Functions of more than one variable
14. Finding a function from its derivatives
13.4. Proof of Clairaut’s theorem. With some algebra we can show that the defini-tion of partial derivatives implies
(95) ∂2f
∂x∂y =
∆xlim→0 lim
∆y→0
f (x + ∆x, y + ∆y)− f(x, y + ∆y) − f(x + ∆x, y) + f(x, y)
∆x∆y while
(96) ∂2f
∂y∂x=
∆ylim→0 lim
∆x→0
f (x + ∆x, y + ∆y)− f(x, y + ∆y) − f(x + ∆x, y) + f(x, y)
∆x∆y
So it’s a maer of showing that one can switch the two limits. We won’t go into the details here, but the hypothesis that fxyis continuous implies that we are indeed allowed to switch the limits.
14. Finding a function from its derivatives
We now look at integrating the partial derivatives of a function, which looks out of place here (this being a chapter on derivatives and not on integrals), but Clairaut’s eo-rem actually turns out to play a role.
If we have the derivative f′(x)of some function of one variable then we know how to recover the function f(x): we integrate, i.e.
f (x) =
∫
f′(x)dx + C.
Furthermore, any (continuous) function can be the derivative of a function, because, if someone gives us a continuous function f(x), then
F (x)def=
∫ x a
f (t)dt is a differentiable function whose derivative is F′(x) = f (x).
What about functions of more than one variable? Suppose we know the partial deriva-tives
(97) ∂f
∂x = P (x, y)and ∂f
∂y = Q(x, y)
of a function of two variables, can you then find the function f(x, y)?
e answer is “yes, you can find f by integrating, if it exists, but not every pair of functions P and Q are the partial derivatives of some function.”
e following two examples are typical of what can happen.
14.1. Example. Does there exist a function f(x, y) of two variables such that
∂f
∂x = x3− 2xy, and∂f
∂y = 3y2
both hold? e answer is no, such a function cannot exist, and here is the reason: if there were such a function, then we could compute
∂2f
∂y∂x =∂(x3− 2xy)
∂y =−2x, and ∂2f
∂x∂y =∂(3y2)
∂x = 0.
By Clairaut’s eorem both computations should give us the same answer, but they don’t.
erefore the function f whose partials are as above cannot exist.
14.2. Example. Does there exist a function f(x, y) of two variables whose deriva-tives are
∂f
∂x = x3− 2xy, and∂f
∂y =sin πy− x2? Let’s check Clairaut’s condition:
∂2f
∂y∂x = ∂(x3− 2xy)
∂y =−2x, and ∂2f
∂x∂y = ∂(sin πy− x2)
∂x =−2x.
is time both computations gave us the same answer, so Clairaut’s theorem does not rule out the existence of the function f that we are looking for. We can try to compute it by integrating both partial derivatives. ere is a systematic way of doing this that usually leads to the answer.
We first integrate fxwhile treating y as a constant:
f (x, y) =
∫
{x3− 2xy} dx =14x4− x2y + C(y).
e “constant” is only a constant in the sense that it does not depend on x. It may depend on y, and that is why we wrote it as C(y). To find C(y) we differentiate this result with respect to y:
sin πy− x2= fy =∂{1
4x4− x2y + C(y)}
∂y =−x2+ C′(y).
So we see that C′(y) = sin πy, and hence C(y) = −1πcos πy + K, where K is a real constant (K depends neither on x nor on y).
We find that the following function has the prescribed partial derivatives f (x, y) = 14x4− x2y−π1cos πy + K
where K is constant, i.e. where K depends on neither x nor y.
e method used in this example always works, and we summarize this fact in the following theorem.
14.3. eorem. Suppose P (x, y) and Q(x, y) are two functions that are defined on a rectangular domainR = {(x, y) : a < x < b, c < y < d}, and suppose that they have continuous partial derivatives on this domain.
If a function f(x, y) exists such that (97) holds onR, then
(98) ∂P
∂y = ∂Q
∂x must hold onR.
Conversely, if P and Q satisfy (98) then there is a function f defined onR that satisfies (97).
To prove this theorem we need to understand integrals of functions of several variables, and Green’s theorem in particular, so this will have to wait until the end of the semester.
See § VII.11.
It should be noted that the assumption above that the functions P and Q be defined on a rectangle is important: the theorem is no longer true if the domain of P and Q “has holes.” See problem15.16.
15. PROBLEMS 81
15. Problems
1. Find all first and second partial
deriva-tives of x3y2+ y5. •
2. Find all first and second partial deriva-tives of 4x3+ xy2+ 10. • 3. Find all first and second partial
deriva-tives of xsin y. •
4. Find all first and second partial deriva-tives ofsin(3x) cos(2y).
5. Find all first and second partial deriva-tives of ex+y2.
6. Find all first and second partial deriva-tives ofln√
x3+ y4.
7. Find all first and second partial deriva-tives of z with respect to x and y if x2+ 4y2+ 16z2− 64 = 0. (Hint: solve for z or use implicit differentiation…)
8. Find all first and second partial deriva-tives of z with respect to x and y if xy + yz + xz = 1. (Hint: solve for z or use im-plicit differentiation…)
9. How many different second partial derivatives does a function of two variables have? What about a function of three vari-ables? How many derivatives of third degree does a function of two variables have? • 10. Derive the formulas(95) and (96) from the definition of partial derivatives(51) and (52).
11. The equation which describes the vibrat-ing strvibrat-ing (as in a guitar, piano, or violin string) is
(99) ∂2f
∂t2 = c2∂2f
∂x2
where c > 0 is some constant. The equation is called the wave equation. It is an example of a partial differential equation.
Note : this problem looks like a prob-lem about differential equations, but to an-swer the following questions you really only have to compute partial derivatives of cer-tain functions, and solve some (easy) alge-braic equations.
(a)For which values of the constant v is a
“traveling wave with velocity v and profile
F (x)” a solution of the wave equation (99)?
Does it maer which profile F is used here?
(For the terminology used here, revisit problem5.16in Chapter III, §5.2.)
(b)Suppose the string is clamped down at its ends, and that its length is L. For which values of the constants A and α is
f (x, t) = Asin(αt) sinπx L
a solution of the wave equation? (Assume A̸= 0).
(c)Same question for
g(x, t) = Bsin(βt) sin2πx L . (d)Describe the movies that go with the so-lutions you found in (b) and (c). Which of the two graphs moves faster?
(e)Show that h(x, t) = f(x, t) + g(x, t) is again a solution of the wave equation, where fand g are as above. (Don’t use the formu-las for f and g: it is easier to prove a more general fact, namely, if two functions f and gsatisfy (99), then so does their sum f + g.) (f)Describe the movie that goes with the function h(x, t) (it is probably beer to use a graphing application likegrapher.appon Mac OS X,graphcalc.exeon Windows or Linux). which a does there exist a function f(x, y) such that P = fxand Q = fy?
14. Suppose P (x, y) = x2 − 2xy3 and Q(x, y) = (xy)2. Does there exist a func-tion f(x, y) such that P = fxand Q = fy?
15. Suppose x = u + v, y = u− v, and suppose f(x, y) = g(u, v). Then compute (a)∂2g
∂u2 •
(b)∂2g
∂v2 •
(c) ∂2g
∂u∂v •
(d)∂2g
∂u2 −∂2g
∂v2 •
(e)∂2g
∂u2 +∂2g
∂v2 •
16. [For discussion] Let P (x, y) = −y
x2+ y2, Q(x, y) = x x2+ y2.
(a)What is the domain of P and Q?
(b)Show that
P = ∂θ
∂x, Q = ∂θ
∂y
where θ is the angle variable from polar co-ordinates.
(c)Show that P and Q satisfy the condition (98). (You don’t have to compute the deriva-tives to check this, although you could.) (d)Is there a function f such that (97) holds?
CHAPTER 5