The discussion revolves around the application of the chain differentiation rule for functions of multiple variables, specifically focusing on the composition of functions and the interpretation of partial derivatives in this context. Participants explore the mathematical formulation of derivatives when dealing with vector-valued functions and their implications.
Participants express differing views on the interpretation of the partial derivatives and the correctness of the proposed formula. There is no consensus on the validity of the formula or the proper interpretation of the variables involved.
There are unresolved issues regarding the definitions of the functions involved and the assumptions made about their differentiability and variable dependencies.
arildno said:Well, the total derivative of g is a matrix, whereas the total derivative of f is a vector. Together, they yield a vector.
Let's take a concrete example:
[tex]h(x,y)=f(u(x,y),v(x,y)), g(x,y)=(u(x,y),v(x,y))[/tex]
We have therefore, for example:
[tex]\frac{\partial{h}}{\partial{x}}=\frac{\partial{f}}{\partial{u}}\frac{\partial{u}}{\partial{x}}+\frac{\partial{f}}{\partial{v}}\frac{\partial{v}}{\partial{x}}[/tex]
This is then one of the two partial derivatives of h, the other being differentiation with respect to y.
It means the partial derivative of f with respect to u, of course. What else could it mean?frb said:what does
[tex]\frac{\partial{f}}{\partial{u}}[/tex]
mean?
It doesn't make sense. If f(u(a,b),v(a,b)) makes any sense then f is a function of u and v, not x and y. You must mean [tex]\frac{\partial f}{\partial u}[/tex] not [tex]\frac{\partial f}{\partial x}[/tex].Is the following formula correct?
[tex]\frac{\partial{h}}{\partial{x}}(a,b)=\frac{\partial{f}}{\partial{x}}(u(a,b),v(a,b))\frac{\partial{u}}{\partial{x}}(a,b)+\frac{\partial{f}}{\partial{y}}(u(a,b),v(a,b))\frac{\partial{v}}{\partial{x}}(a,b)[/tex]