Advanced Mathematical Techniques in Chemical Engineering

Prof. S. De
Department of Chemical Engineering
Indian Institute of Technology, Kharagpur

Lecture No. # 01
Introduction of Vector Space

Good morning everyone. So, welcome to the course of advanced mathematical methods
in chemical engineering. We require several mathematical techniques, for solving of
various chemical engineering problems.

Now, let us first try to understand, why do we require the mathematical methods? Why it
is necessary for a chemical engineer? Now, any chemical process, it is if you would like
to look into the production of any particular species, that will be involving lots of
manpower material and there are several lots of fixed equipment.

So, to conduct one experiment successfully, you need to have huge investment. Now, if
there is something wrong in the production, in the quality of the product, the whole batch
will be waste. And the money involved in the manpower, money involved in the running
the experiment, etcetera will be spoiled. So, there is a need of a modeling and simulation
in actual chemical engineering process. So, let us try to understand, what is modeling and
what is simulation and how it is related to mathematical methods in chemical
engineering?

The modeling is basically required, when you would like to express - one as the
particular process by writing the some characteristic of the process can be written in the
form of some equations. The mathematical expression of the process is known as the
modeling. So, once the model is properly done, then you must be simulating the process;
so, you have to solve these equations. So, that you will be getting the output results in
terms of (()) - let us say, conversion or in a temperature profile or some particular quality
of the product so and so forth.

So, in order to solve these mathematical equations of modeled equation, you need to
require the mathematical techniques. So, it is not possible to conduct the experiments
number of times. So, that you can come to a conclusion, that this should be the
appropriate operating conditions. So, that you will be getting the particular desired
product in the output. So, in order to avoid that one has to conduct limited number of
experiments and express the mathematical process in terms, express the chemical
engineering process in terms of mathematical expression and try to solve them. And
check whether these equations in the results of this simulation modeling and simulation
by using the mathematical techniques whether, they are matching with the experimental
data or not. If that is the case, then you can conduct a number of virtual experiments on
the computer by simulation modeling and simulation and you can scale it up for a higher
order of process. So that you can tell later on, that these are the typical operating
conditions for which i will be getting the product, the quality of this the product of this
quality.

So, therefore the mathematics will be used as a tool to solve the model equations of any
chemical engineering processes. So, with this background, we start this chemical
engineering course, the advanced mathematical techniques for chemical engineers and
we will be starting with the vectors spaces and I will be mostly dealing with the
analytical mathematics that will be most helpful for any chemical engineer in an actual
plant operation.

Because there is another parallel category that is the numerical techniques, but that
requires a different expertise and different setup in the computer etcetera. So, therefore, I
will be mostly stressing on the analytical mathematics which will be very essential tool
for the chemical engineers for solving the model equations for any chemical engineering
process.
(Refer Slide Time: 04:19)

So, we will start with the vector spaces. This is also known as a linear space. Now, we
talk about earlier, we know, we learned about the two dimensional vector space three
dimensional vector space in this case we will be talking about N dimensional vector
space.

So, what is the definition? The definition is ordered n-tuple of numbers is called an
element in n dimensional space. The notation is that means, there are n numbers of
elements in a vector. So, the notation is X is n-dimensional vector, if X is n-dimensional
vector, then it will be having the elements x i small x is so x is are the elements and x i
is the ith element. So, we write X is comprised of a set of element x i.

So, next we talk about the definition of what is an equal vector? We called the two
vectors X and Y equal, if and only if all the corresponding elements of these two vectors
will be identical. So, the notation i f f means, if and only if x i is equal to y i that means
ith vector, ith element of the vector X and ith element of the vector Y. They are identical
and equal.

Then, we have another notation, that is, X belongs to R is to the power n. This simply
means, vector X belongs to n-dimensional real space, n-dimensional space - where
elements are real.
So, vector X contains different elements which they which each every element is
basically a real number. So, that is, why we called the real space and the vector X is
called n-dimensional real space.

(Refer Slide Time: 08:10)

Next, we talk about, what is a complex space, if x belongs to C superscript n. This

simply means, that vector X belongs to n-dimensional space. This n stands for dimension
of the space. Dimension means, you will be having the contribution in n number of
dimensions, which are mutually orthogonal to each other, vector X belongs to n
dimensional space - where elements are complex.

Next, we call about a talk about a zero vector, what is a zero vector? The definition of a
zero vector is that all elements of these vectors are identical to zero, then if all the
elements are zero, then the constituted vector is called a zero vector.

Then, let us a rebrush refresh our ideas, our earlier definitions of vector laws, which most
of us have studied during your ten plus two days and it may be earlier also that the sum
of vectors X and Y that means X plus Y. This simply indicates that it is a new vector and
the each every element of the new vector will be constituted by the corresponding
elements of the two vectors and summing them up; that means, every element of this
vector X new vector X plus Y is basically nothing but the summation of the
corresponding ith vector of the individual vectors, then that will this that the set of these
elements will constituted the vector X plus Y.
So, what are the laws, X plus Y is identical equal to Y plus X and this is known as
commutative law.

Next, vector law is X plus Y plus Z. Suppose, there are three vectors X ,Y and Z and
then these will be identical to X plus Y plus Z. This is known as associative law.

Next, vector law is if minus X is a vector, then minus X plus X will give rise to a 0
vector that means every element of minus X will be nothing but equal and opposite in
sign to the corresponding elements of the X vector. So, when you add them up the whole
addition will give rise to a vector where every element is zero; that is, why you will be
landing up with a zero vector.

(Refer Slide Time: 11:49)

Then, we talk about the scalar multiplication, these simple operations and definitions will
be quite useful, whenever we talk about, we go along the course of these particular
slavers. K times X is nothing; X is the vector and K is the scalar multiplier, then every
element of the X vector will be multiplied by the scalar K or let us say, X is the n-
dimensional vector, then every element of the vector X will be multiplied by the scalar K
and K is the arbitrary scalar.

Then, we talk about the three most important vector spaces in linear vector operations.
First one is the metric space, so with these definitions, you will be able to solve lots of
problem in chemical engineering problems and every case will be looking into some of
the examples at the end of our lecture; so that it will be absolutely clear to the students,
that how these mathematical techniques will be utilized for the chemical engineering
applications for solving the actual chemical engineering processes.

So, what is a metric space? Metric is defined in an n-dimensional space, so we talk about
a most generalized n-dimensional space. Suppose, there are n components in a chemical
engineering process, then the space will be talking about is a n-dimensional space.

So, there is the most generalized one, if someone if in a system there are three
components present, so it will be basically three dimensional space, if there are ten
components present it is ten dimensional space. So, we talk about the most general, in
generalized n dimensional space. So that the mathematical prob[lem]- problem the
techniques that will be covering in this course, will be having a most generalized
framework and any subset can be easily derivable out of these derivations.

So, what is the metric space? A metric space is defined in an n-dimensional space and
the notation is d of X and Y, if X and Y are two vectors, then the metric between the two
is denoted by d of X and Y. So, this is known as the metric between X and Y.

So, what is the physical significance of this metric? The physical significance of the
metric is that it denotes displacement between vectors X and Y. In other words, it is
nothing but the distance between vectors X and Y. So, that is the physical significance of
the metric. Therefore, because of this definition one can understand that the displacement
or distance cannot be a negative quantity, it will be a measurable positive quantity;
therefore, d will be always greater than or equal to 0. If two vectors are identical the
metric between the two vectors will be equal to 0, if not then there will be a definite
metric existing between the two vectors X and Y.
(Refer Slide Time: 16:26)

Now, let us look into some of the properties of the metric. So, the metric is supposed to
satisfy the following axioms or set rules, what are these rules - the first axiom is that as i
said earlier that metric is between two vectors X and Y will be always positive, always
greater than 0, if X is not equal to Y; that means, if they are not identical vectors, then
there will be a definite distance or displacement present between the two vectors X and
Y.

Metric between X and X will be always equal to 0, this simply means that the distance
between the X and X the same vector will be identically equal to 0; that means, the two
vectors we are taking about in these cases are same.

Metric between X and Y will be nothing but metric between Y and X. So, this simply
indicate if whatever the order between X and Y since metric basically a scalar quantity it
giving a distance. Therefore, whether we talk about x and y or y and x. So, the distance
between the two will always same.

The fourth will be basically the triangle law. The triangle law is that sum of two sides of
a triangle is always greater than equal to compare to the other side. So, metric between X
and Y will be less than equal to metric between X plus Z plus metric between X and Z
and metric between Z and Y. This is also known as the triangle inequality law.
Therefore, we have understood that metric is nothing but a scalar quantity. So, it is a
scalar characteristic defined between two vector quantities. Now, if there are two vectors,
let us say X which will be constituted of x i and Y, which will be having the elements y i
and i basically it goes up to n. Therefore, we are taking about an n dimensional space
then what is the metric between X and Y, it is defined as under root x 1 minus y 1 square
plus x 2 minus y 2 square and this goes on up to x n minus y n square.

(Refer Slide Time: 20:02)

So, therefore, we can write it as in a short notation we can write the metric as d of X and
between X and Y is nothing but under root summation x i minus y i whole square and
where the index i runs from 1 to n, because the dimension is n and there are n number of
elements presents in the system.

Similarly, we can talk about metric between Y and X is nothing, but under root
summation i is equal to 1 to n y i minus x i square. So, if you look into these two
expression, if the square whether it is a minus b whole square or b minus a whole square
they are identical. Therefore, we prove the assumption the axiom metric between X and
Y is identical with metric between Y and X.

So, that goes for the metric then we talk about normed linear space, next vector space is
normed linear space. This is again, a real norm is nothing but a real valued scalar norm is
real a valued scalar it is a real value scalar from obtained from a vector its physical
significance is it gives norm represents the length of the vector and let us say, what is the
notation? Notation is denoted by the two double bars.

(Refer Slide Time: 22:42)

So, these means we are taking about norm or length of vector. Since, length does not
have any direction, it is a scalar quantity. Again a norm must be satisfying some of set
rules or axioms. Let us note them down, axioms that a norm should satisfy they are as
follows: the first one, since it is a length norm is ever positive - if and only if X is not
equal to 0.If X vector is not a zero vector, then norm of X vector is always greater than 0.
Therefore, the next axiom follows that norm of a 0 vector is nothing but a zero vector.
Third one is that norm of alpha X - where alpha is an arbitrary scalar. Then norm of
alpha X is nothing but mod of alpha times norm of X.

Next assum[tion]- axiom is that norm of X plus Y is nothing but this is less than norm of
X plus norm of Y again. This rule comes from triangle inequality rule. Next, we look
into how the norm is define given a vector suppose, consider a vector X, such that X
belongs to R superscript n that means X belong to a real number of space of dimension n.
Therefore, X will constituting n numbers of real valued element x 1 x 2 up to x n.

So, therefore x will be compri[se]- you will be having n number of elements and every
each one of them is a real valued number. Therefore, norm of X is defined as under root
x 1 square plus x 2 square plus x 3 square plus up to x n square.
So, therefore, these this gives the length of vector X, hence it is a scalar quantity. Now, if
we look into another vector, let us say X and Y are two vectors, each of it is they belong
to the n-dimensional real space that means X belongs to n dimensional real space and Y
it belongs to n-dimensional real space, then we defined a vector X minus Y.

So, therefore, X minus Y is a vector that belongs to n-dimensional real space as well that
means, what is X minus Y? X minus Y is nothing but it will be a vector in n-dimensional
space, where every element of this vector will be nothing but x 1 minus y 1 x 2 minus y 2
like that and up to X n minus Y n.

(Refer Slide Time: 26:59)

So, what will be the norm of the new vector X minus Y. The norm of new vector is
nothing but under root summation and this summation goes from index i equal to 1 to n
and this will be x i minus y i square.

Now, if you look into the definition of metric, what is this under root summation x i
minus y i square. This is nothing but the metric between the two vector X and Y.

So, therefore, if we look into these very closely. So, d metric is called it is a natural
metric, defined from a norm therefore, whenever a norm is defined a metric is generated

So, therefore, whenever we will be defining a norm a metric will be naturally derived
automatically generated. So, therefore, we can come to a conclusion that normed linear
space is nothing but a metric linear space but the reverse is not true.
That means whenever a norm is defined a metric is generated, but whenever a metric is
generated; we cannot define a norm. So, it is not from metric to norm but it is the other
way round it is from norm to metric. So, whenever we will be defined a norm a metric is
automatically generated.

Next, linear vector space we talk about the inner product space. Let us look, into the
definition of inner product - inner product is again a real valued scalar defined in n-
dimensional real space and what is the physical significance of inner product?

The physical significance of the inner product is that physically it indicates the angle
between the two vectors X and Y. So, it gives, an idea on the orientation of two vectors
in more specific terms it gives the angle between them.

(Refer Slide Time: 31:00)

So, let us look into the various axioms that inner product should satisfy or axioms or
properties the inner product should satisfy are as follows : the first one is that inner
product between X and Y is identical between inner product that of inner product
between Y and X that means the orientation of X and Y and orientation of Y and X they
are same and equal identical.

Second one is if the inner product if the vector X is multiplied by a scalar alpha and then
inner product between alpha X and Y is nothing but alpha multiplied by inner product
between X and Y where alpha is a arbitrary scalar.
Next one is if there are three vectors belonging to the n dimensional real space X Y and
Z then inner product between X plus Z and Y should be the inner product between X and
Y plus inner product between Z and Y.

Now, let us look into how to compute inner product between two vectors X and Y. X
belongs to n dimensional space constituted of the elements x 1 up to x n.Similarly, Y is
another vector belongs to n dimensional real space, such that Y is constituted by the
elements y 1 y 2 up to y n.Then inner product between X and Y is nothing but
summation of the corresponding elements to multiply the corresponding elements of the
two vectors X and Y and then and sum the sum them up for all the elements. So, it will
be nothing but x 1 y 1 plus x 2 y 2 up to x n y n.

So, this is nothing but the dot product of two vectors X and Y. So, what is the dot
product of X and Y. This will be nothing but summation x i y i therefore this will be
nothing but norm of X multiplied by norm of Y multiplied by cosine theta - where theta
is the angle between the vectors X and Y.So, therefore, if we know vectors X and Y, then
their dot product or the inner product will be giving you the angle between them cosine
theta - where cosine theta is nothing but the inner product between the two vectors
divided by norm of X multiplied by norm of Y.

(Refer Slide Time: 34:47)

So, if you look into the inner product of the same vector. Let us say X, so inner product
between the same vector is nothing but if you put the substitute, the definition of the
inner product so this will be for summation over all the elements i is equal to 1 to n this
will be X Y Y Y, if we are talking about two vectors X and Y; therefore this will be
nothing but x i multiplied by x i so this will be summation of x i square i is equal to 1 to
n and what is this definition this is nothing but norm of X square of that this is nothing
but square of norm.So, therefore, if you look into this, so what is the interpretation? The
interpretation is that when an inner product is defined, we will be getting a norm is
automatically generated and we have seen earlier that when a norm is automatically
generated, a metric is automatically generated defined. Therefore, whenever we will be
talking about an inner product we will be getting a norm and then we will be getting a
metric.

So, therefore, definition of inner product, simply indicates that a norm is defined and a
metric of a vector is defined. So, therefore, we can talk about that inner product space it
say more general space, then from that we will be getting a normed linear space, then we
will be getting a metric linear space.

So, these three spaces are differ; so basically in inner product, norm and metric are the
three scalar quantities. One will be getting out of the vectors and they will be quite
important. As far as the mathematical treatment of various systems in chemical
engineering processes are concerned, that we will see as we go along the course along
and the various rest portion of this course.

Now, what we have done till now? We have looked into the norm, how to define metric
norm and inner product of vectors. where each of So, vector is nothing but a discrete
system that we have seen, that if a vector is given n-dimensional generalized vector
either in real space or in complex space, then how will you generate the metric norm,
how will you mathematically express the expression of metric norm and inner product?

Now, next we will see that the if the space if your system is not a discrete system, let us
say it is a continuous system and what is a continuous system - represented a continuous
system is represented by continuous functions.
(Refer Slide Time: 39:01)

So, for continuous system it is represented by functions, again these functions can be one
dimensional functions, two dimensional function, three dimensional function.

So, let us consider first a one dimensional function, what is the one dimensional
function? One dimensional function is nothing but the function that it is an expression,
which is a function of only one independent variable let say x.

So, u of x v of x they are basically continuous functions and their examples of

continuous one dimensional function.

Now, let us say x belongs to a and b, this gives the domain or the limit of x that means x
varies between a and b.

So, what is the metric of u and v? In this case, for the continuous function metric
between u and v is defined as under root integral a to b u of x minus v of x square of that
d x. If you remember, if you look into the corresponding definition of metric for discrete
domain, for example, for the case of vectors then d of u and v is nothing but under root
summation u i minus v i square, if u v u v are vectors.

Now, whenever these particular elements they are closed spaced, let us say these vectors
their of particular elements are closed spaced, then they will be represented by the
functions; so you will be getting the summation. In the case of function, this summation
will be represented by this integral over the domain between a and b and you will be
getting these expression.

So, similarly, norm of u is nothing but under root a to b u of x square of that d X, that is,
the definition of norm of u. Similarly, you will be getting norm of v as under root a to b v
of x square multi[plied]- times d X.

Then inner product between u and v is nothing but the integral of u of x v of x d X. So,
for discrete for vectors u and v, the inner product between u and v is nothing but i is
equal to 1 to n summation u i v i, when these i tends to infinity then the n tends to
infinity, then this discrete domain will be simulated by the continuous domain.
Therefore, for functions u and v, which are the continuous functions in x will be getting
the inner product by this expression integral a to b u x v x d X. So, this is the definition
of the three vector space, inner product, norm and the metric, if we have continuous one
dimensional function.

(Refer Slide Time: 43:23)

Now, let us talk about a for two dimensional continuous functions that means, if we have
functions u as a function of x and y v as a function of x and y then metric between u and
v is nothing but under root. Let us say, x lying between a and b y lying between c and d,
then since it is a two dimensional space, two dimensional function. So, therefore, these
integral will be represented by a the single integral replaced by a double integral u minus
v whole square d x d y.
The norm of u will be nothing but integral over x integral over y u of x x and y square of
that d X d y. Similarly, one can define norm of v, as integral over x integral over y v of x
square of that d X d y. And the inner product between u and v will be nothing but double
integral x from a to b y from c to d u of x y multiplied by v of x y d X d y for a three
dimensional problem this function the functions u and v will functions of three
independent dimensions x y and z so in that case the limits of x y and z will be defined
and they will be prescribed.

So, one can get the metric between u and v as under root triple integral. This double
integral will be replaced by the triple integral in that particular case and it will be triple
integral over x over y over z u minus v whole square d X d y d z.

Similarly, norm of u will be defined as triple integral u of x y z square d X d y d z and

inner product between u and v as triple integral u multiplied by v d X d y d z. So, we
have seen that how norm the linear metric, the vector spaces like norm metric and inner
product are defined in one dimensional continuous function, in case of discrete domain
in vectors in continuous domain, one dimensional function, two dimensional function
and three dimensional function.

(Refer Slide Time: 46:47)

Next, we look into some of the typical examples then computation of these things will be
absolutely clear. The first example will be let us consider a vector X as 1 2 3 and Y as 4
5 6. So, basically X and Y are they are vectors they each of they belong to three
dimensional real spaces. So, we compute the metric between the vectors X and Y, we
compute the norm of X and norm of Y and we like to compute the inner product between
X and Y.

So, what is the metric between X and Y it will be simply summation under root
summation between 1 2 3 i is equal to 1 2 3 because there are three elements, x i y i
square of that we just replace the you know write down the definition

So, this will be under root 1 minus 4 square of that plus 2 minus 5 square of that plus 3
minus 6 square of that. So, what will be getting is minus 3 square; so it is 9 plus minus 3
square so it will be 9 plus minus 3 square; so it will be 9 so you will be getting root over
27 or 3 root 3 that is the metric between the vectors X and Y. So, it is the distance
between the vectors two vectors X and Y.

Next, we define what is norm of X, if you look into the definition of norm, it is nothing
but this is for general definition of a norm of n dimensional vector X, it is under root
summation of x i square but the index i runs from 1 to n.

So, in our for this particular example, it is a three dimensional space. So, it will be under
root of 1 square plus 2 square plus 3 square, it will be 1 plus 4 plus 9; so it will be root
over 14 units

Similarly, we can go for norm of vector Y and this will be 4 square plus 5 square plus 6
square. So, this will be root over 16 plus 25 plus 36 and this will be 16 plus 25 is 41. So,
it will be root over 77 units so that will be the norm of Y.
(Refer Slide Time: 50:32)

And similarly one can compute the inner product between X and Y. The inner product
between X and Y, if we write the general formula it will be summation of x i y i, where
the index i goes form 1 to n that means both X and Y are n dimensional vector.

Now, if we look into our example, so this will be x one x 1 y 1 plus x 2 y 2 plus x 3 y 3.
So, it will be 1 into 4 plus 2 into 5 plus 3 into 6 it will be 4 plus 10 plus 18 so as you 14
plus 18 this is of 32 units.

That is how, whenever this simple example gives that if a vector is given finite
dimensional vectors -three dimensional vector, four dimensional vector, then you will be
these are the simple mathematical operations by which one can compute the norm inner
product and the metric between the vectors.

Next, we take up an example of a continuous domain. So, the earlier example is for the
discrete domain in case of vectors next we take up with the for the continuous domain.
Let us say f of x y is 1 plus 2 x plus 2 y and g of x plus y is x minus y and in both the
cases x and y, they are varying between 0 and 1. So, the limits of x and y from 0 and 1.

So, let us look into how will you get the metric between our idea, what is the metric
between f and g, what is the norm of f, what is the norm of g and what is the inner
product between f and g?
Let us first look into what is a norm since it is a two dimensional function then this norm
is nothing but represented by a double integral. So, norm of f square of that is nothing
but integral one integral over x from 0 to 1 another integral over y from 0 to 1 f x y
square of that d x d y. So, this will be basically from 0 to 1 from 0 to 1 1 plus 2 x plus 3 y
square d X d y.

So, basically you have to carry out this integral and find out the answer, so if we really
carry out this integral, so let us see what we get.

(Refer Slide Time: 53:50)

Norm of f square is that 0 to 1 integral 0 to 1 0 plus 2 x plus 3 y square of that d X d y

just open up this square. So, it will be 1 a square plus b square plus c square plus 2 x y so
it will 4 x plus 3 into 2 6 to 12 x y plus 6 y d X d y.

So, d x d y 0 to 1 0 to 1 plus 4 x square d X 0 to 1 d y 0 to 1 plus 9 0 to 1 d X y square d

y 0 to 1 plus 4 x d X 0 to 1 0 to 1 plus 12 0 to 1 x d X 0 to 1 y d y plus 6 y d y 0 to 1
multiplied by d x 0 to 1.

So, the first integral will be 1 multiplied by 1 second integral will be 4 1 upon 3
multiplied by 1 9 into 1 into 1 upon 3 plus 4 into 1 upon 2 into 1 plus 12 into 1 upon 2
into 1 upon 2 plus 6 into 1 upon 2 multiplied by 1.

So, you will be getting 1 plus 4 by 3 plus 3 plus 2 plus 3 plus 3 three 3 3 6 6 2 8 8 3 11
11 1 12 12 plus 4 by 3 so you will be getting a 4 by 3. So, this the norm of the continuous
function f likewise you have to carry out the other parts. and in the I will stop here in the
next class will be solving this problem completely.

Thank you very much.