Intergration : Joacobian problem

[kaleung]

Sorry for my misunderstanding in Joacobian determinant, but I'm really get confuse in following problem, sorry for it is wrote by hand

I don't really understand the notation of ,is it means than the function of x,y divided by function of u,v (i.e. dxdy/drdθ) ?????

 

[daon2]

Sorry for my misunderstanding in Joacobian determinant, but I'm really get confuse in following problem, sorry for it is wrote by hand
View attachment 3216
I don't really understand the notation of View attachment 3217,is it means than the function of x,y divided by function of u,v (i.e. dxdy/drdθ) ?????

No, suppose \(\displaystyle f(x,y)\) and \(\displaystyle g(x,y)\) are given functions. Then

\(\displaystyle \dfrac{\partial(f,g)}{\partial(x,y)} = \left(\begin{matrix} \partial f/\partial x &\partial f/\partial y \\ \partial g/\partial x &\partial g/\partial y\end{matrix}\right)\).

The rows of the matrix are all the partial derivatives for a particular function, the columns are all the partials with respect to a particular variable.

Here, your x and y are treated as functions x(u,v) and y(u,v)

 

[kaleung]

No, suppose \(\displaystyle f(x,y)\) and \(\displaystyle g(x,y)\) are given functions. Then

\(\displaystyle \dfrac{\partial(f,g)}{\partial(x,y)} = \left(\begin{matrix} \partial f/\partial x &\partial f/\partial y \\ \partial g/\partial x &\partial g/\partial y\end{matrix}\right)\).

The rows of the matrix are all the partial derivatives for a particular function, the columns are all the partials with respect to a particular variable.

Here, your x and y are treated as functions x(u,v) and y(u,v)

Thanks...but in this case, what is the meanings of [FONT=MathJax_Main]∂[/FONT][FONT=MathJax_Main]([/FONT][FONT=MathJax_Math]f[/FONT][FONT=MathJax_Main],[/FONT][FONT=MathJax_Math]g[/FONT][FONT=MathJax_Main]) and [/FONT][FONT=MathJax_Main]∂[/FONT][FONT=MathJax_Main]([/FONT][FONT=MathJax_Math]x[/FONT][FONT=MathJax_Main],[/FONT][FONT=MathJax_Math]y.[/FONT][FONT=MathJax_Main]). Also, is ∂([FONT=MathJax_Math]f[/FONT],[FONT=MathJax_Math]g[/FONT])∂([FONT=MathJax_Math]x[/FONT],[FONT=MathJax_Math]y[/FONT])means [/FONT][FONT=MathJax_Main][/FONT][FONT=MathJax_Main]∂[/FONT][FONT=MathJax_Main]([/FONT][FONT=MathJax_Math]f[/FONT][FONT=MathJax_Main],[/FONT][FONT=MathJax_Math]g[/FONT][FONT=MathJax_Main]) divided by [/FONT][FONT=MathJax_Main] [/FONT][FONT=MathJax_Main]∂[/FONT][FONT=MathJax_Main]([/FONT][FONT=MathJax_Math]x[/FONT][FONT=MathJax_Main],[/FONT][FONT=MathJax_Math]y.[/FONT][FONT=MathJax_Main])?

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[daon2]

Thanks...but in this case, what is the meanings of [FONT=MathJax_Main]∂[/FONT][FONT=MathJax_Main]([/FONT][FONT=MathJax_Math]f[/FONT][FONT=MathJax_Main],[/FONT][FONT=MathJax_Math]g[/FONT][FONT=MathJax_Main]) and [/FONT][FONT=MathJax_Main]∂[/FONT][FONT=MathJax_Main]([/FONT][FONT=MathJax_Math]x[/FONT][FONT=MathJax_Main],[/FONT][FONT=MathJax_Math]y.[/FONT][FONT=MathJax_Main]). Also, is ∂([FONT=MathJax_Math]f[/FONT],[FONT=MathJax_Math]g[/FONT])∂([FONT=MathJax_Math]x[/FONT],[FONT=MathJax_Math]y[/FONT])means [/FONT][FONT=MathJax_Main]∂[/FONT][FONT=MathJax_Main]([/FONT][FONT=MathJax_Math]f[/FONT][FONT=MathJax_Main],[/FONT][FONT=MathJax_Math]g[/FONT][FONT=MathJax_Main]) divided by [/FONT][FONT=MathJax_Main]∂[/FONT][FONT=MathJax_Main]([/FONT][FONT=MathJax_Math]x[/FONT][FONT=MathJax_Main],[/FONT][FONT=MathJax_Math]y.[/FONT][FONT=MathJax_Main])?

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You cannot separate the symbols that way, it is just another name for the jacobian which specifies your functions and variables.