Ivancevic_Applied-Diff-Geom

90 Applied Differential Geometry: A Modern Introduction2.3.1.5 Compositions of MapsGiven two maps f and g, the composite map f ◦ g (also called the compositionof f and g) is defined by(f ◦ g)(x) = f(g(x)).The (f ◦ g)−machine is composed of the g−machine (first) and then thef−machine [Stuart (1999)],x → [[g]] → g(x) → [[f]] → f(g(x))For example, suppose that y = f(u) = √ u and u = g(x) = x 2 + 1. Since yis a function of u and u is a function of x, it follows that y is ultimately afunction of x. We calculate this by substitutiony = f(u) = f ◦ g = f(g(x)) = f(x 2 + 1) = √ x 2 + 1.2.3.1.6 The Chain RuleIf f and g are both differentiable (or smooth, i.e., C k ) maps and h = f ◦ gis the composite map defined by h(x) = f(g(x)), then h is differentiableand h ′ is given by the product [Stuart (1999)]h ′ (x) = f ′ (g(x)) g ′ (x).In Leibniz notation, if y = f(u) and u = g(x) are both differentiable maps,thendydx = dy dudu dx .The reason for the name chain rule becomes clear if we add another linkto the chain. Suppose that we have one more differentiable map x = h(t).Then, to calculate the derivative of y with respect to t, we use the chainrule twice,dydt = dydududxdxdt .2.3.1.7 Integration and Change of Variables1–1 continuous (i.e., C 0 ) map T with a nonzero Jacobian∣∣ ∂(x,...)∂(u,...)∣ thatmaps a region S onto a region R, (see [Stuart (1999)]) we have the followingsubstitution formulas: