Divergence and curl of vector fields are complementary concepts from vector calculus.

Divergence quantifies the rate at which a field flows outward from a point, and curl represents rotation.

\(\nabla\cdot\mathbf{F}\) denotes divergence (expansion/contraction)

\(\nabla\times\mathbf{F}\) denotes curl (rotation)

Left: A radial source field with pure divergence and no curl.

Centre: A rotational field with pure curl and no divergence.

Right: A spiral source field with both divergence and curl.

Credit: Alec Helbling

#Divergence #Curl #Vector #VectorFields #VectorCalculus

#Vector / #Multivariable #Calculus
#VectorCalculus #MultivariableCalculus

This year, Simon Prince, Professor of Computer Science at UCL, published a series of tutorials on ordinary differential equations (ODEs) and stochastic differential equations (SDEs) in machine learning for RBC Borealis. These are intended for readers with no background in these areas and require only basic calculus.

Article 1 describes what ODEs and SDEs are and their applications in machine learning.

https://rbcborealis.com/research-blogs/odes-and-sdes-for-machine-learning

Article 2 describes ODEs, vector ODEs and PDEs and defines associated terminology. They develop several categories of ODE and discuss how their solutions are related to one another. They discuss the necessary conditions for an ODE to have a solution.

https://rbcborealis.com/research-blogs/introduction-ordinary-differential-equations

Article 3 describes methods for solving first-order ODEs in closed form. They categorise ODEs into distinct families and develop a method to solve each family.

https://rbcborealis.com/research-blogs/closed-form-solutions-for-odes

For many ODEs, there is no known closed-form solution.

Article 4 considers numerical methods, which can be used to approximate the solution of any ODE regardless of its tractability.

https://rbcborealis.com/research-blogs/numerical-methods-for-odes

This concludes their treatment of ODEs. In the coming weeks, we will focus on SDEs. They will describe stochastic processes and SDEs, and show how to solve SDEs using either direct stochastic integration or Ito's lemma. They will introduce the Fokker-Planck equation, which transforms a stochastic differential equation into the PDE governing the evolving probability density of the solution. They also consider Andersen's theorem, which allows us to reverse the direction of SDEs.

#ODEs #PDEs #SDEs #ODE #PDE #SDE #Calculus #ML #DL #VectorCalculus #LectureSeries #Tutorials

If you teach multivariable calculus, do you incorporate differential forms as an option to traditional vector calculus? Just curious. #ITeachMath #Calculus #VectorCalculus #DifferentialForms

starting to get excited about this vector calculus project

#updateshere #vectorcalculus

#Mechanical #Circuits

As an EE, I take pride in the fact that, although ME came before us and CE before them, much of the #engineering formalism arose in the 19th Century in the electrical context. Much of engineering theories are shared between #EE, #ME, and #CE, and at the core is #VectorCalculus and #AbstractAlgebra. This video shows one such connection.

https://youtu.be/Zv9Q7ih48Uc

GENERALIZED STOKES THEOREM:
The integral of a differential form \(\omega\) over the boundary \(\partial\Omega\) of some orientable manifold \(\Omega\) is equal to the integral of its exterior derivative \(d\omega\) over the whole of \(\Omega\).
\[\displaystyle\int_{\partial\Omega}\omega=\int_\Omega d\omega\]
#VectorCalculus #DifferentialGeometry #MultivariateCalculus #Calculus #StokesTheorem #GeneralizedStokesTheorem #Calculus #FundamentalTheorem #Manifold #Boundary #ExteriorDerivative #Stokes

The fundamental theorem of #vectorcalculus states that any vector field can be expressed as the sum of an #irrotational and a #solenoidal field. The condition of zero #divergence is satisfied whenever a vector field v has only a vector potential component

The fundamental theorem of #vectorcalculus states that any vector field can be expressed as the sum of an #irrotational and a #solenoidal field. The condition of zero #divergence is satisfied whenever a vector field v has only a vector potential component