ASSIGNMENT-1 GURU JAMBHESHWARUNIVERSITY OF SCIENCE & TECHNOLOGY, HISAR DIRECTORATE OF DISTANCE EDUCATION
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1 ASSIGNMENT-1 GURU JAMBHESHWARUNIVERSITY OF SCIENCE & TECHNOLOGY, HISAR Paper Code: MAL-631 Nomenclature of Paper:TOPOLOGY Q.1 Prove that in a topological space, E E d E E X. Q.2Characterize topology in terms of Kuratowski closure operator. Q.3 Show that a topological space X is connectediff no non-empty proper subset which is both open and closed.
2 ASSIGNMENT-2 GURU JAMBHESHWARUNIVERSITY OF SCIENCE & TECHNOLOGY, HISAR Paper Code: MAL-631 Nomenclature of Paper:TOPOLOGY Q.1 Show that a topological space is compact iff any family of closed sets having FIP has a non-empty intersection. Q.2 Prove that every second countable space is separable. Show by means of anexample that the converse need not be true. Q.3Prove that a topological space is completely normal iff every subspace ofx is normal.
3 GURU JAMBHESHWARUNIVERSITY OF SCIENCE & TECHNOLOGY, HISAR Paper Code: MAL-632 Nomenclature of Paper:Partial Differential Equations (Assignment-I) (iii) (iv) 1. State and prove mean value theorem for Laplace s equation. 2. State and prove Harnack s inequality for harmonic functions. 3. Find the solution of boundary value problem u in B (, r) ; u g on B(, r) using Green s function. Paper Code: MAL-632 Nomenclature of Paper:Partial Differential Equations (Assignment-II) 1. Derive the D Alembert formula for 1-D wave equation. n 2. Find the solution of the Hamilton Jacobi equation u H( Du) in R (, ) where H is the Hamilton function, by the method of separation of variables. 3. Solve the partial differential equation u u f in n R where 2 n f C ( R ) using Fourier transform. t
4 GURU JAMBHESHWARUNIVERSITY OF SCIENCE & TECHNOLOGY, HISAR Paper Code: MAL-633 Nomenclature of Paper:MECHANICS OF SOLIDS-I (Assignment-I) (v) (vi) 1. Define isotropic tensors ij and ijk. Also show that ijm klm 2. Show that the divergence and curl of a vector field is a scalar and vector respectively. 3. Prove that principal strains are real and principal directions of strain are orthogonal to each other. ik jl il jk Paper Code: MAL-633 Nomenclature of Paper:MECHANICS OF SOLIDS-I (Assignment-II) 1. Show that stress tensor is symmetric. 2. Discuss Mohr s diagram for finding maximum shearing stresses. Let the stress components at a point P are given by xx 2, yy 4, zz, xy, xz 3, yz where is a constant. Determine the stress at the point P on a plane with normal in the direction (2, 2, 1). Also determine the normal and shearing stress at P on this plane. 3. Define Poisson s ratio and Modulus of Rigidity for an isotropic elastic medium. Prove that if 1 / 2, then E /3,, K, e jj u i, i 1 and Hooke's law is ij 2 eij ij. Show that in thiscase u j, ij 3 x i
5 GURU JAMBHESHWAR UNIVERSITY OF SCIENCE & TECHNOLOGY, HISAR Paper Code: MAL-634 Total Marks = 15 Nomenclature of Paper: Fluid Mechanics (Assignment-I) (vii) (viii) 1. Define vorticity equation for the flow of incompressible fluid in the presence of conservative body force. 2. Derive equation of continuity and hence deduce the condition for incompressible fluid flow. 3. Derive equation of streamline, pathline and streakline. Find equation of streamline, pathline for the flow field defined by q ( x 2 (1 2t), yt 2,) Paper Code: MAL-634 Total Marks = 15 Nomenclature of Paper: Fluid Mechanics (Assignment-II) 1. Two sources are placed at (a, ) and (- ai, ) and two sinks are placed at (, ia), (,- a). All the sources and sinks are of equal strengths, then show that the circle through these four points is a stream line. 2. State and prove Blasius theorem for force and moment on a rigid body placed in a flowing fluid and find force in a cylinder of radius 2 c.m. in the stream flow past with velocity 5 m/sec. 3. State and prove Milne- Thomson circle theorem and apply it to find the image of a source relative to a circle.
6 GURU JAMBHESHWARUNIVERSITY OF SCIENCE & TECHNOLOGY, HISAR Paper Code: MAL-635 Nomenclature of Paper: Advanced Discrete Mathematics (Assignment-I) (ix) (x) 1. Explain AND, OR and NOT logical operations of propositions. Construct the truth table of ( p ( q r)) ( p ( q r)) where p, q, r are the negations of p, q and r. 2. Define a lattice. Give an example. Draw the Hasse diagram of D 3, D 6, D State and prove Idempotent laws, Boundedness laws and Involution law of a Boolean algebra. Paper Code: MAL-635 Nomenclature of Paper: Advanced Discrete Mathematics(Assignment-II) either in person or through Speed Post 1. State and prove First theorem of graph theory. Also prove that there must be an even number of vertices of odd degree in a given graph. Is there a graph with 8 vertices if degree 2, 2, 3, 6, 5, 7, 8, 4? 2. Define Euler s path and circuit. Prove that if a graph has an Euler circuit, then every vertex of the graph has even degree. Is the converse true? If not, give an example. 3. Prove that a connected graph is a tree if only if there is unique and simple path between any two vertices.
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