partialderivatives topology

7/29/2019 Partialderivatives Topology

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Partial Derivatives

1. Open Sets in Rn.When we consider a derivative of a function f of one real variable at a given point x0,

we provide all our considerations in a small interval (x0 , x0 + ) containing x0. Now,we need an analogue of this open interval. We start with the definition of an open ball

in Rn

that is also called an open disc in case of n = 2.We also identify points (x1, . . . , xn) in Rn with vectorsx = x1, . . . , xn, as usual.

Definition 1 The open ball of radius r > 0 and centerx0 is the set of all pointsxinRn such that the distance betweenx and x0 is strictly less than r, i.e.,

Br

x0

:=x Rn :

x x0 < r .Definition 2 Let U Rn (means that U is a subset inRn). We call U an open setwhen for any pointx 0 U there exists some r > 0 such that Br

x0

U, i.e., U

contains some open ball with the center atx0

.

This definition is crucial for differentiation of functions of several variables.

Example 1 Any open ball is an open set.

Proof. Let us consider an arbitrary open ball Br (a ), r > 0. Having taken an arbitraryx0 Br (a ), we have

x0 a < r. Let us take r0 := r x0 a . Since x0 a < r,

r0 > 0. Now, consider Br0

x0

.

Ifx Br0

x0

, then using the triangle inequality, we get

|x a | x x0 + x0 a < r0 +

x0 a = r x0 a + x0 a = r.Thus,x Br (a ). Sincex was chosen arbitrarily, it means that

Br0

x0

Br (a ) .

Now, sincex0 Br (a ) was chosen arbitrarily, we get that Br (a ) is an open set. Intuitively, open sets are the sets that do not contain their boundary, i.e., they do notcontain any boundary point. So, the following definition may also be useful in the future.

Definition 3 Let us consider a set D Rn. A pointx 0 is called a boundary point ofD if for any r > 0, the open ball Br

x0

contains at least one point belonging to D and

one point that does not belong to D.The set of all boundary points of D is called the boundary of D and is denoted by

D.

Note thatx0 may belong or not belong to D.

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Definition 4 The set E is called a closed set inRn if its complement

Ec = {x Rn :x E}

is open.

Thus, it is easy to see that any closed set contains its boundary unlike an open set that

does not contain any boundary point.

Example 2 Find the boundary of the following rectangle

A ={

(x, y) R2 : 0 < x 1, 1 y 1} .Definition 5 The closure of a set D Rn is the union of D and its boundary. It isdenoted by D. Thus,

D = D D.It is clear that D is a closed set. Moreover, it is the smallest (according to inclusions)closed set containing D.1

Definition 6 For a set E Rn, its interior is defined as a set of all pointsx E suchthat there exists some r > 0 with Br (x ) E. It is denoted by Eo.

It is easy to see that Eo = E\ E. Furthermore, E is open if and only if E = Eo.Finally, any open set containingx0 is called a neighborhood of x0.Now, we are ready to introduce differential operations in Rn.

2. Partial Derivatives.If we remember the univariate case, then the notion of derivative reflects nothing else

than the velocity of changing the function in a small neighborhood of a given point. Thefaster growth of a function, the larger value of its derivative and vice versa. Of course,negative values of the derivative show how fast the function decreases. The same ideamay be considered in the multivariate case. However, we cannot apply it directly. Tounleash the possible problems, let us consider the following example in R2.

Example 3 Let us consider the function

f(x, y) = x2 y2.

First of all, the domain of f is the whole plane R2. Now, let us consider the behaviorof this function at the origin (x0, y0) = (0, 0). If we consider changing only y from 0, thenthe values of f(0, y) = y2 will be negative, and the function g(y) := f(0, y) will have alocal maximum (even global) at 0. Thus, if we take any y = 0, the value off(0, y) will bestrictly less than f(0, 0). The same for h (x) := f(x, 0) but this function has a minimumat 0.

1The last sentence is the most general definition of a closure that usually considered in general topology.

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So, how to characterize the growth of f in a small neighborhood of the origin?What actually do we have? This function has a local maximum in the direction of the xcoordinate axis, and a local minimum at the direction of y coordinate axis.

So, we can only characterize the growth in a given direction.Q: What object actually has defined direction?A: A non-zero vector!!!

Thus, we can consider the question of characterization of a functions growth in thedirection determined by some vector. To do that, we need some proper tools. So, firstof all, we need to find the proper characteristics in the directions of the basis coordinateaxis. It is being done by fixing all the variables except one, i.e., considering a functionF (xj) = f

(x01, . . . , x

0j1, xj, x

0j+1, . . . , x

0n

). In this way, we naturally come to the definition

of a partial derivative.

Definition 7 LetU Rn be an open set and f : U R be a real-valued function. Then,f/x1, ...,f/xn, thepartial derivatives of f with respect to the first, second,...,n-th

derivative, are the real functions of n variables, which, at the point x0 = x01, . . . , x0n U,are defined by

f

xj

(x01, . . . , x

0

n

)= lim

h0

f(

x01, . . . , x0j1, x

0j + h, x

0j+1, . . . , x

0n

) f(x01, . . . , x0n)h

=

limh0

f

x0 + h ej

f

x0

h

if the limit exists. Here, j = 1, 2, . . . , n, and ej is the unit vector in the direction of thej-th coordinate axis, i.e., the unit basis vectorej = 0, . . . , 0, 1, 0, . . . , 0, with 1 in thej-th slot.2

The domain of the derivative fxj is the set of pointsx Rn for which the limitexists.Note. In the case when n = 2 or n = 3, we will consider functions f(x, y) and f(x,y,z),respectively. And, we also may use special notations for the partial derivatives on x, y,and z, namely

fx =f

x, fy =

f

y, fz =

f

z

instead of fx1

, fx2

, and fx3

, respectively.We will also use the following notation

f

xjx1=x01,...,xn=x0n

=f

xj(x

0

1, . . . , x0

n) .

Finally, we can easily write fxj (x1, . . . , xn) that means partial derivative in the direction

of j-th coordinate axis at the point (x1, . . . , xn), or just omit the point at all:fxj

if it is

essentially variable or understood.

2In the space, i.e., in R3, we already considered such basis vectors: i, j and k. In our general notations,

e1 = i, e2 = j , e3 = k.

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Geometrically, a partial derivative defines the direction of the tangent line to itsgraph (surface or hyper-surface) built in the direction of j-th coordinate axis. You couldread more detailed explanation and some illustration in the textbook. The reason forsuch geometrical means is the fact that partial derivative reflects changing of a functionin a given direction. Thus, slicing you graph at a given point by the plane parallel to the

j-th coordinate axis yields a graph (curve) in this plane. The usual tangent line to this

graph is that we are talking about.

Example 4 Find f/x if f(x, y) = xy/

x2 + y2.

Solution. First of all, let us define the functions domain. Obviously, it is R2 withoutthe origin. This set is open (you could prove it directly)m and we can consider partialderivatives there.

To find f/x, we need to consider y a constant and differentiate our function accord-ing to x. So, applying the quotient rule for the derivatives, we have

f

x(x, y) =

y

x2 + y2 xy 2x2

x2+y2

x2 + y22 = y (x

2 + y2) yx2

(x2 + y2)3/2=

y3

(x2 + y2)3/2.

As we can see, the domain of f/x in this example is again R2 \ {(0, 0)}. However,in general situation, the domain for the partial derivative may be less than the functionsdomain, because the partial derivative may not exist at some points.

Example 5 Find f/x if f(x, y) =

x2 + y2.

Solution. The functions domain is obviously R2. Thus, we may try to find a derivativeat any point. If we do formal differentiation on x, we get

f

x(x, y) =

xx2 + y2

.

So, all is clear when the denominator is non-zero. What about the origin? Is the partialderivative defined there. To check this, we need to remember what the partial derivativeis, and consider a function

h(x) = f(x, 0) =

x2 = |x| .

Now,f

x (0, 0) = h(0)

that does not exist. Thus, f/x(0, 0) does not exist. Therefore,

f

x(x, y) =

xx2 + y2

, (x, y) R2 \ {(0, 0)} .

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3. Higher Order Derivatives.In fact, the partial derivatives f/xj are also functions of n variables. Thus, we

can also differentiate them in the interiors of their domains. Actually, this is a generalapproach: if a functions domain D is not open but has nonempty interior Do, we canconsider partial derivatives in Do.

Definition 8 LetUR

n be an open set and f : UR be a real-valued function. If the

partial derivative f/xj is defined on some open set D U, then the second partialderivatives are defined by

2f

xixj(x ) =

xi

f

xj

(x ) , i = 1, 2, . . . , n

if the last partial derivative exists.If i = j, we write

2f

x2j=

2f

xjxj.

In case of a function of two variables we will also use notations

fxx =2f

x2, fxy =

2f

xy, fyx =

2f

yx, fyy =

2f

y2.

The case n = 3 is treated in the same way.If we have a partial derivative 2f/xixj with i = j, it is also called a mixed deriva-

tive. So, fxy and fyx are mixed derivatives.Note. The same approach may be used for defining partial derivatives of higher order.

Definition 9 Let U Rn be an open set and f : U R be a real-valued function.If the partial derivatives of (m 1)-th order exist, we can define the m-th order partialderivatives as

mf

xkmxkm1 . . . xk1(x ) =

xkm

m1f

xkm1 . . . xk1

(x ) , all kj = 1, 2, . . . , n .

Example 6 Find all the second order partial derivatives of the function

f(x, y) = exey

.

Solution. The functions domain is R2 since the exponential function is defined every-where. Let us find the first partial derivatives. All that we need is just the chain rule.

fx =

x(

exey)

= exey

(xey

)x = exey

ey

= ey+xey

,

fy =

y

(exe

y)= exe

y

(xey)y = exeyxey = xey+xe

y

.

Now, differentiating these guys on x, we have:

fxx =

x

(ey+xe

y)= ey+xe

y

(y + xey)x = ey+xeyey = e2y+xe

y

,

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fxy =

x

(xey+xe

y)= ey+xe

y

+ xey+xey

(y + xey)x = ey+xey + xey+xe

y

ey = ey+xey

(1 + xey) ,

fyx =

y

(ey+xe

y)= ey+xe

y

(y + xey)y = ey+xey (1 + xey) ,

fyy =

y (xey+xe

y

)= xey+xe

y

(y + xey)y = xey+xey (1 + xey) = ey+xe

y

(x + x2ey

).

As you could note, fxy = fyx in this example, and it is not accidental. The follow-ing theorem could make life easier, because you may choose which mixed derivative tocalculate.

Theorem 1 (Clairauts Theorem) If a function f is defined in some neighborhood ofa point (a, b), and the second derivatives fxy and fyx are both continuous at this point,then fxy(a, b) = fyx(a, b).

Of course, this is also true for higher order derivatives. Namely, the following results holdstrue.

Theorem 2 Suppose that a function f is defined in some neighborhood of a pointa , andtwo mixed partial derivatives of order m exist in some neighborhood ofa and are contin-uous ata . If the only difference between the derivatives is the order of the derivatives,then their values ata coincide.

4. Partial Differential Equations.Partial differential equations are equations involving partial derivatives of a func-

tion. The order of equation is the higher order of the partial derivatives involved. Forexample, the following Laplaces Equation

2

ux2

+ 2

uy2

= 0

is a second order partial differential equation. It plays an exceptional role in problems ofheat conduction, fluid flow, and electrical potential. The solutions of this equation areharmonic functions that are also very important object of modern complex analysis.

Another second order partial differential equation describes the motion of a waveform

2u

t2= a2

2u

x2.

It is called the wave equation.Let us note that the majority of physical processes are described in terms of some

partial differential equations. It is not a surprise, because in fact the functions involvedare dependent on coordinates, time, temperatures, etc., so, they are in fact functions ofseveral variables. Hence, partial derivatives play the same role as a derivative of a functionof one variable in classical problems of physics and mechanics.

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Supplement: Derivatives of Elementary FunctionsThe following derivatives are derivatives of functions of one variable. Nevertheless,

they may be helpful when calculating partial derivatives of functions of several variables.If f(x) c = const, then f(x) = 0.

(xn) = nxn1, x R, n N.(x) = x1, x > 0, R.

(ax) = ax ln a, x R, a > 0, a = 1.(ex) = ex, x R.

If a > 0, a = 1, then

(loga x) =

1

x ln a, x > 0; (loga |x|) =

1

x ln a, x = 0.

(ln x) =1

x, x > 0; (ln |x|) = 1

x, x = 0.

(cos x) = sin x, x R.(sin x) = cos x, x R.

(tan x) =1

cos2 x, x =

2+ k, k Z.

(cot x) =1

sin2 x, x = k, k Z.

(arcsin x) =1

1 x2 , |x| < 1.

(arccos x)

= 1

1 x2 , |x| < 1.

(arctan x) =1

1 + x2, x R.

(arccot x) =1

1 + x2, x R.

(sinh x) = cosh x, x R.(cosh x) = sinh x, x R.

(tanh x) =1

cosh2

x

, x

R.

(coth x) =1

sinh2 x, x = 0.

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partialderivatives topology

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