Math 18.0A Gradients, Chain Rule, Implicit Dierentiation, igher Order Derivatives These notes ocus on our things: (a) the application o gradients to ind normal vectors to curves suraces; (b) the generaliation o the Basic Chain Rule to more general compositions o unctions o several variables; (c) a new more general approach to implicit dierentiation; (d) the deinition meaning o nd derivatives o unctions o several variables. Rate o change o a unction along a parameteried curve the Basic Chain Rule We have shown that i a unction (, ) is given a parameteried curve is described b r () t t (), t (), then we can determine the rate o change o the unction (, ) as we travel along this parameteried curve b the Basic Chain Rule. I we think o this as a composition, we have: t ( t ( ), t ( )) ( t ( ), t ( )) We showed that d [ ( t ( ), t ( ))] dt d d + dt dt. This is the Basic Chain Rule. The same construction can be done with a dierentiable unction (,, ) a parameteried curve r () t t (), t (), t () to give the rate o change d d d d [ ( t ( ), t ( ), t ( )) dt ] + + dt dt dt Chain Rule in this contet. as the Basic In either case (or in an even more general contet, we see that d d v where the gradient vector dt, in the ormer case,, in the latter case. It gives a vector at ever point, i.e. a vector ield. We investigated the geometr o the gradient ound that at ever point the gradient vector will be perpendicular to the level set o the given unction passing through an given point. This gives us a remarkabl simple wa to determine normal vectors to curves suraces, with this, a simple wa to determine equations or tangent lines to curves tangent planes to suraces. Eample 1: ind an equation or the tangent line to the curve deined b the equation at the point (,1). + 8 Solution: I we let (, ) +, then this curve is, in act, the 8 level set (level curve or contour). (You ma want to veri that (,1) 8.). We calculate the gradient vector +, + 6. At the point (,1) this gives the vector (,1) 6,16,8, we know that this must be perpendicular to the 8 level set at this point. We can thereore take n,8 as a normal vector to the line tangent to this level set. Using the relation n ( 0) 0, we have,8, 1 0 or ( ) + 8( 1) 0 or + 8 14. Eample : ind an equation or the tangent plane to the surace deined b the equation at the point (,,1). + 10 Solution: I we let (,, ) +, then this surace is, in act, the 10 ma want to veri that (,,1) 10.). We calculate the gradient vector level set (level surace). (You +,, + 6. At the 1 Revised November 1, 016
point (,,1) this gives the vector (,,1) 5,,18, we know that this must be perpendicular to the 10 level set at this point. We can thereore take n 5,,18 as a normal vector to the plane tangent to this level set. Using the relation n ( 0) 0, we have 5,,18,, 1 0 or 5( ) + ( ) + 18( 1) 0 or 5+ + 18 4. Note: In order to use this method to ind normal vectors to curves or suraces, ou ma have to transpose an variables in a given equation to one side o the equation leaving onl a constant on the other side beore deining a unction b the variable epression on one side o this equation. The General Chain Rule In general, the chain rule is an algebraic rule that describes how to calculate rates o change o unctions built rom other unctions through composition. or eample, in a irst semester calculus course we learn that i u ( ) u u ( ), then we can calculate d d d du b the chain rule:. In a multivariable setting, we d d du d d d d might have (, ) t (), t (). We then have + b the basic chain rule. dt dt dt The chain rule gets more interesting when ou appl it to situations where there are more input variables output variables. or eample, let us suppose we have a situation where there are two parameters, s t, st (,) that or an s t we have equations giving st (,). Let us urther suppose that or an choices o the st (,) u u(,, ) variables,, we have two other variables, u v, deined b equations. v v(,, ) In this case we can think o this unctionall as: (, ) G st (,, ) ( uv, ). I we var s onl (holding t constant) onl ocus on how the output variable u will change, the Basic Chain u u u u Rule gives that + +. Note that all o the derivatives are now partial derivatives. s s s s We can do the same b selectivel varing either s or t ocusing selectivel on the output variables u or v. Revised November 1, 016
We calculate: u u u u u u u u + + + + s s s s v v v v v v v v + + + + s s s s There is nothing especiall diicult about these seemingl complicated relationships. We simpl selectivel ocus on a particular output variable (u or v), then calculate partial derivatives (with respect to either s or t) b treating the other one as though constant. Unlike the Basic Chain Rule, all derivatives are now partial derivatives because all unctions are unctions o several variables. In each case there are as man terms as there are variables in the middle o the composition. These equations can be organied into a statement about the Jacobian matrices o the two unctions o their composition. A Jacobian matri ma be thought o simpl as an arra o (partial) derivatives o the various output variables with respect to the various input variables, where the outputs are listed rom top to bottom the inputs are listed rom let to right. I ou know about matri multiplication, we have: u s v s u v u u u v v v s s s or, more succinctl, J G JJG. Note: It s worth mentioning that the rows o each matri look like gradient vectors, the columns look like velocit vectors. This view o the Chain Rule can be eplained in terms o how incremental vectors or tangent vectors in the original domain are transormed to their counterparts in the image spaces. It is reall a statement o how the composition dierentiable unctions can be approimated b a composition o the linear transormations deined b the respective Jacobian matrices. To picture what this is telling us, let s speciicall look at the situation where φ θ represent latitude longitude with the minor change that latitude will be measured rom the north pole as 0, the equator as 90, the south pole as 180. We can then describe a sphere o radius R b the parametric equations R cosθ sinφ R sinθsinφ. R cosφ Let us urther suppose that the variables u v measure, or eample, temperature barometric pressure at an point (,, ) in R, in particular, at points on this parametried sphere in R. θ v φ u Revised November 1, 016
We might ask questions about how temperature would var as we change latitude or longitude, or how barometric pressure would var as we change latitude or longitude. These are the quantities in the Jacobian u u u u u φ θ matri J G. The rows o the Jacobian matri J are just the gradient vectors v v v v v φ θ (in R ) o the temperature barometric pressure unctions. (Note that these are unctions deined on R not just on the spherical surace.) φ θ The two columns o the Jacobian matri J G represent velocit vectors tangent to the longitudes φ θ φ θ (φ varing) latitudes (θ varing). These two column vectors are tangent to curves ling in the sphere are thereore tangent to the sphere. The are, essentiall, the south vector the east vector at an point o the sphere (ecept at the poles). You might urther observe that their cross product will be normal to this spherical surace at an given point a act which will be useul later in this course when we look at surace integrals. u u φ θ The two columns o the Jacobian matri J G represent vectors in the (u, v) plane indicate the v v φ θ directions o change i we slightl var the latitude or the longitude. Implicitl Deined unctions Implicit Dierentiation Oten it is the case that an equation (or several equations) relate some variables we wish to consider one variable (or several) as depending on the rest. or eample, given the equation o a circle + 16 we ma wish to consider ( ). I we solve eplicitl, we get either 16 or 16 whose graphs are, respectivel, the upper lower semicircles. Though we could calculate the derivatives directl, there is an alternate approach. We can think o as a parameter use it to parametrie either one o the semicircles as (, ( )), where the dependence o on is deined implicitl b the given curve (semicircle). I we let (, ) +, then we can view the circle as just the 16 contour, or level set, o the unction. Composing these unctions, we have: Appling the chain rule ( using (, ( )) (, ( )) constant to denote the partial derivatives o ), we have: d d (, ( )) 1 + 0 d d 4 Revised November 1, 016
ere we used the act that d 1 that the composite unction was constant everwhere on this level set. d Solving or d, we get that d d. So, as long as we avoid those places where 0 (where the two d semicircles meet), we have a valid ormula or calculating d. In the above eample, this gives d d d. This ma be used or either the upper or the lower semicircle. Note: The epression d d can also be derived geometricall b noting that, point, give a vector perpendicular to the level set, so the slope o the normal line will be tangent line will thereore be given b its negative reciprocal, i.e. d d. will, at an the slope o the This ormulation will be valid whenever we have a relation o the orm (, ) constant, where is a dierentiable unction where we can consider ( ) as being implicitl deined b the equation. The onl eception is at those points where 0, i.e. at points where the tangent line to the relation is vertical. This same approach can be used or relations o the orm (,, ) constant, where we ma wish to consider one o the variables as being dependent on the others. or eample, i we choose to think o (, ) deined implicitl b the given relation, then it is useul to consider as parameters to ormulate the situation as (, ) (,, (, )) (,, (, )) constant ere we can think o the relation as a surace in R, what this is saing is that b choosing (, ) we ma ind one point (or several points) on the graph. We can appl the chain rule to calculate the partial derivatives o the composition with respect to the parameters. What makes this a bit trick is the act that are plaing dual roles as parameters as coordinates in R. Nonetheless, we have (,, (, )) 1+ 0+ 0 (,, (, )) 0+ 1+ 0 (*) which enable us to solve or. These epressions will be valid wherever is dierentiable where 0. It should be relativel clear that this same ormulation could be done or relations with an number o variables would give analogous epressions or the partial derivatives o the implicitl deined unctions. 5 Revised November 1, 016
Note: The equations (*) above can also be interpreted in terms o dot products perpendicularit. The give that,, 1, 0, v 0,, 0,1, v 0 where ou should recognie the vectors v v as tangent vectors to the (, ) graph surace. Once again, we see that the gradient vector is perpendicular to tangent vectors to the (,, ) constant level surace thereore perpendicular to this level surace at an given point on the surace. Note: ad we instead chosen to deine (, ) as being deined implicitl b this relation, we would have similarl obtained the epressions, these epressions will be valid wherever is dierentiable where 0. We might also have chosen to deine (, ) as being deined implicitl b this relation, we would then obtain the epressions. These epressions will be valid wherever is dierentiable where 0. Second derivatives higher order derivatives o unctions o several variables I we think o partial derivatives as rates o change, then we can sa that or a unction (,, ) rate o change o the values o with respect to (increasing) -slope rate o change o the values o with respect to (increasing) -slope Continuing with these interpretations, we can deine ( interpret) nd derivatives as ollows: rate o change o the -slopes with respect to (increasing) rate o change o the -slopes with respect to (increasing) rate o change o the -slopes with respect to (increasing) rate o change o the -slopes with respect to (increasing) I we organie these nd derivatives into a matri, we call this the essian matri o this unction: 6 Revised November 1, 016
The irst last o these nd derivatives are relativel simple to interpret as concavities o the cross-sectional curves in the graph surace just as was the case with nd derivatives o unctions o a single variable. The mied partial derivatives are a little more diicult to interpret. You ma ind it helpul to imagine a graph surace use a straight object to represent a tangent vector imagine how the -slope might change as ou move laterall in the -direction, e.g. i there was a little twist in the graph surace. This is what the mied partial derivative would measure. Similarl, ou can interpret b imagining how the - slope might change as ou move laterall in the -direction. In act, these two mied partial derivatives are generall equal, though this is certainl not obvious. Indeed, this is the essence o Clairaut s Theorem. Clairaut s Theorem: I a unction o two or more variables is dierentiable i its irst second derivatives are continuous, then mied partial derivatives are equal. In the case o a unction o two variables, this simpl means that. or unctions o three variables, we can deine 9 second derivatives organie them into a essian matri: Clairaut s Theorem in this case gives that,,. In either this case or the previous case, Clairaut s Theorem means that the essian matri is a smmetric matri. We could also consider higher order derivatives, Clairaut s Theorem would continue to appl (assuming all derivatives are continuous unctions. or eample, or a unction (,, ) there would be 8 third partial derivatives:,,,,,,. Tr calculating some o these to see Clairaut s Theorem in action.,. owever, b Clairaut s Theorem, we would have or a unction o three variables (,,, ) there would be irst partial derivatives, 9 second partial derivatives, 7 third partial derivatives, but b Clairaut s Theorem the mied partial derivatives would be equal. Eample: or the unction (, ) +, the two irst partial derivatives are just the components o the gradient vector, +, + 6. The essian matri will then be: + 6 + 6 1 Eample: or the unction (,, ) +, the three irst partial derivatives are just the components o the gradient vector,, +,, + 6. The essian matri will then be: 0 + 6 0 + 6 1 Notes b Robert Winters 7 Revised November 1, 016