I have a function $(x_1, x_2)\mapsto g(x_1, x_2)$ where $x_1$ and $x_2$ are both 3D vectors.

I would like to integrate function $g$ over the whole space but with some constraints on $x_1$ and $x_2$ which are represented by following functions : $$x_1 = f_1(x)~,~~x_2 = f_2(x)~,~~x\in\mathbb{R}^3$$ Then integration could be written as follows :

$$ I= \iint_{x_1 = f_1(x)~,~x_2 = f_2(x)~,~x\in\mathbb{R}^3} g(x_1, x_2)\,dx_1\,dx_2 $$

Variables $x_1$ and $x_2$ are linked by $x$, so I would like to replace this double integral over space by a single one over $x$ : $$ J= \int_{x\in\mathbb{R}^3} g[f_1(x), f_2(x)]\,dx $$

But I am really not sure about $J = I$.

If there was only one variable, say $x_1$, then to integrate over $x$ instead over $x_1$ comes to a change of variable : $$ I= \int_{x_1 = f_1(x)~,~x\in\mathbb{R}^3} g(x_1)\,dx_1 = \int_{x\in\mathbb{R}^3} g[f_1(x)]|\det(J_1)|\,dx$$ where $J_1$ is the jacobian matrix of function $f_1$.

How this applies to my case ?

Thanks for your attention.



1 Answer 1


I answeer to my own question because I think I have understood what was wrong in my first post.


First, I take a very close example to my qestion borrowed from line integration.

Consider a surface function $(x, y)\mapsto f(x, y)$ with both $x,~y\in\mathbb{R}$.

Then, you want to compute the integral of this function on a given contour $\mathcal{C}$ parametrically defined by $x = h(t)~,~~y = g(t)$.

Integration over this contour is defined by : $$\int_{\mathcal{C}} f(x,y)\,ds \triangleq \int_a^b f[h(t), g(t)]~~||r'(t)||\,dt$$

where $||r'(t)||$ is the norm of vector $r'(t) = (h'(t), g'(t))^T$

So one gets : $$\int_{\mathcal{C}} f(x,y)\,ds \triangleq \int_a^b f[h(t), g(t)] \sqrt{h'(t)^2 + g'(t)^2}\,dt$$


Coming back to my earlier question and trying to apply this above development, the "I" integral in previous post should be defined as follows :

$$ I = \int_{\mathcal{C}}f(x_1,x_2)\,ds$$ where $\mathcal{C}$ is a "contour" defined by $x_1 = f_1(x)~,~~x_2 =f_2(x)$

Following the analogy, $I$ becomes : $$ I = \int_{x\in\mathbb{R}}f[f_1(x),f_2(x)]~||r'(x)||\,dx$$

However, the definition of vector $r'(x)$ is not straightforward. For sure, we have : $$\,dx_1 = J_{f_1}\,dx~~\text{and}~~\,dx_2 = J_{f_2}\,dx$$

Then, I can only infer the norm of this vector is : $$||r'(x)|| = \sqrt{\det(J_{f_1})^2 + \det(J_{f_2})^2} $$

so that, when you keep only variable, say $x_1$, you fall back on the former formula for changing of variables.

So, the formula I was looking for is :

$$I = \int_{x\in\mathbb{R}}f[f_1(x),f_2(x)]~\sqrt{\det(J_{f_1})^2 + \det(J_{f_2})^2} \,dx$$

Do you agree with this reasoning ?



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