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jjou
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[SOLVED] partial derivatives - verify solution?
Let f:\mathbb{R}^3\rightarrow\mathbb{R}, g:\mathbb{R}^2\rightarrow\mathbb{R}, and F:\mathbb{R}^2\rightarrow\mathbb{R} be given by
F(x,y)=f(x,y,g(x,y)).
1. Find DF in terms of the partial derivatives of f and g.
2. If F(x,y)=0 for all (x,y), find D_1g and D_2g in terms of the partial derivatives of f.
My solution:
1. DF=D_1F+D_2F=(f_1+f_3g_1)+(f_2+f_3g_2)
2. If f_3\neq0, then we have the partials of F being zero, so:
g_1=-f_1/f_3 and g_2=-f_2/f_3. However, if f_3=0 then we have f_1=f_2=0.
My concern is with the last part of 2. If f_3=0, then I cannot make any statement about the partials of g. Am I doing something wrong?
NOTE: f_1 refers to differentiation of f by the first variable.
Let f:\mathbb{R}^3\rightarrow\mathbb{R}, g:\mathbb{R}^2\rightarrow\mathbb{R}, and F:\mathbb{R}^2\rightarrow\mathbb{R} be given by
F(x,y)=f(x,y,g(x,y)).
1. Find DF in terms of the partial derivatives of f and g.
2. If F(x,y)=0 for all (x,y), find D_1g and D_2g in terms of the partial derivatives of f.
My solution:
1. DF=D_1F+D_2F=(f_1+f_3g_1)+(f_2+f_3g_2)
2. If f_3\neq0, then we have the partials of F being zero, so:
g_1=-f_1/f_3 and g_2=-f_2/f_3. However, if f_3=0 then we have f_1=f_2=0.
My concern is with the last part of 2. If f_3=0, then I cannot make any statement about the partials of g. Am I doing something wrong?
NOTE: f_1 refers to differentiation of f by the first variable.
