#vectorcalculus — Public Fediverse posts
Live and recent posts from across the Fediverse tagged #vectorcalculus, aggregated by home.social.
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Frenet–Serret Formula ✍️
It explains how a curve reveals its hidden geometry by tracking the way it bends and twists through space. Imagine tracing the path of a roller coaster, a winding river, or the spiral of a DNA strand. At every point along the path, the curve is constantly changing direction, and the Frenet–Serret formulas provide a precise way to describe that change.
They do this by attaching a moving frame of three special directions to each point on the curve. The first points forward along the path, showing where the curve is heading. The second points inward, toward the direction of bending. The third stands perpendicular to both, capturing how the curve twists out of its plane. Together, they form a local coordinate system that travels with the curve itself.
As you move along the curve, these three directions rotate and evolve. The formulas measure this evolution using two key quantities: curvature and torsion. Curvature tells how sharply the path bends, while torsion tells how strongly it twists into three dimensions. If curvature vanishes, the path becomes straight; if torsion vanishes, the curve lies flat in a plane.
Mathematicians and physicists use the Frenet–Serret formulas to study motion, design smooth paths in engineering, understand particle trajectories, and analyze natural shapes. They transform a simple line into a rich geometric story, revealing exactly how space is being navigated at every step.
#FrenetSerretFormula #DifferentialGeometry #Geometry #Mathematics #Math #PureMathematics #AppliedMathematics #MathematicalPhysics #Physics #STEM #ScienceEducation #MathEducation #Curvature #Torsion #SpaceCurves #VectorCalculus #Calculus #LinearAlgebra #GeometricAnalysis
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Frenet–Serret Formula ✍️
It explains how a curve reveals its hidden geometry by tracking the way it bends and twists through space. Imagine tracing the path of a roller coaster, a winding river, or the spiral of a DNA strand. At every point along the path, the curve is constantly changing direction, and the Frenet–Serret formulas provide a precise way to describe that change.
They do this by attaching a moving frame of three special directions to each point on the curve. The first points forward along the path, showing where the curve is heading. The second points inward, toward the direction of bending. The third stands perpendicular to both, capturing how the curve twists out of its plane. Together, they form a local coordinate system that travels with the curve itself.
As you move along the curve, these three directions rotate and evolve. The formulas measure this evolution using two key quantities: curvature and torsion. Curvature tells how sharply the path bends, while torsion tells how strongly it twists into three dimensions. If curvature vanishes, the path becomes straight; if torsion vanishes, the curve lies flat in a plane.
Mathematicians and physicists use the Frenet–Serret formulas to study motion, design smooth paths in engineering, understand particle trajectories, and analyze natural shapes. They transform a simple line into a rich geometric story, revealing exactly how space is being navigated at every step.
#FrenetSerretFormula #DifferentialGeometry #Geometry #Mathematics #Math #PureMathematics #AppliedMathematics #MathematicalPhysics #Physics #STEM #ScienceEducation #MathEducation #Curvature #Torsion #SpaceCurves #VectorCalculus #Calculus #LinearAlgebra #GeometricAnalysis
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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
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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
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If you teach multivariable calculus, do you incorporate differential forms as an option to traditional vector calculus? Just curious. #ITeachMath #Calculus #VectorCalculus #DifferentialForms
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If you teach multivariable calculus, do you incorporate differential forms as an option to traditional vector calculus? Just curious. #ITeachMath #Calculus #VectorCalculus #DifferentialForms
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starting to get excited about this vector calculus project
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starting to get excited about this vector calculus project
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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.
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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.