Function of several variables

The temperature at a point in space depends on three coordinates, the area of a rectangle on two measurements, and the cost of a product on many inputs. With \(n=2\) one writes \(z=f(x,y)\) and with \(n=3\), \(w=f(x,y,z)\). The set of values \(f\) takes is its range, and the set of points where it can be evaluated is the domain.

When only a formula is given, the domain is the largest set where the formula makes sense: denominators nonzero, square roots with nonnegative argument, logarithms with positive argument. For \(f(x,y)=\sqrt{x-y}\) the domain is the half-plane \(x\ge y\), and for \(\ln(xy)\) it is the points with \(xy>0\), the first and third quadrants. Sketching the domain in the plane is usually the first step of any exercise.

The graph of \(f(x,y)\) is the surface \(z=f(x,y)\) in \(\mathbb R^3\), made of the points \((x,y,f(x,y))\). For \(f(x,y)=x^2+y^2\) it is a paraboloid. With three or more variables the graph lives in \(\mathbb R^4\) or higher and cannot be drawn, so tools such as level curves, studied next, are used instead.