INTRODUCTION TO PARTIAL DIFFERENTIATION

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CONTENT

PARTIAL DIFFERENTIATIONMATRIX AND DETERMINANTSEIGEN VALUE AND EIGEN VECTORS

INTRODUCTION TO PARTIAL DIFFERENTIATIONIn engineering, it sometimes happens that the variation of one quantity depends on changes taking place in two, or more, other quantities. For example, the volume V of a cylinder is given by V=r2h. The volume will change if either radius r or height h is changed. The formula for volume may be stated mathematically as V=f (r, h) which means V is some function of r and h. Some other practical examples include:

time of oscillation, t=2l/g i.e. t = f (l, g).torque T =I, i.e. T =f (I, ).(iii) pressure of an ideal gas p= mRT/V i.e. p=f (T, V).

When differentiating a function having two variables, one variable is kept constant and the differential coefficient of the other variable is found with respect to that variable. The differential coefficient obtained is called a partial derivative of the function.

INTRODUCTION TO MATRIX AND DETERMINANTSIntroduction to Matrices You can view a matrix simply as a generalization of avector, where we arrange numbers in both rows and columns. Let's keep the number of rows and columns arbitrary, lettingmbe the number of rows andnthe number of columns. We refer to such a matrix as anmnmatrix.The arrangement of a matrix in rows and columns is more than just to make it look pretty. The structure of a matrix allows us to define a fundamental operation on matrices:multiplication. This multiplication forms the basis of linear algebra. In particular, this matrix multiplication allows matrices to representlinear transformations(or linear functions) that transform vectors into other vectors. (A simple example of a linear transformation is the rotation of a vector.) Other uses of matrices involvecalculating their determinant.

Introduction to Determinants

For any square matrix of order 2, we have found a necessary and sufficient condition for invertibility. Indeed, consider the matrix

The matrixAis invertible if and only if . We

called this number thedeterminantofA. It is clear from this, that we would like to have a similar result for bigger matrices (meaning higher orders). So is there a similar notion of determinant for any square matrix, which determines whether a square matrix is invertible or not?

In order to generalize such notion to higher orders, we will need to study the determinant and see what kind of properties it satisfies. First let us use the following notation for the determinant

INTRODUCTION TO EIGEN VALUES AND EIGEN VECTORS For any square matrix A, if A-I=0 then the value of is called eigen values of the matrix.For each eigen value, if (A-I)X=0, then the matrix X is called the eigen vector of that eigen values .Eigen values and eigen vectors are very important in some engineering field .For example ,in civil engineering, Eigen vector are used in mass calculation while making structures in blocks.

QUESTIONS AND ANSWERSQuestion.1 Find (a) AB (b) BA by using matrix.

Solution.1

Question.2 Find the eigen values of

two eigen values: 1, 2 Note: The roots of the characteristic equation can be repeated. That is, 1 = 2 == k. If that happens, the eigen value is said to be of multiplicity k.Solution.2

Question 3: Laplace Equation

Laplace:

Substitute with the Central divided differences and assuming that Dx = Dy = h

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xy

Finite Difference Gridi,ji-1,ji+1,ji,j+1i,j-1

-41111

i,j

At i and j:

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REAL WORLD APPLICATION ON MATRICES AND DETERMINANTS

BERMUDA TRIANGLEThe Bermuda Triangle is a large triangular region in the Atlantic ocean. Many ships and airplanes have been lost in this region. The triangle is formed by imaginary lines connecting Bermuda, Puerto Rico, and Miami, Florida. Use a determinant to estimate the area of the Bermuda Triangle.

EWNSMiami (0,0)Bermuda (938,454)Puerto Rico (900,-518)

...

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SOLUTIONThe approximate coordinates of the Bermuda Triangles three vertices are: (938,454), (900,-518), and (0,0). So the area of the region is as follows:

Hence, area of the Bermuda Triangle is about 447,000 square miles.

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MATRICES APPLIED TO ELECTRIC CIRCUITS

There are two closed loops in the above circuit. loop 1: e1, R1 and R3 and loop 2: e2, R2 and R3. e1 and e2 are sources of voltages. R1, R2 and R3 are resistors. i1 is the current flowing across R1 and i2 is the current flowing across R2. We now apply Kirchhoff's law to each loop.

loop 1: e1 = R1 i1 + R3 (i1 - i2)

loop 2: e2 = R2 i2 + R3 (i2 - i1)

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Question: If e1, e2, R1, R2 and R3 are known, how do you calculate i1 and i2? This circuit is simple and involves only two equations. However electric circuits can be much more complicated that the one above and matrices are suitable to answer the above question. Let us group like terms in the abovesystem of equations

e1 = i1 (R1 + R3) - i2 R3

e2= -i1 R3 + i2(R2 + R3)

and then write it in matrix form as follows

The above is a matrix equation that may be solved using any known method to solve systems of equations. Let e, R and i be matrices given by

The solution to the above matrix equation is given by

where R-1is the inverse matrix of R and is given by.

GRAPHIC USES OF MATRIX MATHEMATICS

Graphic software uses matrix mathematics to process linear transformations to render images. A square matrix, one with exactly as many rows as columns, can represent a linear transformation of a geometric object. For example, in the Cartesian X-Y plane, the matrix reflects an object in the vertical Y axis. In a video game, this would render the upside-down mirror image of a castle reflected in a lake.If the video game has curved reflecting surfaces, such as a shiny silver goblet, the linear transformation matrix would be more complicated, to stretch or shrink the reflection.

REAL WORLD APPLICATIONS ON EIGEN VALUE AND EIGEN VECTOR

Eigenvectors and eigen values are important for understanding the properties ofexpander graphs, which I understand to have several applications in computer science (such as derandomizing random algorithms). They also give rise to agraph partitioning algorithm.

PRuler is an iPhone app that lets you measure objects using a credit card and your iPhone camera. I'm told it uses a Singular Value Decomposition (which is very closely related to eigenvalues and eigenvectors)

Example 1

Example 2Plot the sequence of approximations of the maximum eigenvector for the matrixStarting with the initial vector:These are shown in Figure 1. The sequence of approximations is shaded from blue to red. The last plotted red vector is quite close to the actual eigenvector of 9.54 (0.763, 0.646)T(9.54 being the corresponding eigen value).

Figure 1. The sequence of approximations of the maximum eigenvector with the initial vectorv= (-3.1, 5.2)Tin black.

REAL WORLD APPLICATION ON PARTIAL DIFFERENTIATION

Shape Processing using PDEsShape processing refers to operations such as denoising, fairing, feature extraction, segmentation, simplification, classification, and editing. Such operations are the basic building blocks of many applications in computer graphics, animation, computer vision, and shape retrieval.Many shape processing operations can be achieved by means ofpartial differential equationsor PDEs. The desired operation is described as a (set of) PDE(s) that act on surface information, such as area, normals, curvature, and similar quantities. PDEs are a very attractive instrument: They allow complex manipulations to be described precisely, compactly, and measurably, and come with efficient and effective numerical methods for solving them.We present several applications of PDEs in shape processing.

Partial Derivative in Economics:In economics the demand ofquantity and quantity supplied are affected by severalfactors such as selling price, consumer buying power andtaxation which means there are multi variable factors that affect the demandand supply. In economics marginal analysis is used to find outor evaluate the change invalue of a function resulting from 1-unit increase inone of its variables. For example Partial derivative is used in marginal Demand to obtain condition for determining whether two goodsare substitute or complementary. Two goods aresaid to besubstitute goods if an increase in the demandfor either result in a decrease for the other. While two goods are said to be complementary goods if a decrease of either result in a decrease in the demand. Example ofcomplementary goods are mobile phones andphone lines. If there is more demand formobilephone, it will lead to more demand for phone line too.

In image processing edge detection algorithm is usedwhich uses partial derivatives to improve edge detection. Grayscale digital images can be considered as 2D sampled points of a graph of a function u(x,y) where the domain of the function is the area ofthe image.Partial Derivative in Engineering:

From learning the applications of the eigen vectors and eigen values, we came to know that the eigen vectors and values is having very much importance in engineering and in other fields also, but the thing is how we connect that concept. Even these are very less applications on eigen values and vectors and a lot of work can be done on that because the innovation is limitless.Conclusion

REFERENCE BOOKAUTHORTITTLE PUBLICATIONSITEERWIN KREYSZIGADVANCED ENGGINEERING MATHEMATICSJOHN WILEY & SONS, INC.www.wileyplus.com

JOHN BIRDHIGHER ENGGINEERING MATHEMATICSJOHN BIRD. PUBLISHED BY ELSEVIER LTD.www.elsevier.comwww.bookaid.orgwww.sabre.org JOHN BIRDENGGINEERING MATHEMATICSLIBRARY OF CONGRESS CATALOGUING IN PUBLICATION www.books.elsevier.com MICHAEL BATTYESSENTIAL ENGGINEERING MATHEMATICSDOVER PUBLICATIONSwww.bookboon.com H.K. DASSADVANCED ENGGINEERING MATHEMATICSS CHAND PUBLICATIONShttps://www.schandpublishing.com

REFERENCE SITESwww.wileyplus.comwww.elsevier.comwww.sabre.orghttp://math.stackexchange.com/questions/82413/practical-applications-of-eigenvalues-eigenvectors-in-computer-sciencehttp://www.springer.com/in/book/9780857297839https://www.chalmers.se/en/departments/cee/research/Graduate-education/graduate-courses-and-course-descriptions/Pages/Mathematical-modelling-in-civil-engineering-applications.aspx