Consider the function $f(x) :\mathbb{R}\rightarrow \mathbb{R}$, such that $f(x)\geqslant 0\; \forall x\in \mathbb{R}$, and has a set of extremum points at $x_{j}$.
Consider the integral :
$$\int_{\bar{x}}^{g_{c}(\bar{x})}f(x)dx=c$$
where c is a constant, and $g_{c}(\bar{x})$ is a function of $\ba...
I will think of $ \mathbb{R}^{n+m}$ as $\mathbb{R}^n \times \mathbb{R}^m$.
Let $ V \subset \mathbb{R}^{n+m}$ be open and $g:V \to U \subset \mathbb{R}^{n+m} $ be a $C^1$ diffeomorphism. For a fixed ${y} \in \mathbb{R}^m$, the image $g(\mathbb{R}^n \times \{y\})$ is an $n$-dimensional $C^1$ man...
I will think of $ \mathbb{R}^{n+m}$ as $\mathbb{R}^n \times \mathbb{R}^m$.
Let $ V \subset \mathbb{R}^{n+m}$ be open and $g:V \to U \subset \mathbb{R}^{n+m} $ be a $C^1$ diffeomorphism. For a fixed ${y} \in \mathbb{R}^m$, the image $g(\mathbb{R}^n \times \{y\})$ is an $n$-dimensional $C^1$ man...
