Lecture 1

Lecture 1

Differentiations

Given a function f

( )

x , applying differentiation to it would yield a product called derivative.

Depending on what the function represents, its derivative can be interpreted as slope of the function, rate of change of the function with respect to certain quantity, or magnitude of effect that a certain variable has on the function. While the term “differentiation” essentially refers to the process of finding “derivative”, they are often used interchangeably.

In science and engineering, quantities are usually represented by functions. For instance, a particle’s velocity and energy at any given time are represented by functions v

( )

t and E

( )

t respectively.

Activity of a radionuclide at any given time is usually written as a function A

( )

t . Derivatives of these functions v

( )

t , E

( )

t , and A

( )

t would give rates of change of these quantities at time t. Note that in this case, the time t is the independent variable.

Differentiation is often defined as

Differentiation of f

( )

x with respect to x

[ ( ) ] ( ) ( )

h x f h x x f

dx f d

h

−

= +

= lim→0 ,

which suggests that the derivative is a localized change in a sense that it is only defined between x and x + h, where h is infinitesimally small; as opposed to the average change which is calculated between the beginning and the ending values as shown in Figure **1.

y

x dx dy

y

x

Figure **1 Derivative vs. average change.

Throughout your career, you may encounter different ways people choose to represent derivatives.

For instance,

[

f

( )

x

]

dx

d , f′(x), and d_xf

( )

x

are identical, and they represent first-order derivative. Higher-order derivative is the result of applying differentiation to the derivative that is one order lower, e.g. second order derivative is calculated by applying differentiation to the first order derivative. The n^th-order derivative is represented by

[

f

( )

x

]

dx d

n n

, f ′′^...

( )

x , or d_xx...f

( )

x .

Based upon the definition of differentiation given above, the following derivatives are obtained:

(a)

[ ]

x^{n =}nxⁿ⁻¹ dx

d .

(b)

[ ]

c =0 dx

d ; c is a constant number, or c is not a function of x.

(c)

[ ( )

x

] ( )

x dx

d sin =cos .

(d)

[ ( )

x

] ( )

x dx

d cos =−sin .

(e)

[ ]

x x dx

d 1

ln = .

(f)

[ ]

e^x e^x dx

d = .

Some very useful differentiation properties include:

(a) Linearity property:

[ ( ) ( ) ] [ ( ) ] [

g

( )

x

]

dx x d dx f x d g x dx f

d + = + =>

[ ( ) ] [

f

( )

x

]

dx c d x dx cf

d = .

(b)

[ ( ) ( ) ] ( ) [ ( ) ] ( ) [

f

( )

x

]

dx x d g x dx g x d f x g x dx f

d = + .

(c)

( )

( )

( ) [ ( ) ] ( ) [ ( ) ] [

g

( )

x

]

²

x dx g x d f x dx f x d g x g

x f dx

d −

=



 



 .

(d)

[ ( ( ) ) ] [ ( ( ) ) ] [

g

( )

x

]

dx x d g dg f x d g dx f

d = , also known as “chain rule”.

With help of the derivatives and the differentiation properties above, more complex derivatives can be derived.

Example: Find

[

^tan

(

^x2+1

) ]

dx d

[

^tan

( ) ] [

²⁺¹

]

= x

dx u d du

d # set x²+1≡u, and use chain rule

[ ( )

u

]

du x d tan

=2

( ) ( )

^_^



= 

u u du x d

cos

2 sin # write tan() in term of sin() and cos()

[ ( ) ] ( ) [ ( ) ]

( )

_^













 −

= u

du u u d du u

u d

x ₂

cos

cos sin

sin ) cos(

2

( ) ( ) ( )

^_^



 +

= u

u u u

x u ₂

cos

) sin(

sin cos

)

2 cos( # cos²

( )

u +sin²

( )

u =1

( )

x x

( )

u x ₂ u 2 2 sec²

cos

2 1  =



 



= 

(

¹

)

sec

2 ^{2 2}+

= x x

In some cases, a function can have more than one independent variable, e.g. f

( )

x,t depends on both variables x and t, and both variables are independent of each other. Partial differentiation of a function is the result of taking derivative of the function with respect to one of its independent variable. Symbol “∂” is often used in place of “d” in the representation of partial differentiation, e.g.

( )

^{x t t}

f ∂

∂ , / .

In nuclear engineering, differentiation often appears in discussion about rate of change of certain quantity, e.g. radioactive activity, neutron flux, and core temperature. In such cases, derivatives are often performed with respect to time t. There are some useful interpretations of the first and the second derivatives of such quantities. At a given time t0,

(a) If the first derivative is positive, then the quantity is increasing in time.

a. If the second derivative is positive, then the quantity will increase faster.

b. If the second derivative is negative, then the quantity will increase slower.

c. If the second derivative is zero, then the quantity will keep increasing at the same rate.

(b) If the first derivative is negative, then the quantity is decreasing in time.

a. If the second derivative is positive, then the quantity will decrease slower.

b. If the second derivative is negative, then the quantity will decrease faster.

c. If the second derivative is zero, then the quantity will keep decreasing at the same rate.

(c) If the first derivative is zero, then the quantity is neither increasing nor decreasing at t0. a. If the second derivative is negative, then the quantity changes from increasing to

decreasing around t0.

b. If the second derivative is positive, then the quantity changes from decreasing to increasing around t0.

c. If the second derivative is zero, it is inconclusive.

From such behaviors, further conclusions can be made:

(a) Local minimum occurs at the time when the first derivative is zero, and the second derivative is positive.

(b) Local maximum occurs at the time when the first derivative is zero, and the second derivative is negative.

(c) When both the first and the second derivatives are zero, it is necessary to look at higher derivatives to justify the behavior of the quantity. If f′(t₀)= f′′

( )

t₀ =K= f⁽ⁿ⁾

( )

t₀ =0_, but f⁽ⁿ⁺¹⁾

( )

t₀ ≠0, then

a. If n is odd and f⁽ⁿ⁺¹⁾

( )

t₀ >0, the point is a local minimum.

b. If n is odd and f⁽ⁿ⁺¹⁾

( )

t₀ <0, the point is a local maximum.

c. If n is even, then the point is a saddle point.

Notice that (a), (b), and (c) can be used to find local minimum, local maximum, or saddle point of a function, and is known as extremum test. The word “local” indicates that these are merely localized values. That is, the actual (i.e. global) maximum or minimum of the function can be somewhere else.

Integrations

Also known as anti-differentiation, integration is a mathematical process used to convert a derivative back to its original function. If f

( )

x is the derivative of F

( )

x , i.e. F′

( )

x = f

( )

x , then

Integration of ^f

( )

^x with respect to x =

∫

f

( )

x dx=F

( )

x +C^,

where C is called constant of integration. This type of integration is called indefinite integration. In order to find the exact value of C, it is important to have extra information such as value of F

( )

x +C at a specific point, and perform back-calculation from there.

There is yet another type of integration called definite integration. This type of integration is applied over a specific range, and is represented by

( )

xdx F

( )

b F

( )

a

bf

a = −

∫

^.

The integration product of a function, usually referred to as integral, can be interpreted as the net area between the curve of the function and its reference axis (e.g. x axis) as shown in Figure **2. One can view this area as a combination of infinite number of very narrow rectangles whose height extends from the reference axis to the corresponding value on the curve of the function, and whose width is infinitesimally small. Hence, integral can be calculated from summation of the areas of these rectangles. Note that area under x-axis has negative value, while area over x-axis has positive value.

y

x y = f (x)

Figure **2 Area under curve of function y = f (x) representing integral of f (x).

Similar to the terms differentiation and derivative, the terms integration and integral are often used interchangeably. It should be straightforward to find the integrals of the derivatives previously presented:

(a) C

n dx x x

n

n +

= +⁺

∫

₁^{1 ; n≠−1.}

(b)

∫

cdx=cx+C ; c is a constant number, or is not a function of x.

(c)

∫

^sin

( )

xdx=−^cos

( )

x +C^.

(d)

∫

^cos

( )

x dx=^sin

( )

x +C^.

(e) dx x C

x = +

∫

^{1 ln .}

(f)

∫

e^xdx=e^x+C^.

Some integration can be derived from applying the following integration properties to basic integrals:

(a)

[

^f

( )

^{x dx}

]

^f

( )

^x

dx

d

∫

=

(b)

∫ [

^f

( ) ( )

^{x +g x}

]

^dx=

∫

^f

( )

^{x dx+}

∫

^g

( )

^{xdx =>}

∫

^cf

( )

^{x dx=c}

∫

^f

( )

^{x dx}

(c)

∫ ( )

⁼⁻

∫

b^a

( )

b

a f x dx f x dx

(d)

∫

_^f

( )

^{u du}_dx^dx=

∫

^f

( )

^{u du} ; known as “substitution of variable” method

Example: Find

∫

^2x

(

^x2+1

)

^50dx

Let u=x²+1, then x dx

du =2 <=> du=2xdx.

Thus,

( ) ( )

x C u C

du u dx x

x + +

= +

=

=

+

∫

∫

^{2 1} _{51 51}¹

2 51 51

50 50

2 .

(e)

∫

^udv=uv−

∫

^vdu ; known as “integration by part” method where u= f

( )

x , v=g

( )

x ,

( )

x dx f

du= ′ , and dv=g′

( )

xdx Example: Find

∫

a^b

xdx xe

Let u=x, and dv=e^xdx.

Thus, du=dx, and ^v=

∫

^dv=

∫

^exdx=ex.

Then,

[ ]

^ba

x b x

a b x a b x

a

xdx xe e dx xe e

xe = −

∫

= −

∫

^.

For other integrals, one can refer to tables of integrations containing various complex integrals, and are available in many mathematical textbooks or internet resources.

There are some integration techniques which are often useful in solving scientific and engineering problems:

(a) Integrations of odd function A function ^f

( )

^x is an odd function when ^f

( )

^{−x =−f}

( )

^{x .}

Integrations of any odd functions over symmetric range −a≤x≤a always yield zero result. To see this, one can write

( ) ∫ ( ) ∫ ( )

∫

− ⁼ − ⁺

a a

a

a f x dx f x dx f x dx

0 0

.

By doing substitution of variable x in the second integral on the right hand side by −x, one find that

( ) ∫ ( ) ( ) ∫ ( ) ∫ ( ) ∫ ( )

∫

− −

−

− − − =− − = − =−

= ^{0 0}

0 0

0 a a

a a

a

dx x f dx

x f dx x f x

d x f dx x f

Thus, substituting the term back into the original equation, one get

( )

=

∫

⁰

( )

−

∫

⁰

( )

=0

∫

− −a −a

a

a f x dx f x dx f xdx .

Examples of odd functions include ^sin

( )

^{x , sin}

( ) ( )

^{x cos x , and ax2n+1} where n is any integer number.

(b) Integration of sinusoidal function Integrations of function sin

( )

x or cos

( )

x over their periods yield zero result since both functions go through the same negative and positive values over one (or multiple) period, e.g.

∫

a^a+2πn/bsin(bx)dx⁼0.

(c) Integration on other coordinate We have so far discussed about integration on Cartesian coordinate (x, y, z). On some occasions, using other coordinate may be easier for solving problem. The most commonly used coordinates (other than the Cartesian coordinate) are cylindrical and spherical coordinates as shown in Figure **3.

For cylindrical coordinate, the axis consists of (r, θ, z) where θ

cos r

x= , y=rsinθ, and z=z

For spherical coordinate, the axis consists of (r, θ, φ) where θ

φcos sin r

x= , y=rsinφsinθ, and z=rcosφ

r z

y z

x

(a) (b)

Figure **3 (a) Cylindrical coordinate and (b) spherical coordinate. (Figures from Wolfram Math World. URL: http://mathworld.wolfram.com)

(d) Surface integration For two- or three-dimensional objects, their surface areas can be calculated using double integrals. The simplest case is a rectangular area between

b x

a≤ ≤ and c≤ y≤d:

Rectangular area ⁼

∫ ∫

_a^b _c^ddydx=

∫

_a^b

[ ]

^yd_c ^dx=

∫

_a^b

(

^d−c

)

^dx=

(

^d−c

)

^xb_a

(

d−c

)(

b−a

)

=

This result should not be a surprise to anyone since it is from basic algebra.

For a more complex case, consider surface area of a cylinder of radius R and a length which extend from z = -a to z = a. The surface area can be divided into the side and the two ends of the cylinder. We shall use cylindrical coordinate for the side of the cylinder, and spherical coordinate for the two end of the cylinder:

Surface area of cylinder (the side)

[ ] ∫

∫

∫ ∫

− = − = −

= ^a

a a

a a

a ^2πrdθdz rθ ²₀^π dz 2πrdz

0

Ra ra

rz^a_a π π

π 4 4

2 = =

= ₋ since r = R

Surface area of cylinder (each of the two ends)

[ ] ∫ [ ]

∫

∫ ∫

^{= =}

= ^R rd dr ^R r dr ^R r dr

0 0

2 0 0

2

0^π θ θ ^π 2π

2 0

2 R

r ^R π

π =

=

Total surface area of cylinder = 4πRa+2πR²

Notice the r in the integral for the cylinder’s side, and the rsinφ in the integral for the cylinder’s end. The presences of these factors are due to the fact that in cylindrical and spherical coordinates the axis variables are not completely exclusive of each other, i.e.

change in one variable results in change in the others. It is a good practice to remember the following line element:

For cylindrical coordinate: ds=drrˆ+rdθθˆ+dzzˆ For spherical coordinate: ds=drrˆ+rdθθˆ+rsinφdφφ^ˆ

When calculating surface area of the cylinder’s side, for example, we need to integrate over dθ and dz. Thus, we take the last two terms of the cylindrical line element: rdθdz. When calculating surface area of the cylinder’s two ends, we need to integrate over dr and dθ. Thus, we take the first two terms of the spherical line element: rdrdθ ^.

Surface integration is quite useful for calculating change on the surface of a volume. For example, heat flux along a metal wire may be defined as Q

(

r,θ,z

)

. One can then find out the total heat flux along the surface of the wire by calculating

∫∫

^Q

(

^r,θ,z

)

^rdθdz.

Often, you may find notation like

∫

S

[ ]

dS. Such notation represents surface integration, and you still need to perform double integrations.

(e) Volumetric integration Triple integrations are needed to calculate volumes of various geometries. The line elements introduced earlier shall be used when working in cylindrical or spherical coordinates. For instance, volume of a half sphere of radius R can be calculated from integrating over all three of its coordinates in spherical

coordinate. Thus, we take all three terms of the spherical line element: r²sinφdrdθdφ^. Volume of half sphere ⁼

∫ ∫ ∫

0 ^/2

2

0 0

2sin

π π

φ θ

R φ

d drd r

∫ ∫

∫ ∫

_^{ =}









=  ^/2

0 2 0 2 3

/ 0

2 0

0 3

3 sin 3 sin

π π

π π

φ θ φ φ

θ

φ R d d

d r ^R d

[ ] ∫

∫

⁼

= ^/2

0 2 3

/ 0

2 0 3

3 sin 2 3 sin

π

π π

φ φ π

φ φ

θ R d

R d

[

0^/2

]

³

3

3 cos 2

3

2π R ₋ φ^π ₌ π R

=

One of the uses of volumetric integration in nuclear engineering is for finding total number of particles in a volume. Density of particles may have certain distribution instead of being uniform over a whole space. The density distribution can be of the form, for instance, n

(

r^,θ,φ

)

. Thus, to find total number of particles, one needs to perform the following integration:

Total number of particles ⁼

∫∫∫

ⁿ

(

^r,θ,φ

)

^{r2sinφdθdφdr.}

Flux calculation may be calculated in similar manner if flux density is known. In certain cases, such as when we consider flux density of a beam, performing integration in cylindrical coordinate may be more suitable for the problem.

One of the popular notations for volumetric integration is

∫

V

[ ]

dV . It should not come as a surprise that this is a short hand for triple integration.

Logarithmic and Exponential Functions

Logarithmic function is a “computational tool” which has been created as an alternative way to deal with exponentiation. Suppose we have a function

bx

y= ,

its equivalent logarithmic function is x

b y= log

where b is called base. It is important to remember that in real number analysis the logarithmic function above is only defined when b>0 since negative value of b can lead to imaginary number, e.g.

( )

−2^1/2 = −2 =i 2, and b=0 would yield trivial result. It then follows that y>0. In complex number analysis, however, logarithm of a negative number is defined. While this point shall be covered in more detail later on, it should be noted here that the properties of real logarithm mentioned below can be applied to complex logarithm as well. For now, there should be no confusion between the two.

There are several important logarithmic properties which can be derived directly from properties of exponentiation. They include:

(a) log_b1=0,

(b) b

a a

c c

b log

log =log => log_bb=1,

(c) log_bac=log_ba+log_bc => log_ba^r =rlog_ba,

(d) a c

c a

b b

b log log

log = −



 



 => c

c ^b

b 1 log

log =−



 



 .

Example: Solve 1

2 =

− ^−x

x b

b for x

=2

− ^−x

x b

b # multiply through by 2

1 2

=

− _x

x

b b # rewrite b^−x

x

x b

b² −1=2 # multiply through by b^x

0 1

2^x−2b^x− =

b # rearrange terms

2 2 1

4 4

2± + = ±

=

bx # solve the binomial equation

(

^{1 2}

)

log log₁₀b= ₁₀ ±

x # take logarithm on both sides

( )

x b

10 10

log 2 1 log ±

= # rearrange for x

While the base of logarithm can be any numbers, there are in general two base numbers which are widely supported by modern scientific calculators. They are base 10 and base e. Notation-wise, the base-10 logarithm is usually written without base number as logx, while the base-e logarithm is usually written as lnx. The logarithmic property (b) above is useful for changing logarithm of other base into either base 10 or e. In engineering, base-e logarithm, a.k.a. natural logarithm, is used quite often.

The numerical value of e is approximately 2.71828. This value is discovered by Leonard Euler as the horizontal asymptote of function

x

y x



 

 +

= 1

1

as shown in Figure **4.

Figure **4 Plot of function y= 1+1 x



  

 

x .

Since such function approaches its horizontal asymptote as x→ ±∞, the definition of e can be written as

x

x x

e 



 

 +

= →+∞

1 1

lim or

x

x x

e 



 

 +

= →−∞

1 1

lim ,

and with appropriate substitution, one can also find that

( )

^x

x x

e ^1/

01 lim +

= → . The existence and the significance of e has to do with the derivative of logarithm. That is, we can apply definition of differentiation to logarithm:

[

x

]

dx d

logb

( )

h

x h

x _b

b h

log limlog

0

−

= +

→



 



=  +

→ x

h x h ^b

h 1log

lim0 # use property of log



 

 +

= → x

h h ^b

h 1log 1

lim0

(

v

)

vx ^b

v +

= → 1 log 1 lim

0 # write v as h/x

(

v

)

v

x^{v b} +

= → 1log 1 1lim

0

( )

^v

v b v

x

/ 1

0log 1

1lim

+

= →

( )

[

_v ^v

]

b v

x

/ 1 01 lim 1log

+

= → # log is a continuous function

Notice that the term in the bracket is the definition of e. Thus,

[ ]

b e x

x x dx

d

b

b ln

log 1

log =1 =

and when b is replaced by e, d[ln x]/dx = 1/x which is exactly what was presented earlier. This derivative is one of the reason why natural logarithm and e are important quantities in calculus.

Exponential function or e^x is probably one of many, if not the most fundamentally, important functions in science and engineering. The following examples are just a few of many phenomena which can be directly or indirectly described using exponential function. The first example is the decay of temperature in a bar or a wire of length L due to cooling at both ends. The temperature profile T

( )

x,t is given as

( ) ∑

^∞

=

 −



 



= 

1

sin 2

,

n

t

n e

L x A n

t x

T π ^λ

,

where the term e^−λ2t represents the change in temperature as a function of time. Another example is the rate of change in activity ^A

( )

^t of a radioactive sample which is given as

( )

t Ae ^t A = ₀ ^−λ .

Both examples suggest processes which are decaying in time. Since the values of λ² and λ in the first and the second examples, respectively, are both positive, the exponential terms go to zero as time increases. Naturally, exponential function does not just represent decay. It can also represent growth.

One of such processes is the time-dependent neutron flux in a nuclear reactor n t

( )

which is of the form

( )

t Ae^{t T} n = ₀ ^/ .

Here, the exponential term goes to infinity as time increases. Of course, this is not the complete solution of neutron flux in a nuclear reactor as such equation would suggest that the neutron production goes out of control. But that is not our main concern at this point.

An interesting property of exponential is x

e^lnx = => e^xlnb =b^x.

This property is sometimes useful for eliminating natural logarithm and exponential from equation.

There are also various functions which are constructed from exponential function. For instance,

sinh 2

x

x e

x e

− −

= and

cosh 2

x

x e

x e + −

= ,

which are known as hyperbolic functions. It follows that

[

x

]

x

dx

d sinh =cosh and

[

x

]

x

dx

d cosh =sinh .

We can write the ^sin

( )

^{x and} cos

( )

x in term of exponential functions. That is,

( )

i

e x e

ix ix

sin 2

− −

= and

( )

cos 2

ix ix e x e

+ −

= .

On the other hand, we can also write x

i x

e^ix =cos + sin .

We will cover these last two transformations in more detail when we discuss about complex number analysis.