B2.1MATH10121 B2functionsinmorethanonedimensionEquations, such asxyza “saddle”z= 1 +x2−y2describe a surfacein space, whilealso, defininga functionz=f(x, y)wherefmaps(x, y)to1 +x2−y2with domainR2andrangeR(a ‘real-valued function of two independent variables’)More examples:excosymapsR2→R11 +x2+y2+z2mapsR3→Rsin(r−t)mapsR2→Rtcosti+tsintjmapsR→R2k×rmapsR3→R3(the last two are called ‘vector-valued functions’)B2.2functions fromRntoRmFunctions may have domains inRnand ranges inRmA real-valued function has its range inRA vector-valued function has range inRmform >1A vector-valued function with range inRmcan be thought of asmreal-valued functions(each representing one coordinate of a vector)f1(x1, x2, . . . xn), f2(x1, . . . xn), . . . fm(x1, . . . xn)A function of (say) two variables is often written asf(x, y)orf(·,·)where the order of the variables is important, that isf(x, y)=f(y, x)Example:ifg(x, y) =excosywhat isg(y, x)?Functions of any number of variables can arise, e.g.S(t, x, y, z, P, T)mappingR6→RAs for functions of a single-variable:•a function isdefined for all elementsin its domain.•a function maps any one element in its domaintoonly one elementin its range.