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Math Help - show the Jacobian is zero

  1. #1
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    show the Jacobian is zero

     F: R^{2}\rightarrow R^{2} , \ \ \ J(F) is Jacobian.

    Suppose F maps into a curve in the plane, in the sense tht F is a composition of two smooth mappings,  R^{2}\rightarrow R\rightarrow R^{2}

    show that the Jacobian of F is identically zero.
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  2. #2
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    Quote Originally Posted by silversand View Post
     F: R^{2}\rightarrow R^{2} , \ \ \ J(F) is Jacobian.

    Suppose F maps into a curve in the plane, in the sense tht F is a composition of two smooth mappings,  R^{2}\rightarrow R\rightarrow R^{2}

    show that the Jacobian of F is identically zero.
    so F(x,y)=fg(x,y), where g: \mathbb{R}^2 \longrightarrow \mathbb{R} and f: \mathbb{R} \longrightarrow \mathbb{R}^2. let g(x,y)=z. then F(x,y)=f(z)=(u(z),v(z)). thus: J(F)=\begin{vmatrix} u_x & u_y \\ v_x & v_y \end{vmatrix}= \begin{vmatrix} u_z z_x & u_z z_y \\ v_z z_x & v_z z_y \end{vmatrix}= u_zz_xv_zz_y - u_zz_yv_zz_x = 0.
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