We can calculate the gradient of a tangent to a curve by differentiating. In order to find the equation of a tangent, we: Differentiate the equation of the curve Substitute the \(x\) value into the ...

The study of Lagrangian equations and gradient estimates occupies a critical niche at the intersection of partial differential equations, differential geometry, and variational calculus. Lagrangian ...

Before reading this guide, you may find it helpful to read the guide on graphs from Module 7 (M7) and the guide on indices from Module 8 (M8). In Module 7 (M7) Graphs, you learned that simultaneous ...

The yield spread between long-term and short-term Treasury securities is known to be a good predictor of economic activity, particularly of looming recessions. One way to learn more is through a ...

It is conventional wisdom that the reduced form Phillips curve has become flatter in recent decades. Accordingly, we show that the statistical relationship between changes in U.S. inflation and ...

Elastic flows and gradient flows in planar curves form a vibrant area of research in geometric analysis, where curves evolve in time under rules dictated by an energy functional. Central to this ...

In August, a pair of mathematicians discovered an exotic, record-breaking curve. In doing so, they tapped into a major open question about one of the oldest and most fundamental kinds of equations in ...