(a) Consider the analytic function f(z) = x − 2x 2 2y 2 i(y − 4xy) This function defines two families of level curves u(x, y) = c 1 and v(x, y) = c 2 Sketch or plot these two families of level curves on the same axes (See for example Slides 40 and 41 in the Differentiation notes) Verify that the families are orthogonalGiving us 1 2 (1−cos(1))≤S 1 0 S 0 sin(x) 1(xy)4 dxdy≤1 5512 Find the volume of the solid bounded by x2 2y2 =2;z=0;and xy2z=2 x2 2y2 =2 de nes a vertical cylinder that crosses the xyplane in an ellipse z=0 is the xyplane Since the ellipse in the xyplane given by x2 2y2 =2 does not intersect the line xy=2, the plane xy2z=2 crosses the cylinderFrom Equation (2118), dy/dx = v/u = x/y, and vdxudy=0 (2118) y dy = x dx Integrating, we obtain y^{2} = x^{2} c where c is a constant of integration For the streamline through (0, 5), we have 5^{2}=0c or c=25 Thus, the equation of the streamline is x^{2} y^{2} = 25
Solved 21 Find The Jacobian Of The Transformation From The Chegg Com
