Application of mathematics
The content and character of mathematics needed in applications are changing rapidly. Differential equations are of fundamental importance because many physical laws and relations appear mathematically in the form of such equations. Meridians and parallels on the globe are orthogonal trajectories of each other. The Gauss elimination is a standard method for solving linear systems. Rank of matrix answers questions of existence and uniqueness of solution of a system of linear equations. Determinants were introduced for solving linear systems. Cramer’s rule gives solutions of linear systems as quotient of determinants. Eigen value problems are useful to the Engineers and Physicist. Complex square matrices are important in quantum mechanics. Inner products are used in work done by a force. Vector products are used in velocity of a rotating body. Scalar triple product is the volume of the parallelopiped. The volume of the tetrahedron is 1/6 of that of the parallelopiped. Curves play an important role in mechanics as paths of moving bodies. The divergence measures outflow minus inflow. Double integrals are used to calculate (i) Area (ii) Volume (iii) Centre of gravity (iv) Moments of inertia. Fourier series is used in solving ordinary and partial differential equations. Most problems in fluid mechanics lead to partial differential equations. Complex analysis is used in (i) Heat conduction (ii) Fluid flow (iii) Electrostatics. Conformal mapping can help in solving boundary value problems. Cauchy’s integral formula which is useful for evaluating integrals. Residue theorem yields a very elegant and simple method for evaluating certain classes of complicated real integrals. Numerical methods are used to solve problems on computers or calculators. Trapezoidal rule and Simpson’s rule are used for numerical integration. Fitting straight line or parabolas are of great practical importance. They use the method of least squares by Gauss. Runge - kutta method is of great practical importance. Linear programming solves problems in production, airplane scheduling. Simplex method is an iterative method. Graphs and diagraphs are powerful tools in electrical and civil engineering, communication networking, industrial management, operations research, computer science, economics, management science and in other areas. Trees can be used to exhibit, organize, or analyze electrical networks. Mathematical statistics has various engineering applications for instance in testing materials, robotics and automization. For large sets of data, histograms are used. Probability theory provides mathematical models. Permutation and combination help in finding probabilities. The binomial distribution occurs in games of chance. Random variables occurs more frequently in engineering. Samples are selected from populations. Correlation analysis deals with: relation between wear X and wear Y of the front tiers of cars, between grade X and Y of students in mathematics and in physics, respectively, between the hardness X of steel plates in the centre and the hardness Y the edges of plates, etc.
Prof. Yashpal Sharma
Mathematics Dept.
Nothing is permanent in this world, not even our troubles. . . . !
The most wasted day in life, is the day in which, we have not laughed.
When any one lies to you..its not his/her fault
Actually its yours because
you didn't give him/herenough space to tell the truth.
Wo khud hi tay karta hai manzil asmano ki,
Parindon ko nhi di jati taleem udano ki,
rakhta hai to hausala asmaa chune ka,
usko nahin hoti parwah gir jane ki..
Saurabh Kumar Srivastava I Lecturer I CSIT Deptt. I
MGM's CoET, Noida I