$12\mathrm{.}2\mathrm{.}2$ Let $u(x,y)=y^3-3x^{2}y$ and $v(x,y)=x^3-3x{y}^2.$

$(a)$ Show that $\frac{de{l}^{2}u}{del{x}^2}+\frac{de{l}^{2}u}{del{y}^2}=0on{R}^2.$

$(b)$ Show that $\frac{de{l}^{2}v}{del{x}^2}+\frac{de{l}^{2}v}{del{y}^2}=0on{R}^2.$

$(c)$ Show that $\frac{\text{d}\text{e}\text{l}\text{u}}{\text{d}\text{e}\text{l}\text{x}}=\frac{\text{d}\text{e}\text{l}\text{v}}{\text{d}\text{e}\text{l}\text{y}}$ and $\frac{\text{d}\text{e}\text{l}\text{u}}{\text{d}\text{e}\text{l}\text{y}}=-\frac{\text{d}\text{e}\text{l}\text{v}}{\text{d}\text{e}\text{l}\text{x}}on{R}^2.$