# Multiple integration with constraints on variables

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.

Regards

## 1 Answer

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

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

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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 ?

Regards