高専の数学解説

第6章 6.20

(1)

u=log⁡(x3+y3+z3−3xyz)u = \log(x^3 + y^3 + z^3 - 3xyz)

ux=3x2−3yzx3+y3+z3−3xyzu_x = \frac{3x^2 - 3yz}{x^3 + y^3 + z^3 - 3xyz}

(2)

uy=3y2−3xzx3+y3+z3−3xyzuz=3z2−3xyx3+y3+z3−3xyzu_y = \frac{3y^2 - 3xz}{x^3 + y^3 + z^3 - 3xyz} \qquad u_z = \frac{3z^2 - 3xy}{x^3 + y^3 + z^3 - 3xyz}

ux+uy+uz=3(x2+y2+z2−xy−yz−zx)x3+y3+z3−3xyz=3(x2+y2+z2−xy−yz−zx)(x+y+z)(x2+y2+z2−xy−yz−zx)u_x + u_y + u_z = \frac{3(x^2 + y^2 + z^2 - xy - yz - zx)}{x^3 + y^3 + z^3 - 3xyz} = \frac{3(x^2 + y^2 + z^2 - xy - yz - zx)}{(x + y + z)(x^2 + y^2 + z^2 - xy - yz - zx)}

=3x+y+z= \frac{3}{x + y + z}

解説: r31bn1z