Partial Differentiation L. M. Kalnins, January 2010
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1 Partial Differentiation L. M. Kalnins, Januar 2010 Just as ordinar derivatives enable us to stud the rate of change of a function of one variable, partial derivatives enable us to stud the rates of change of a function of multiple variables. If f is a function of, it will have two independent rates of change, one describing how f changes with a change in with held constant one describing how f changes with change in with held constant. These are the partial derivatives f f. Even though is held constant, the value of ma depend on that constant value. For eample, whether food is spoils or not depends on ( f both temperature time, but a small increase in time will increasing the odds of spoiling much more if the temperature is high than if it is low; even though the temperature itself is constant, it will affect the value of ( spoiling time temp. Notation: d,, what it means when ou have half a derivative With ordinar derivatives, we said that df d was the rate of change of f with respect to. The same is true with partial derivatives: is still the rate of change of f with respect to. The smbol ( f is used to indicate that f does not depend onl on, i.e. there will be other rates of change for f with respect to the other variables. The subscript tells us that is held constant, meaning that we don t have f(; we have f(,. The subscripts should be used whenever it might be ambiguous which other variables are involved. Likewise, f is seldom used with partial derivatives because it is not clear what variable ou are differentiating b. (There are tpes of notation to get around this, but we don t use them in this course. So if ou write d dt ou are impling that is a function onl of t. If ou write t, ou are impling that is a function of several variables, ou are considering how changes when onl one of them, t changes. When working with derivatives, we often encounter what looks like half a derivative, just d on its own. d is a tin change in (rigourousl, an infinitesimal change. So, broken down, the derivative d dt is a tin change in divided b a tin change in t. On a graph of as a function of t, dt would be the movement along the t ais, d the movement along the ais, d dt the slope of the line. Taking a Partial Derviative Partial derivatives follow the same rules as ordinar derivatives - the product rule, the chain rule, the quotient rule, etc. The major difference is the need to pa attention to which variables are being held constant, thus must be treated like an other constant, for this differentiation. Eample 1: Calculate ( f if f(, 2 sin(2 cos( 3 4 Answer: f 2 2 cos(2 cos(3 4 Note the similarit to f( a 2 b sin(2, d f d 2a 2b cos(2 As with ordinar derivatives, it is not necessar to have the equation in a form where the dependent function is alone on the left h side while the right h side contains onl the independent variables. 1
2 You can differentiate both sides of the equation, so long as ou are careful to remember what is an independent variable should be treated as a constant what is a dependent variable, where ou will need to use the chain rule. Eample 2: Consider two possible sets of variables for a function, (, (u, v. The two sets of variables are related b 2 sin(u 2 cos(v 2uv. Differentiate with respect to. Answer: Consider dependent versus independent variables with respect to, the variable b which we are differentiating. is independent; u v are dependent. Thus we get: 2 cos(u u 2 cos(v 2 sin(v 2 u v 2u 2uv N.B. Knowing which variable is being held constant is particularl important if ou want to use the reciprocal of a derivative. Just as du d 1 d, u 1. However, looking at the variables in this du ( u eample, if we differentiate b u, we should be holding v constant, u 1. ( u v Choosing Dependent Independent Variables In man cases, the problem is constructed such that the choice of variables is fied, though ou ma have to determine what is being said or implied about the dependence or independence of each variable. In other cases, the choice ma be more open ou ma be able to choose to describe an object s motion in Cartesian or polar coordinates, or to pick which two variables to treat as independent in a thermodnamics question. If ou are choosing variables, consider 1 if the end goal requires (or is at least aided b a particular choice of variables if ou need to find a partial derivative with respect to P with T held constant, ou re going to need to treat P T as independent at some point 2 what makes the equations eas to set up an object moving in a circle has a simpler description in polar coordinates than in Cartesian ones. Simultaneous Equations in Partial Derivatives Again consider two possible sets of variables (, (u, v. Sa ou are asked to find the partial derivative u. If ou can write an epression of the form u f(,, i.e. u as a function of the variable of differentiation ( here variables independent of it ( here, calculating the partial derivative is relativel straightforward, analogous to eample 1. Often, though, it is not eas (or even impossible to write u f(,. It ma be difficult to isolate u on the left-h side, or it ma be difficult to eliminate v. In this case, ou will have to take a derivative analogous to eample 2. However, ou can still find epressions for u. In order to relate (, to (u, v, ou will need two equations (two unknowns, hence two equations. B differentiating both equations with respect to, ou will create two equations in u. You now have simultaneous equation in partial derivatives. As ou are just rearranging equations, the partial derivatives behave just like an other algebraic quantit, the equations can be solved with an of the usual techniques for simultaneous equations. Eample 3: Find u for (, (u, v related b u v v 2 u 2 3 2
3 Answer: u u 2v 2 u 2u 0 u u ( u 2v 2 u 2u u 0 ( 2 v u 3 u 2u 0 ( 3 2u 2 u 2 (v u u 2 (v u 3 2u 2 u 2(v u 3 2u 2 The Total Derivative When ou have (or need an equation involving multiple d s, ds, dt, dp, etc., it s important to remember what those quantities mean. You re looking for an equation that describes, for eample, how the change in f depends on the change the change in. If f, then we might notice that if ou change b a little bit, the effect of that change on f will be magnified b whatever the current value of is, because f, likewise for a small change in, giving us df d d. More formall, what ou are looking for in this case is the total derivative of f, the epression that describes how f changes as a function of the change in each of the variables on which f depends. For each variable, this will be equal to the change in that variable multiplied b the rate of change of f with respect to that variable. Thus, df d d for an f(, Note that this gives the same answer as we got above if f. The total derivative epression can be eped for as man variables as necessar, for eample, df d,z d,z dz for an f(,, z z, Combining Differentials using Geometr In cases where the variables are spatial coordinates, ou can sometimes deduce similar relationships using geometr, for eample when determining arc length. The differential arc length, dl, can be treated as a straight line. We thus have (dl 2 (d 2 (d 2, where d d are the differential 3
4 lengths in associated with a segment of arc. This can then be rearranged to give dl ( 1 d 2d. d Make sure ou underst this rearrangement. What is the equivalent epression for an integral in terms of rather than? Eact Ineact Differentials An epression M(, d N(, d is eact if there eists an f such that M f N f, e.g. if there eists a function such that this epression is its total derivative. As we will see at the end of Hilar Term, this becomes important when working with differential equations, when an eact differential can be solved directl whilst an ineact one requires an integrating factor. In order to test for eactness, we use the fact that 2 f 2 f for an analtic function f.1 Therefore, an epression M(, d N(, d is eact iff 2 f 2 f. (N.B. Derivatives are read right to left, e.g. the left-h side describes differentiating first with respect to second with respect to. The Chain Rule in Multiple Variables Just as with ordinar derivatives, the chain rule can be used to epress a derivative in terms of a different set of variables. If F can be epressed in terms of either (, or (u, v, then we can relate the partial derivatives of F in one set of variables to partial derivatives in the other set of variables as follows (Variables held constant shown onl for the first equation. For conciseness, these are usuall omitted if doing so doesn t introduce ambiguit.: ( F F ( F u u F v u F u u F F u F u F u F F F Essentiall, the first equation sas, if I change a little bit, that will change both u v. I can find how fast F changes when I change b multipling how fast F changes when u changes b how fast u changes when I change plus the same for v. 1 All the nice, normal functions ou are likel to think of are analtic. Eamples of functions whose mied derivatives ma not be equal include 1 functions whose derivatives are discontinous ( has a discontinuous derivative, for eample, though since it has onl one variable, mied partial derivatives aren t relevant 2 functions which are path-dependent, i.e. the value of the function depends not onl where ou are, but on how ou got there. A somewhat trivial eample of a path-dependent function would be the orientation of a ball rolling on a tabletop. If ou put a dot on the ball started at the origin of the table with the dot directl on the table, the position of the dot b the time ou got to some second point on the table would depend on the path ou took as ou rolled the ball, i.e. it s a path dependent function. 4
5 Eample 4: The position function P has been epressed in terms of Cartesian coordinates, but ou need the partial derivatives with respect to polar coordinates r θ. r θ r r θ θ r cos θ, r sin θ so r θ (cos θ ( r sin θ b the chain rule (sin θ (r cos θ Higher Order Derivatives of Arbitrar Functions Frequentl ou will be asked to find not just first order partial derivatives such as f but also higher order derivatives such as 2 f or 2 f 2. If ou are given a specific function, e.g. f 2 2, this is straightforward: to find 2 f 2 to get 2 f take f 2 differentiate again b to get 2 f 2. Similarl, 2, differentiate f first b then b. If ou told to find a partial derivative of f, where f can be an arbitrar function, the process is still the same, but it is important to remember that differentiation is an operator it performs some action on the epression in its brackets. This means ou must be careful to differentiate the quantit in the brackets, not tr to multipl out using as one of the terms. Eample 5: Consider a function f such that f f 2 2 sin(. Find an epression for 2 f. 2 Answer: Differentiate the given epression for f, using the product rule chain rule as necessar. 2 f 2 ( 2 f 2 sin( 2 f 2 2 f 2 sin( 2 cos( Choosing a Technique for a Problem There are several cues in the problem that can help ou determine which technique to use. Does it tell ou? Please don t ignore instructions telling ou to use, sa, a total derivative. 5
6 Do ou need to find partial derivatives? Your choices here are essentiall either to rearrange equations then differentiate or differentiate then rearrange equations. If ou can easil isolate the variable ou want to differentiate, the first option is usuall easier. Otherwise, differentiate the equations as the are, use simultaneous equations in partial derivatives. Consider our variables. Do ou have two completel independent sets of variables? If so, the chain rule is probabl going to be involved in relating derivatives in one set to derivatives in the other set. Do ou have a set of variables that are all connected, e.g. ou can choose an two or three to be independent, but then all the others are dependent? Total derivatives are often useful here, although the chain rule can be used so long as ou are ver careful to write down which variables are being held constant for each ever partial derivative. Consider the form of what s being asked. Especiall useful in questions that ask ou to show something. Can ou recognise the outline of the chain rule or a total derivative in the epression ou are tring to derive? Tr differentiating see what happens. If ou cannot see what technique to use, tr differentiating b the required variable, then see what problems ou encounter. If ou end up with u in our epression when ou onl wanted, then ou now have a hint that simultaneous equations in partial derivatives are required. If ou encounter problems with our function being in terms of variables that depend on the variable b which ou re differentiating, then ou probabl need the chain rule (could potentiall also be done with total derivatives. Eample 6: Consider ( a function F (, ( where can ( be epressed as a function of z. Find epressions relating F to F z F z to F. Answer: This can be solved either with total derivatives, or more concisel but with more care required, with the chain rule. Total derivatives: F F df d d F F df d dz z z d d dz z z [ ( F ] [ ] F F df d dz z z F F F F F z z z z Chain rule: Remind ourself of the sets of variables ou are using, (, (, z. Note down which variables are being held constant on each partial derivative. F F F z z z F F F z z z 6
7 However 1 0 z z F F F z z F F z z 7
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