\fADVANCED CALCULATION I SKILADEMI 7 - SOLUTIONS Example1 The expressions for cylindrical coordinates are given by the formulas G el y(p, $,2) = psin AR Find the Jacobian of the transformation and hence the formula for dV in cylindrical coordinates. Example 2 Find the center of mass with density p and with a uniform distnbution of density p if the mass is such that the density passes through the middle of the cylinder. The solution to the problem is in the form I = a M R? where M = pV is the mass used and V the volume of it. The same result is obtained if all the mass M is concentrated on the edge of the cylinder with radius R. Example 3 Consider F : R- - R2 given by F(x, y) = x2i + (1+y?)j. (a) Find the streamlines of the vector field that pass through the point (1,1). (b) Draw or describe how the composition of the streamlines can appear in the same image, i.e., the task in Matlab/Octave, where 0