Exercise 1.7: Properties of inline_formula not implemented
This exercise asks us to prove various products of the variation operator inline_formula not implemented. This is a sort of higher-order derivative operator. Apply it to a higher order function inline_formula not implemented, and you'll get a function back that returns the sensitivity of inline_formula not implemented to fluctuations in its input path function. (Confusing? Check out the textbook.)
Variation Product Rule
The product rule for variations states that:
formula not implementedWrite out the left side explicitly, using the definition of inline_formula not implemented:
formula not implementedMake the inspired move to add and subtract inline_formula not implemented inside the limit, rearrange and factor out the terms that have appeared in common. (Stare at this for a moment to make sure the steps are clear.)
formula not implementedYou might recognize that we've now isolated terms that look like inline_formula not implemented and inline_formula not implemented, as inline_formula not implemented approaches 0. Notice that as this happens, inline_formula not implemented, and the whole expression evaluates to the product rule we were seeking:
formula not implementedVariation Sum Rule
The sum rule is easier. Our goal is to show that:
formula not implementedExpand out the definition of the variation operator, regroup terms, allow inline_formula not implemented and notice that we've recovered our goal.
formula not implementedDone!
Variation Scalar Multiplication
We want to show that inline_formula not implemented preserves multiplication by a scalar inline_formula not implemented:
formula not implementedExpand out the definition of the variation operator:
formula not implementedDone, since the limit operator preserves scalar multiplication.
Chain Rule for Variations
The chain rule for variations states that:
formula not implementedExpand this out using the definition of inline_formula not implemented:
formula not implementedNow multiply the term inside the limit by inline_formula not implemented and factor out the new, more recognizable product that forms:
formula not implementedThe remaining term inside the limit has the form of a derivative of some function inline_formula not implemented evaluated at a point inline_formula not implemented.
formula not implementedWhere inline_formula not implemented and inline_formula not implemented. As inline_formula not implemented, inline_formula not implemented. We know this because we showed that inline_formula not implemented exists and factored it out.
Remember that that this is all function algebra, so composition here is analogous to function application; so inline_formula not implemented is indeed the inline_formula not implemented in equation \eqref{eq:var-chain-proof3}, and the remaining term collapses to inline_formula not implemented evaluated at inline_formula not implemented:
formula not implementedinline_formula not implemented commutes with inline_formula not implemented
We need to show the derivative can commute with a normal derivative of the function that inline_formula not implemented returns after it's passed a path:
formula not implementedExpand the left side by the definition of inline_formula not implemented:
formula not implementedThe derivative inline_formula not implemented is a linear operator, so we can move it in to the limit and distribute it over subtraction:
formula not implementedOur goal is achieved.