fR2R GR2R2 are defined by fxyxy Guvuvuv hR2R is defined to b

f:R^2->R, G:R^2->R^2 are defined by f(x,y)=xy G(u,v)=(u+v,uv) h:R^2->R is defined to be foG. a) Find dh(u,v) - using the chain rule. b) find dh(u,v) by calculating h(u,v) (find a \"formula\" for h(u,v) that involves only the variables u,v), then differentiating that function. (compare to answer from a).

Solution

Functions which have more than one variable arise very commonly. Simple examples are • formula for the area of a triangle A = 1 2 bh is a function of the two variables, base b and height h • formula for electrical resistors in parallel: R = µ 1 R1 + 1 R2 + 1 R3 ¶1 is a function of three variables R1, R2 and R3, the resistances of the individual resistors. Let’s talk about functions of two variables here. You should be used to the notation y = f(x) for a function of one variable, and that the graph of y = f(x) is a curve. For functions of two variables the notation simply becomes z = f(x, y) where the two independent variables are x and y, while z is the dependent variable. The graph of something like z = f(x, y) is a surface in three-dimensional space. Such graphs are usually quite difficult to draw by hand. Since z = f(x, y) is a function of two variables, if we want to differentiate we have to decide whether we are differentiating with respect to x or with respect to y (the answers are different). A special notation is used. We use the symbol instead of d and introduce the partial derivatives of z, which are: • z x is read as “partial derivative of z (or f) with respect to x”, and means differentiate with respect to x holding y constant • z y means differentiate with respect to y holding x constant Another common notation is the subscript notation: zx means z x zy means z y Note that we cannot use the dash 0 symbol for partial differentiation because it would not be clear what we are differentiating with respect to.