Consider a sinusoidal coordinate system
(u, w). The transformation of the coordinates cartesian (x, y) to
parabolic coordinates are given by:
u(x,y) = x, q(x, y) = y - a sin (bx), with a and b constants.
(a) Obtaining the inverse transformation, from get the metric in
the sinusoidal system.
(b) Assumes that an observer moves with constant velocity v those
components are
v^x = v and v^y = 0. What is the speed of the observer in the
system (u,w)?
(c) Samples that the speed component v^w it is not
independent of time to weigh that the magnitude of v is constant.
Explain why v^w is not constant despite that the vector v always
points in the same direction and its magnitude is constant.
Consider a sinusoidal coordinate system (u, w). The transformation of the coordinates cartesian (...
11. Consider the parabolic coordinate system (u, v) related to the Cartesian coordi- nates (r, y) by х — 2иv, y — u? — u? for (и, v) € [0, оо) х [0, оо) 1 u = 1, u 2' (a) Sketch in the ry-plane the curves given u = 2. Then sketch in 1 v = 1, v = 2. Shade in the region R the xy-plane the curves given v = 2' bounded by the curves given by...
34. Consider a two-dimensional Cartesian coordinate system and a two-dimensional uv-system with the coordinates related by y = (1/2)(112-U2). In general, Laplace's equation in two dimensions can be written as with ох ду Zi (a) In the xy-plane, sketch lines of constant u and constant v. (b) Express Laplace's equation using the uw-coordinates. (c) Use the method of separation of variables to separate Laplace's equation in the v-system and obtain the general solution for ų'(u, u). 34. Consider a two-dimensional...
Problem 4 The parabolic cylindrical coordinates , , u) are related to the Cartesian coordinat es (x,y, z) by the transformat ion a) The line-element in Cartesian coordinates is given by d82-dr2+dy2+d22-De- termine the lne-elemen expressed in terms of the parabolic cylindrical coordinates b) Given F = 211,2) of the equation V22) F e where F depends only nu. Find the explicit form F-x F kF c) Solve the equation fro b) to find F Useful formulas: Given any ort...
Coordinates and metric 6.1 Hartle's Equation (7.7) shows an example where finite coordinate distance corresponds to infinite proper distance. In his Problem 7.1 you work out the proper distance from r' = 0 to, say, r1 at fixed d. It's a2 A.S=/ds= dr' r/2 s' lo which is infinitely greater than the coordinate distance of 1 Let's do the opposite, look at an infinite coordinate distance which is a finite proper distance. In Cartesian coordinates and flat space it takes...
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BOX 5.1 The Polar Coordinate Basis Consider ordinary polar coordinates r and 0 (see figure 5.3). Note that the distance between two points with the same r coordinate but separated by an infinitesimal step do in 0 is r do (by the definition of angle). So there are (at least) two ways to define a basis vector for the direction (which we define to be tangent to the r = constant curve): (1) we could define a basis vector es...
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Consider the following class Point which represents a point in Cartesian coordinate system. The class has two instance variables x and y which represent its position along the x-axis and y-axis. The constructor of the class initializes the x and y of the point with the specific value init; the method move() moves the point as specified by its parameter amount. Using the implementation of the class Point information and write its specification. The specification of the class includes the...
MARK WHICH STATEMENTS BELOW ARE TRUE, USING THE FOLLOWING, Consider Vf(x, y, z) in terms of a new coordinate system, x= x(u, v, w), y=y(u, v, w), z=z(u, v, w). Let r(s) = x(s) i+y(s) + z(s) k be the position vector defining some continuous path as a function of the arc length. Similarly for the other partial derivatives in v and w. For spherical coordinates the following must also be true for any points, x = Rsin o cose,...