#euler equation

Euler Equation of Motion in Fluid Dynamics — Complete Mind Map

For an inviscid fluid,

ρ D𝐕/Dt = −∇p + ρ𝐅

This mind map connects the complete Euler-equation concept: the material derivative, local and convective acceleration, pressure and body forces, Cartesian component equations, steady-flow reduction, streamline form, assumptions and physical interpretation.

For steady, inviscid, constant-density flow under gravity, the streamline form provides the route to Bernoulli’s equation.

Useful as a compact revision resource for UPSC CSE Mathematics Optional and IFoS Mathematics Optional.

Complete explanation: https://mathsoptional.com/social-updates/euler-equation-of-motion-in-fluid-dynamics/?utm_source=tumblr&utm_medium=social&utm_campaign=euler_equation_fluid_dynamics&utm_content=mind_map

Euler Equation of Motion in Fluid Dynamics — UPSC & IFoS Mathematics Optional

The Euler equation of motion describes the motion of an inviscid fluid:

ρ D𝐕/Dt = −∇p + ρ𝐅

The material derivative is

D𝐕/Dt = ∂𝐕/∂t + (𝐕·∇)𝐕

and shows that the acceleration of a fluid particle can arise both from change with time at a fixed location and from movement through a spatially varying velocity field.

For steady flow, the local acceleration ∂𝐕/∂t vanishes, although convective acceleration may remain.

This visual revision sheet summarises:

• Euler equation of motion • Material derivative • Meaning of each term • Cartesian component form • Steady-flow form • Physical interpretation • Relationship with Bernoulli’s equation

Useful for UPSC CSE Mathematics Optional and IFoS Mathematics Optional preparation.

Complete post: https://mathsoptional.com/social-updates/euler-equation-of-motion-in-fluid-dynamics/?utm_source=tumblr&utm_medium=social&utm_campaign=euler_equation_fluid_dynamics

If the Euler equation is not verified in the real world, as various papers claim (see http://noahpinionblog.blogspot.it/2014/01/the-equation-at-core-of-modern-macro.html), as consumption is shown to be actually high when interest rates are high, I think we should give some thought to the below observation:

The Beautiful Equation: Euler’s Identity

Also called Euler’s relation, or the Euler equation of complex analysis, this bit of mathematics enjoys accolades across geeky disciplines.

Swiss mathematician Leonhard Euler first wrote the equality, which links together geometry, algebra, and five of the most essential symbols in math -- 0, 1, i, pi and e -- that are essential tools in scientific work.

Theoretical physicist Richard Feynman was a huge fan and called it a "jewel" and a “remarkable” formula. Fans today refer to it as “the most beautiful equation."