.
when looked at closely, this also can be phrased as implying that the integral measuring the flow of a "vector field" with components (M,N) around the boundary curve of a closed region R, equals the double integral of Nx-My taken over R, (where Nx and My are partial derivatives of N,M wrt the variables x,y).
that is called greens theorem.
thus the subject consists in defining analogs in several variables of the old friend ideas: namely derivatives and integrals; i.e. tools for measuring how a function changes, and for averaging its values; and then giving techniques for reducing the calculations of these concepts to the old calculations in one variable.
a great technical convenience is acquired by studying how these calculations change when we change variables. for derivatives these are called chain rules, and for integrals are called change of variable formulas for integrals. the chain rule for derivatives basically says the deriv of a composite is computed by taking dot products, or matrix products of the successive derivatives. for integrals, volume changes by multiplying by the determinant of the linear approximation of the transformation.
e.g. polar coords or cylindrical coords, or spherical coords are useful when studying objects of those shapes. in my class i also taught elliptical coords but these are not usually mentioned. still computing the volume of an ellipsoid is easiest in elliptical coords, just as computing volume of a sphere is easiest in spherical coords.
i have covered most of the topics listed in post 7 above. (since those topics listed after green are just higher dimensional, or rotated, or curved, versions of green.)