向量分析求解

1. 每個分量各自作偏微分便可。

A = (3xy - y^2) i + cosxy j + (x + 3y) k

Ax = 3y i - ysinxy j + k

Ay = (3x - 2y) i - xsinxy j + 3k

Axx = (Ax)x = -y^2cosxy j

Axy = (Ax)y = 3i - (sinxy + xycosxy) j

Ayy = (Ay)y = -2i - x^2cosxy j


2. fA = (x^2y^2z^2) i + (xy^3z^2) j + (x^2y^2z^2) k

(fA)xxy = [(2x^2yz^2) i + (3xy^2z^2) j + (2x^2yz^2) k]xx

= [(4xyz^2) i + (3y^2z^2) j + (4xyz^2) k]x

= (4yz^2)(i + k)

所以在(1, 1, 1)的值為 = 4(1)(1)^2(i + k) = 4i + 4k