ADDITIVELY SEPARABLE FUNCTIONS

Transcription

1 ADDITIVELY SEPARABLE FUNCTIONS FUNCTIONS OF THE FORM F (, ) =f()+g() STEVEN F. BELLENOT Abstract. A sample project similar to the project for Spring 25. The picture below has a hortizontal space coordinate s and vertical time coordinate t. We see two waves, a raised hump moving to the left and a depression moving to the right. At t =,the middle hortizontal bold black line, the two waves cancel each other briefl. From another viewpoint, the red line is the -aes and the red curve above it is f() =ep( 2 ), the blue line is the -aes and the blue curve below it is g() = ep( 2 ). The surface is the graph of F (, ) =f()+g(), a additivel separable function; but also a solution to awaveequation.

2 2 Additivel Separable Functions Introduction We consider a simple special class of functions of two variables, those that can be written in the form F (, ) =f()+g(). (Your project will be about functions that can be written in the form F (, ) =f()g().) This is perhaps the simplest collection of functions of two variables that depends on both variables. Our first step is to give a name to the collection, since this collection is not usuall studied, we get to make up a name. Definition. A function of two variables F (, ) will be called additivel separable if it can written as f()+g() for some single-variable functions f() and g(). After a definition one usuall adds some simple observations to aid the reader to get a feel for the topic. Note that constant functions like F (, ) =5orfunctionsofonevariable F(, ) =h() are additivel separable. Indeed, let f() =andg()=5org()=h()in the definition. But not all functions are additivel separable, later we will see F (, ) = is not additivel separable. Finall, the functions f() andg() in the definition are never unique, since if F (, ) =f()+g(), then also F (, ) =(f()+c)+(g() C) for an constant C. This collection is surprisingl rich. We start with a few eamples, Figure shows that such functions can look full three dimensional. This is surprising, since for fied = a, each cross section F (a, ) isg()+c a translate of the function g() withc=f(a). That is the sections = a for different parameters a all wave to the same time steps. The tangent plane to z = h(, ) atanpoint(a, b, h(a, b)) is additivel separable since is alwas of the form z = h (a, b)( a) +h (a, b)( b) +h(a, b). (So we could take f() =h (a, b)( a) andg()=h (a, b)( b)+h(a, b). Remember a, b, h(a, b), h (a, b) and h (a, b) are all constants and not variables.) The three basic etrema, local minimums, local maimums and saddle points, are often drawn like in Figure 2. These are graphs of the functions F (, ) = 2 2, showing a local maimum at (, ), F (, ) = 2 2, showing a saddle point at (, ), and F (, ) = 2 + 2, showing a local minimum (, ). All three functions are additivel separable. The last function in polar coordinates is just r 2 which is curious. A function of one variable in one coordinate sstem which is a nice additivel separable in a radicall different coordinate sstem. Neither r = 2 + 2, r 3, nor even r = is additivel separable. Rectangle Conditions When is a function F (, ) additivel separable? A simple necessar condition is obtained b assuming F (, ) =f()+g() and then plaing with its values at the four corners of Figure. The graph of sin + sin (left) and selected cross sections (right)

3 Additivel Separable Functions Figure 2. Local Ma, Saddle and Local Min a rectangle {(a, b), (c, b), (c, d), (a, d)} (see Figure 3). We have F (c, b)+f(a, d) F (a, b) =(f(c)+g(b))+(f(a)+g(d)) (f(a)+g(b)) = f(c)+g(d) =F(c, d) which we can write as a theorem. Theorem 2. If F (, ) is additivel separable, then for all a, b, c, d, F (c, b) +F(a, d) F (a, b) =F(c, d). Using a = b =andc=d=weseef(, ) = is not additivel separable. as the left hand side is zero, while the right hand side is one. Theorem 2 has a converse which is also true. Let f() =F(, ) and g() =F(,) F(, ) and use the theorem with = c, = d and a = b =. Wehave F(, ) =F(, ) + F (,) F(, ) = f()+g() which we can write as another theorem. Theorem 3. If for all a, b, c, d, F (c, b)+f (a, d) F (a, b) =F(c, d), thenf(, ) is additivel separable. Derivative conditions Obviousl if F (, ) =f()+g() then the partial derivatives of F are eas to compute. F = f () sinceg() is constant as a function of. Similarl F = g (), F = f (), F = g () and more importantl F = F =. Again lets make this a theorem. Theorem. If F (, ) is additivel separable, then = Again we see F (, ) = is not additivel separable since F =. Theorem sas all additivel separable functions satisf a simple PDE (Partial Differential Equation). The converse is also true. First remember from one variable that f () = (a, d) (c, d) (a, b) (c, b) Figure 3. The sum of the values at the diagonall opposite corners are equal

4 Additivel Separable Functions means that f() is a constant. For a function of two variables G (, ) = means that G(, ) is a constant function of but can be an arbitrar function of. SoF =(F ) = means F = h() some arbitrar function of and hence F (, ) =f()+g()whereg() is some arbitrar function of and f () =h()(sof()= a h(t)dt). Lets state this as another theorem Theorem 5. If F (, ) satisfs the PDE then F (, ) is additivel separable, = The Wave Equation It turns out this PDE is reall a famous PDE in disguise. With a a change of coordinates, this PDE becomes the one dimensional wave equation. The solutions to the one dimension wave equation can have two components, a wave function moving to the right (the g()) and another wave function moving to the left (the f()). Figure shows the function f() =ep( 2 ),g()= ep( 2 ),F(, ) =f()+g() along with the curve at several times. The t = graph is the bold straight line in the middle. Times before zero are below andinfrontoft= and times after zero are above and behind. So time increase as we go up Figure The space dimension shows the leftward motion of the wave. This is too important a fact to not show the connection, and it is a good application of the chain rule for several variables, but this section is not used in the sequel. The graph in Figure (also on the cover) page shows raised wave moving to the left and depression moving to the right. The -aisisred,the-ais is blue and the function is F (, ) =ep( 2 ) ep( 2 ). Time increases as one goes up the page and the one space dimension is hortizontal. The change of coordinates is given b t = + (time) and s = is the one space dimension. Inverting this transformation gives =(s+ t)/2and=(t s)/2. Using the chain rule F s = F s + F ( F s = F t = F t + F t = F ) 2 ( F + F ) 2 To take second partials F ss, the equation becomes mess since we have to appl the chain rule to F and F since the are functions of both and as well as appling the product rule. s 2 = ( 2 s + 2 F Since F = ss = ss = This simplifs to ) s s + F 2 ( 2 ) s 2 + F s + 2 F 2 s s + F 2 s 2 Similarl s 2 = 2 F 2 ( ) F s 2 t 2 == ( F 2 ) ( ) 2 =( 2 F s F 2 )

5 Additivel Separable Functions 5 Figure. The space dimension is hortizontal, the time vertical, the red line is the -aes and the blue line is the -aes and we have the one dimensional wave equation s 2 = 2 F t 2 The two forms of the wave equation are equivalent, although we have shown onl one direction. Usuall the wave equation has a constant c which represents the speed of the waves, in our case this c =. It is more common to see the wave equation written as s 2 = c 2 t 2 Critical Points and local etrema Here is one of sections that ou will need to do for our project on functions of the form f()g(). We want to characterize the local etrema of additivel separable functions F (, ) =f()+g() in terms of the critical points of f() andg(). From the derivatives section we have Theorem 6. The point (a, b) is a critical point of F (, ), ifandonlifboth=ais a critical point of f() and = b is a critical point of g(). To use the several variables classification function, big D = F F F 2, weusef = and the other second partials from the derivatives section to compute big D = f ()g (). As long as big D is not zero, namel when both f (a) g (b). We cover all cases in the table below which is onl for critical points (a, b).

6 6 Additivel Separable Functions Figure 5. The four basic cases: when big D f (a) g (b) big D Classification Local Min + Saddle + Saddle + Local Ma Clearl these crition are nicel illustrated b sin + sin in Figure at the points (π/2,π/2), (π/2, 3π/2), (3π/2,π/2) and (3π/2, 3π/2). The additivel separable functions are simple enough we can handle the general case with Talor series. If a is a critical point for f() thenf (a)=,letf (n) (a)=a n bethefirst non-zero higher derivative of f at a. (Similarl let m be the first non-zero higher derivative of g at b and let g (m) (b) =b m.) Talor sas f() f(a)+a n ( a) n. So if n is odd this is like 3 at, the critical point = a is neither a local min nor a a local ma. This is because a n ( a) n will be positive on one side of a and negative on the other side of a. If nis even, the point is a local min if a n > andalocalmaifa n <. Our more complete collection is summarized in Table below. n a n m b m Classification even + even + Local Min even + even Saddle even + odd ± Saddle even even + Saddle even even Local Ma even odd ± Saddle odd ± even + Saddle odd ± even Saddle odd ± odd ± Saddle But a better statement is the following theorem Theorem 7. The critical point (a, b) of the additive separable function F (, ) =f()+g() is a local ma [respectivel local min], if and onl if, both a is a local ma [respectivel local min] for f() and b is a local ma[respectivel local min] for g(). Otherwise the point (a, b) is a saddle point for F (, ). Basicall this sas if F (, ) F (a, b) for(, ) nearb(a, b) thenf()+g(b)=f(, b) F (a, b) =f(a)+g(b)sof() f(a) and similarl g() g(b). Conversel if f() f(a) and g() g(b) thenf()+g() f(a)+g(b)orf(, ) F (a, b). Theorem 7 is false for functions in general. The function F (, ) = has both F (, ) = andf(,)=sothat= is both a local min and local ma for F (, ) = and =

7 Additivel Separable Functions 7 Figure 6. Some of the odd power cases: note big D = is both a local min and local ma for F (,)=andet(,) is clearl a saddle point for F. Closure and Uniqueness Properties This is a bonus section and is off the topic. Closure properties sa what can ou do a class or collection and still remain in the collection. For eample, if F (, ) andg(, ) are both additivel separable then so is F (, )+G(, ), but not ever product F (, )G(, ) is additivel separate. Multiplication b a constant C is ok, since CF(, ) is additivel separable whenever F is additivel separable. Also partial derivatives and integrals of additivel separable functions are additivel separable. We summarize in the net theorem. Theorem 8. The collection of additivel separable functions is closed under the following operations. (The first two make the collection a Linear space.) () addition (2) multiplication b constants (3) partial derivatives with respect to or () integrals with respect to d or d Here is a uniqueness question, let L be an line in the -plane which is not parallel to the or ais. If the additivel separable function F (, ) is zero on all points on L and the directional derivative of F in the direction normal to L is also zero on L, then must F (, ) be identicall zero everwhere? The picture on the cover page, shows it is not enough for F to be zero on one line.

8 8 Additivel Separable Functions Your optional group assignment This is a group project. Groups are 3- people ecept in unusual cases. Individual projects are not accepted. The work on the project must be a true collaboration in which each member of the team will carr their own weight. It is NOT acceptable for the team members to split the project between them and work on them independentl. Instead all members must activel work together on all parts of the assignment. The fact that group members have to work together means that ou need to carefull consider a potential group member s schedule before forming the group. You cannot be a group if ou cannot find large chunks of time to spend together. Your assignment is to write in prose, in understandable English, a tped report about separation of variables, that is functions of the form F (, ) = f()g(). There are three parts to the assignment. First ou must do the local etrema for functions of the form f()g(), note that this is harder than the case for functions of the form f() +g() done above. Second, ou must find something interesting about about such functions, unfortunatel one cannot just take logs and translate a additivel separable result via ln(f()g()) = ln f() +lng() since logs are onl defined for positive values. This fact must be strong enough to show that + cannot be written in the form f()g(). Third, ou must research and find a place where separation of variables is used outside the ordinar differential equation case ou might have learned in Calculus 2. The project has two due dates: A hardcop outline which lists who is in our group and what our project is going to contain is due on Frida 25 March at 3pm and the completed hardcop project itself is due on Frida 8 April also at 3pm. (The project must be bound or stapled and not loose, nor dog-eared. Part of the grade will be based on presentation and clarit. There are a few bonus points available for wow value.) In addition to the project, EACH team member will write a separate evaluation of the relative contributions of all the team members.