Discover Top Posts Tagged with #parameterization

Some line integral examples of the Fundamental theorem of geometric calculus

[Click here for a PDF version of this post] On my discord server, Frank asked about his attempt to demonstrate an example line integral computation of the fundamental theorem of geometric calculus. Before working through his example, and some others, it is first worth restating the line integral specialization of the \textit{Fundamental theorem of geometric calculus}: Theorem 1.1: Fundamental…

#Fundamental Theorem of Geometric Calculus #Geometric Algebra #gradient #line integral #parameterization #projection #pseudoscalar #tangent space #vector derivative

Area within closed boundary

[Click here for a PDF version of this post] Motivation. On vacation I was reading some more of [1]. It was mentioned in passing that the area contained within a closed parameterized curve is given by \begin{equation}\label{eqn:containedArea:20} A = \inv{2} \int_{t_0}^{t_1} \lr{x y’ – y x’} dt, \end{equation} where \( x = x(t), y = y(t), t \in [t_0, t_1] \). This has the look of a Stokes theorem…

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#dot product #fundamental theorem of integral calculus #Green&039;s theorem #Jacobian #parameterization #Stokes&039; theorem #wedge product

Vector Area

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One of the results of this problem is required for a later one on magnetic moments that I’d like to do.

Question: Vector Area. ([1] pr. 1.61)

The integral

\begin{equation}\label{eqn:vectorAreaGriffiths:20} \Ba = \int_S d\Ba, \end{equation}

is sometimes called the vector area of the surface \( S \).

(a)

Find the vector area of a…

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#cross product #Griffiths electrodynamics #Jacobian #parameterization #vector area

Πλατφόρμα Παραμετροποίησης του Κοινού Βίου

Parameterization Platform of Collective Life

The parameterization platform of collective life is an experimental proposal for collective life under urban mobility circumstances, and is encouraged through an architecture production process. Principle of this architecture is the emphasis on spatial qualities that are created by the relationships between different kinds of spaces and all the different situations that give those spaces life.The implementation of collective life is encouraged through three kinds of methods: the minimization of privacy, the expansion of daily activities outside the private spaces, and then the customization of public life.

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