MAT223H5 Lecture 5: Equations, Matrices, and Transformations

Find an equation for the ellipse x2 / 4 + y2 / 16 = 1 in parametric form (x(t) y(t)). (2 sin t, 4 cost), 0 le t le pi (2sin t, 4cost), 0 le t le 2pi (2sin2t, 4cos3t), 0 le t le 2pi / 3 (2 cos2t, 4 sint), 0 le t le 2 pi (2sin 2t, 4cost), 0 le t le pi

Let W = [0, 2] times [0, 2] times [0, 2] = {(x, y, z)|0 le x le 2, 0 le y le 2, 0 le le 2}, and let S be the boundary of W. Suppose the vector field F is defined by F(x, y, z) = 10(y - x)i + (28x - xz)j + (xy - 8z/3)k. Evaluate the flux s F middot ndS.

Let T : D* rightarrow D be defined by T (s, t) = (s/3 - 2t / 3, (s + t) / 3), where D* = {(s, t) : 2 le s le 4, -3 le t le 0. Evaluate