Yabi 發問時間: 科學數學 · 1 0 年前


Suppose that F(x,y,z)=0 is an equation that can be solved to yield any of the three variables as a function of the other two.Show that


provided that the symbols are interpreted properly.(part of the problem is to say what the proper interpretation is.)

1 個解答

  • 1 0 年前

    Let F(x,y,z)=0 ----(1); x=f(y,z) ----(2); y=g(x,z) ----(3);And z=h(x,y) --- (4), according to the assumptions.

    (1)+(2) --> F(x=f(y,z), y,z)=0 ---(5); d/dy (5) --> F_x*(x_y)+F_y=0 --> x_y=-(F_y/F_x) ---(A) . Here x_y =f_y is the partial derivative of x (hence f) with respect to y.

    Mimicking the last paragraph:

    (1)+(3) --> F(x, y=g(x,z), z)=0 ---(6); d/dz (6) --> F_y*(y_z)+F_z=0 --> y_z=-(F_z/F_y) ---(B) . Where y_z =g_z is the partial derivative of y (hence g) with respect to z. Also

    (1)+(4) --> F(x, y,z=h(x,y))=0 ---(7); d/dx (7) --> F_x+F_z*(z_x)=0 --> z_x=-(F_x/F_z) ---(C) . Here z_x =h_x is the partial derivative of z (hence h) with respect to z.

    Now the desired conclusion follows from (A)*(B)*(C).