The displacement of the object is symmetrical about the $Oz$ axis. Attempt to find the displacement components and strain components in Cartesian coordinates is ().

Options:
A. $$
\left. \begin{array} {l} {{u={\frac{v_{r}} {r}} y=g ( r, z ) x}} \\ {{v={\frac{v_{r}} {r}} x=g ( r, z ) y}} \\ {{w=w ( x, y )}} \\ \end{array} \right\} \begin{array} {l} {{\varepsilon_{x}=g ( r, z )+{\frac{x y} {r}} {\frac{\partial g ( r, z )} {\partial r}}, \quad\varepsilon_{x y}={\frac{y^{2}} {r}} {\frac{\partial g ( r, z )} {\partial r}}}} \\ {{\varepsilon_{y}=g ( r, z )+{\frac{x^{2}} {r}} {\frac{\partial g ( r, z )} {\partial r}}, \quad\varepsilon_{y z}={\frac{1} {2}} {\frac{\partial g ( r, z )} {\partial z}}+{\frac{1} {2}} {\frac{\partial w} {\partial z}} {\frac{x} {r}}}} \\ {{\varepsilon_{z}={\frac{\partial w} {\partial x}} \, \quad\varepsilon_{z x}={\frac{1} {2}} {\frac{\partial g ( r, z )} {\partial z}}+{\frac{1} {2}} {\frac{\partial w} {\partial y}} {\frac{y} {r}}}} \\ \end{array} \}
$$
B. $$
\left. \begin{array} {l} {{u={\frac{u_{r}} {r}} x=f ( r, z ) x}} \\ {{v={\frac{u_{r}} {r}} y=f ( r, z ) y}} \\ {{w=w ( r, z )}} \\ \end{array} \right\} \begin{array} {l} {{\varepsilon_{x}=f ( r, z )+{\frac{x^{2}} {r}} {\frac{\partial f ( r, z )} {\partial r}}, \quad\varepsilon_{x y}={\frac{x y} {r}} {\frac{\partial f ( r, z )} {\partial r}}}} \\ {{\varepsilon_{y}=f ( r, z )+{\frac{y^{2}} {r}} {\frac{\partial f ( r, z )} {\partial r}}, \quad\varepsilon_{y z}={\frac{1} {2}} {\frac{\partial f ( r, z )} {\partial z}}+{\frac{1} {2}} {\frac{\partial w} {\partial r}} {\frac{y} {r}}}} \\ {{\varepsilon_{z}={\frac{\partial w} {\partial z}} \, \quad\varepsilon_{z x}={\frac{1} {2}} {\frac{\partial f ( r, z )} {\partial z}}+{\frac{1} {2}} {\frac{\partial w} {\partial r}} {\frac{x} {r}}}} \\ \end{array} \}
$$
C. $$
\left. \begin{array} {l} {{u={\frac{u_{r}} {x}} y=o ( r, z ) x}} \\ {{v={\frac{u_{r}} {x}} x=o ( r, z ) y}} \\ {{w=w ( x, r )}} \\ \end{array} \right\} \begin{array} {l} {{\varepsilon_{x}=o ( r, z )+{\frac{x^{2}} {r}} {\frac{\partial o ( r, z )} {\partial r}}, \quad\varepsilon_{x y}={\frac{x y} {r}} {\frac{\partial o ( r, z )} {\partial r}}}} \\ {{\varepsilon_{y}=o ( r, z )+{\frac{y^{2}} {r}} {\frac{\partial o ( r, z )} {\partial r}}, \quad\varepsilon_{y z}={\frac{1} {2}} {\frac{\partial o ( r, z )} {\partial z}}+{\frac{1} {2}} {\frac{\partial w} {\partial x}} {\frac{y} {r}}}} \\ {{\varepsilon_{z}={\frac{\partial w} {\partial x}} \, \quad\varepsilon_{z x}={\frac{1} {2}} {\frac{\partial o ( r, z )} {\partial z}}+{\frac{1} {2}} {\frac{\partial w} {\partial x}} {\frac{x} {r}}}} \\ \end{array} \}
$$
D. $$
\left. \begin{array} {l} {{u=u_{r} z=k ( r, x ) z}} \\ {{v=v_{r} y=k ( r, x ) y}} \\ {{w=w ( x, z )}} \\ \end{array} \right\} \begin{array} {l} {{\varepsilon_{x}=k ( r, x )+{\frac{x^{2}} {r}} {\frac{\partial k ( r, x )} {\partial r}}, \quad\varepsilon_{x y}={\frac{x y} {r}} {\frac{\partial k ( r, x )} {\partial r}}}} \\ {{\varepsilon_{y}=k ( r, x )+{\frac{y^{2}} {r}} {\frac{\partial k ( r, x )} {\partial r}}, \quad\varepsilon_{y z}={\frac{1} {2}} {\frac{\partial k ( r, x )} {\partial z}}+{\frac{1} {2}} {\frac{\partial w} {\partial r}} {\frac{y} {r}}}} \\ {{\varepsilon_{z}={\frac{\partial w} {\partial z}} \, \quad\varepsilon_{z x}={\frac{1} {2}} {\frac{\partial k ( r, x )} {\partial z}}+{\frac{1} {2}} {\frac{\partial w} {\partial r}} {\frac{x} {r}}}} \\ \end{array} \}
$$
E. $$
\left. \begin{array} {l} {{u={\frac{x y} {r}} y=m ( x, z ) x}} \\ {{v={\frac{v_{r}} {r}} x=m ( x, z ) y}} \\ {{w=w ( y, z )}} \\ \end{array} \right\} \begin{array} {l} {{\varepsilon_{x}=m ( x, z )+{\frac{x^{2}} {r}} {\frac{\partial m ( x, z )} {\partial x}}, \quad\varepsilon_{x y}={\frac{x y} {r}} {\frac{\partial m ( x, z )} {\partial x}}}} \\ {{\varepsilon_{y}=m ( x, z )+{\frac{y^{2}} {r}} {\frac{\partial m ( x, z )} {\partial x}}, \quad\varepsilon_{y z}={\frac{1} {2}} {\frac{\partial m ( x, z )} {\partial z}}+{\frac{1} {2}} {\frac{\partial w} {\partial x}} {\frac{y} {r}}}} \\ {{\varepsilon_{z}={\frac{\partial w} {\partial y}} \, \quad\varepsilon_{z x}={\frac{1} {2}} {\frac{\partial m ( x, z )} {\partial z}}+{\frac{1} {2}} {\frac{\partial w} {\partial x}} {\frac{x} {r}}}} \\ \end{array} \}
$$
F. $$
\left. \begin{array} {l} {{u={\frac{u_{y}} {r}} x=n ( r, y ) x}} \\ {{v={\frac{u_{y}} {r}} y=n ( r, y ) y}} \\ {{w=w ( r, x )}} \\ \end{array} \right\} \begin{array} {l} {{\varepsilon_{x}=n ( r, y )+{\frac{x^{2}} {r}} {\frac{\partial n ( r, y )} {\partial r}}, \quad\varepsilon_{x y}={\frac{x y} {r}} {\frac{\partial n ( r, y )} {\partial r}}}} \\ {{\varepsilon_{y}=n ( r, y )+{\frac{y^{2}} {r}} {\frac{\partial n ( r, y )} {\partial r}}, \quad\varepsilon_{y z}={\frac{1} {2}} {\frac{\partial n ( r, y )} {\partial z}}+{\frac{1} {2}} {\frac{\partial w} {\partial y}} {\frac{x} {r}}}} \\ {{\varepsilon_{z}={\frac{\partial w} {\partial x}} \, \quad\varepsilon_{z x}={\frac{1} {2}} {\frac{\partial n ( r, y )} {\partial z}}+{\frac{1} {2}} {\frac{\partial w} {\partial y}} {\frac{y} {r}}}} \\ \end{array} \}
$$
G. $$
\left. \begin{array} {l} {{u=u_{r} y=x ( z, y ) y}} \\ {{v=v_{r} x=x ( z, y ) x}} \\ {{w=w ( r, y )}} \\ \end{array} \right\} \begin{array} {l} {{\varepsilon_{x}=x ( z, y )+{\frac{x^{2}} {r}} {\frac{\partial x ( z, y )} {\partial z}}, \quad\varepsilon_{x y}={\frac{x y} {r}} {\frac{\partial x ( z, y )} {\partial z}}}} \\ {{\varepsilon_{y}=x ( z, y )+{\frac{y^{2}} {r}} {\frac{\partial x ( z, y )} {\partial z}}, \quad\varepsilon_{y z}={\frac{1} {2}} {\frac{\partial x ( z, y )} {\partial y}}+{\frac{1} {2}} {\frac{\partial w} {\partial r}} {\frac{y} {r}}}} \\ {{\varepsilon_{z}={\frac{\partial w} {\partial r}} \, \quad\varepsilon_{z x}={\frac{1} {2}} {\frac{\partial x ( z, y )} {\partial y}}+{\frac{1} {2}} {\frac{\partial w} {\partial r}} {\frac{x} {r}}}} \\ \end{array} \}
$$
H. $$
\left. \begin{array} {l} {{u={\frac{x y} {r}} x=h ( x, y ) x}} \\ {{v={\frac{x y} {r}} y=h ( x, y ) y}} \\ {{w=w ( x, y )}} \\ \end{array} \right\} \begin{array} {l} {{\varepsilon_{x}=h ( x, y )+{\frac{x^{2}} {r}} {\frac{\partial h ( x, y )} {\partial x}}, \quad\varepsilon_{x y}={\frac{x y} {r}} {\frac{\partial h ( x, y )} {\partial x}}}} \\ {{\varepsilon_{y}=h ( x, y )+{\frac{y^{2}} {r}} {\frac{\partial h ( x, y )} {\partial x}}, \quad\varepsilon_{y z}={\frac{1} {2}} {\frac{\partial h ( x, y )} {\partial y}}+{\frac{1} {2}} {\frac{\partial w} {\partial r}} {\frac{x} {r}}}} \\ {{\varepsilon_{z}={\frac{\partial w} {\partial y}} \, \quad\varepsilon_{z x}={\frac{1} {2}} {\frac{\partial h ( x, y )} {\partial y}}+{\frac{1} {2}} {\frac{\partial w} {\partial r}} {\frac{y} {r}}}} \\ \end{array} \}
$$
I. $$
\left. \begin{array} {l} {{u=u_{z} x=p ( x, y ) x}} \\ {{v=v_{z} y=p ( x, y ) y}} \\ {{w=w ( y, r )}} \\ \end{array} \right\} \begin{array} {l} {{\varepsilon_{x}=p ( x, y )+{\frac{x^{2}} {r}} {\frac{\partial p ( x, y )} {\partial x}}, \quad\varepsilon_{x y}={\frac{x y} {r}} {\frac{\partial p ( x, y )} {\partial x}}}} \\ {{\varepsilon_{y}=p ( x, y )+{\frac{y^{2}} {r}} {\frac{\partial p ( x, y )} {\partial x}}, \quad\varepsilon_{y z}={\frac{1} {2}} {\frac{\partial p ( x, y )} {\partial y}}+{\frac{1} {2}} {\frac{\partial w} {\partial r}} {\frac{x} {r}}}} \\ {{\varepsilon_{z}={\frac{\partial w} {\partial y}} \, \quad\varepsilon_{z x}={\frac{1} {2}} {\frac{\partial p ( x, y )} {\partial y}}+{\frac{1} {2}} {\frac{\partial w} {\partial r}} {\frac{y} {r}}}} \\ \end{array} \}
$$
J. $$
\left. \begin{array} {l} {{u={\frac{u_{y}} {r}} z=q ( r, x ) x}} \\ {{v={\frac{u_{y}} {r}} y=q ( r, x ) y}} \\ {{w=w ( z, x )}} \\ \end{array} \right\} \begin{array} {l} {{\varepsilon_{x}=q ( r, x )+{\frac{x^{2}} {r}} {\frac{\partial q ( r, x )} {\partial r}}, \quad\varepsilon_{x y}={\frac{x y} {r}} {\frac{\partial q ( r, x )} {\partial r}}}} \\ {{\varepsilon_{y}=q ( r, x )+{\frac{y^{2}} {r}} {\frac{\partial q ( r, x )} {\partial r}}, \quad\varepsilon_{y z}={\frac{1} {2}} {\frac{\partial q ( r, x )} {\partial z}}+{\frac{1} {2}} {\frac{\partial w} {\partial y}} {\frac{x} {r}}}} \\ {{\varepsilon_{z}={\frac{\partial w} {\partial x}} \, \quad\varepsilon_{z x}={\frac{1} {2}} {\frac{\partial q ( r, x )} {\partial z}}+{\frac{1} {2}} {\frac{\partial w} {\partial y}} {\frac{y} {r}}}} \\ \end{array} \}
$$
Answer: B