Coarea Inequality

If $n<m$ and $f:\mathbb{R}^n \to \mathbb{R}^m$ is an injective (and reasonably nice) function then image of $f$ is an $n$-dimensional object sitting inside the larger $\mathbb{R}^m$. Examples: a curve or surface in $\mathbb{R}^3$. But what if we turn the tables and consider functions $f:\mathbb{R}^m \to \mathbb{R}^n$ where $m > n$? There is just not enough room and many points must map to a common target point. In this talk I will explain the precise mathematical meaning of the following and end with the Coarea Inequality:

The Origin of the Logarithm: And how it catalyzed the scientific revolution

The way we learn about logarithms as young mathematical epsilons, is in their relationship to the exponential function. On the other hand, the history of logarithms has no exponents in sight. The invention of the logarithm is more closely related to the invention of a piece of technology like the calculator than it is to the invention a new mathematical function. This new technology emerged simultaneously with the scientific revolution and this was no accident.