Bagh E Jannat Book Free //TOP\\ 89
Bagh E Jannat Book Free 89
Bagh E Jannat. (translation) Quranic Commentaries in Urdu of the most important books of the Qur’aan. (partial e text)
Bagh E Jannat (translation) Quranic Commentaries in Urdu of the most important books of the Qur’aan. Kitab-E-Zindagi. Kitab-E-Marifat. Aqwal-E-Hikmat.Q:
What metric can I use on a “true” vector field to get rid of the length?
Suppose $X$ is some vector field on $\mathbb{R}^n$, and I define the field strength $F$ by the formula $F(p)=\|X(p)\|_2$ where $p$ is a point in $\mathbb{R}^n$. Note that $F(p)$ is a non-negative real number. Now suppose that there exists some point $p$ for which $F(p)$ is infinite. I would like to delete this point from the domain of $X$. Is there a metric on $\mathbb{R}^n$ for which the $F(p)$ is exactly zero for some point $p$? I know that the norm $\|X(p)\|_2$ is a zero of a differentiable function in such a metric, but I don’t know if the length itself is a zero of a function.
A:
If you ask for a metric on $\mathbb{R}^n$ such that $F(p)$ is equal to zero for all $p\in \mathbb{R}^n$, then you ask for a metric of zero volume. For every point in $\mathbb{R}^n$, the volume of a ball of radius $r$ centred at that point is $V_n(r)$, where $V_n(r)= \frac{\pi^{n/2}}{\Gamma(\frac{n}{2}+1)}r^n$.
Clearly $V_n(r)$ is a non-negative, continuous function of $r$. But $V_n(r)$ is also positive, strictly increasing, and unbounded on $\mathbb{R}$. Thus there is no metric on $\mathbb{R}^n$ for which $V_n(r)$ is
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