6.4 全微分

1.

(a) $${\nabla f = (3x^{2},2y), df = 3x^{2}dx + 2y dy}$$

(b) $${\nabla f = (6x-y,-x+1), df = (6x-y)dx -(x-1)dy}$$

(c) $${\nabla z = (2xy^{-2}, -2x^{2}y^{-3}), dz = 2xy^{-2}dx - 2x^{2}y^{-3}dy}$$

(d) $${\nabla z = (2xy, x^{2}), dz = 2xydx + x^{2}dy}$$

(e) $${\nabla z = (e^{x}\cos{y}, -e^{x}\sin{y}), dz = e^{x}\cos{y}dx - e^{x}\sin{y}dy}$$

2.

(a) $${3x + 2y - z = 4, (x,y,z) = (1,1,1) + (3,2,-1)t}$$

(b) $${x + 2y - z = 3, (x,y,z) = (2,1,1) + (1,2,-1)t}$$　

(c) $${3x + 5y - z = 4, (x,y,z) = (1,1,4) + (3,5,-1)t}$$