compute the outward flux of Fxyz xiyjzkx2y2z212 through the

compute the outward flux of F(x,y,z) = xi+yj+zk/(x^2+y^2+z^2)^1/2 through the ellipsoid 4x^2 + 9y^2 +6z^2 = 36

Solution

F( x, y) = (16x + 4y)i + (4x + 2y) j M = -16x + 4y and N = 4x + 2y Take the partial derivative in terms of x and y. (16x + 4y)y = 4 = (4x + 2y)x The field is conservative. Mdx = 8x 2 + 4yx + k ( y) 8x 2 ( + 4yx + k ( y))y = 4x + k \'( y) = 4x + 2y k \'( y) = 2y f ( x, y) = 8x 2 + 4yx + y 2 B. F( x, y) = 8yi 7xj (8y)y = 8 (7x)x = 7 Not conservative. C. F( x, y,z) = 8xi 7yj + k (7y) x (8x) y = 0 It is conservative. F=(8x, 7y, 0) + (0,0,1) f ( x, y,z) = 4x 2 7 2 y 2 + z D. F = (8sin y)i + (8y 8x cos y) j N x M y = 8 cos y (8 cos y) = 0 f = 8x sin y + 4y 2 E. F( x, y,z) = 8x 2 i + 4y 2 j + 1z 2 k f = 8 3 x 3 + 4 3 y 3 +