Distributed Parameters. ECX 5241. Academic year 2003. Prepared by D.A.Mangala Abeysekara. ASSIGNMENT NO.1. ASSIGNMENT NO.2. ASSIGNMENT NO.3. ASSIGNMENT NO.4. ASSIGNMENT NO.1. Symmetry in the physical universe Coordinates systems Operators in vector calculus The theorem of gauss

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ASSIGNMENT NO.1 • Symmetry in the physical universe • Coordinates systems • Operators in vector calculus • The theorem of gauss • The theorem of Stokes

Symmetry in the physical universe • Most of the problems in the physical universe exhibit the spherical or cylindrical symmetry. • Examples • the gravity field of the Earth is to the first order spherical symmetric.

The reason for this is that the theory is usually much simpler when one selects a coordinate system with symmetry properties that are the same as the symmetry properties of the physical system that one wants to study. There are two coordinates systems used for study the symmetric system.

Spherical Coordinates • Cylindrical Coordinates Relationship between the Cartesian coordinates and spherical coordinates

Operators in vector calculus There are several operators can be identified in the vector calculus. They are, ·Gradient (f ) ·Divergent(.v ) ·Curl (v )

Gradient • Lets consider the particle, which moved according to the function f from point A to point B.(f Is a function of x and y) • Grad f =

The divergence of a vector field vx dz dy dx Outward flux through the right hand surface perpendicular through the x-axis vx (x+dx, y, z) dydz

Physical meaning of the curl operator The component of curl v in a certain direction is the closed line integral of v along a closed path perpendicular to this direction, normalized per unit surface area.

hyperbolic Equations It can be shown that, if the equation is hyperbolic if B2>4AC, =2A =2A Satisfies this condition. Hence hyperbolic equations have two characteristic given by,

Parabolic Equation It can be shown that the equation is parabolic, if 4AC-B2=0 The solution of the quadric equation, is If =2A, and =-B, the coefficient of , that is,

IRROTTIONAL FLOW Lets consider the two point in the fluid with distance of dr z B dr A y x

E B dz A D dy dx C Les conceder the two points of A and B as a diagonal of the rectangular pipe. We can write the velocity of the point c can be given in the term of the velocity of point A.

Lets investigate the rate of the angular change of the sides of the rectangular pipe. The average rate of rotation about the z axis of the orthogonal line segments AC and AD is The rate of change of the angle CAD (a right angle at time t ) becomes

Now lets consider the rigid body rotation. Thus, the expression Is actually more than just the average rotation of line segments dx and dy about the z axis . It represents for a deformable medium what we maybe consider as the rigid_body angular velocity z about the z axis. This is, Similarly,

Thus, Hence, At this time ,we define irrotational flow as those for which =0 at each point in the flow.

The velocity potential If velocity components at all points in a region of flow can be expressed as continuous partial derivatives of a scalar function (x,y,z,t) thusly In general we can write