Vector calculus - Revision history

Bros: /* V.3 Analytical representation of a vector */

2024-11-09T19:29:07Z

V.3 Analytical representation of a vector

← Older revision		Revision as of 19:29, 9 November 2024
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	'''Figure V.7''' shows the projection, on two different vector bases, of vector<math>\overline{\textrm{r}}</math>, associated with the radius of a rotating platform with planar motion relative to R. The basis <math>(1,2,3)</math> does not change its orientation with respect to R, whereas the basis <math>(1',2',3')</math> changes its orientation relative to R but not relative to R’ (which is the platform).		'''Figure V.7''' shows the projection, on two different vector bases, of vector <math>\overline{\textrm{r}}</math>, associated with the radius of a rotating platform with planar motion relative to R. The basis <math>(1,2,3)</math> does not change its orientation with respect to R, whereas the basis <math>(1',2',3')</math> changes its orientation relative to R but not relative to R’ (which is the platform).


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	In '''Figure V.6''', the vector basis B = (1,2,3) does not change its orientation relative to R (it is a fixed basis in R), whereas the vector basis B = (1',2',3') does change its orientation relative to R but not relative to R’ (it is a fixed basis in R’ but a moving one in R): <math>\velang{B}{R} = \vec 0</math>, <math>\velang{B'}{R} \not= \vec{0}</math>, <math>\velang{B}{R'}\neq\vec{0}</math>, <math>\velang{B'}{R'}=\vec{0}</math>.		In '''Figure V.7''', the vector basis B = (1,2,3) does not change its orientation relative to R (it is a fixed basis in R), whereas the vector basis B = (1',2',3') does change its orientation relative to R but not relative to R’ (it is a fixed basis in R’ but a moving one in R): <math>\velang{B}{R} = \vec 0</math>, <math>\velang{B'}{R} \not= \vec{0}</math>, <math>\velang{B}{R'}\neq\vec{0}</math>, <math>\velang{B'}{R'}=\vec{0}</math>.

Bros: /* V.2 Operations between vectors with geometric representation */

2024-11-09T19:28:32Z

V.2 Operations between vectors with geometric representation

← Older revision		Revision as of 19:28, 9 November 2024
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	[[File:V-6-eng.png\|thumb\|center\|650px\|link=]]		[[File:V-6-eng-new.png\|thumb\|center\|650px\|link=]]
	<center><small>'''Figure V.6''' Time derivative of a vector relative to a reference frame: general case</small></center>		<center><small>'''Figure V.6''' Time derivative of a vector relative to a reference frame: general case</small></center>

Bros at 16:17, 12 July 2023

2023-07-12T16:17:26Z

← Older revision		Revision as of 16:17, 12 July 2023
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	Vectors may be represented geometrically through an image, showing their '''direction''' with an arrow and their '''value''' with a scalar (positive or negative) ('''Figure V.1'''). Both direction and value may change with time.		Vectors may be represented geometrically through an image, showing their '''direction''' with an arrow and their '''value''' with a scalar (positive or negative) ('''Figure V.1'''). Both direction and value may change with time.

	[[File:V-1-neutre~~-color~~.png\|thumb\|center\|400px\|link=]]
			[[File:V-1-neutre.png\|thumb\|center\|400px\|link=]]
	<center><small>'''Figure V.1''' Geometric representation of the position vector of a point <math>\Os</math> moving along a straight line: <br>(a) geometric definition of a positive value; (b) three particular cases</small></center>		<center><small>'''Figure V.1''' Geometric representation of the position vector of a point <math>\Os</math> moving along a straight line: <br>(a) geometric definition of a positive value; (b) three particular cases</small></center>

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	'''Figure V.2''' summarizes the procedures to carry out the three operations between vectors that involve just one time instant.		'''Figure V.2''' summarizes the procedures to carry out the three operations between vectors that involve just one time instant.

	[[File:V-2-~~cat-color~~.png\|thumb\|center\|550px\|link=]]		[[File:V-2-eng.png\|thumb\|center\|550px\|link=]]
	<center><small>'''Figure V.2''' Geometric calculation of instantaneous operations between vectors </small></center>		<center><small>'''Figure V.2''' Geometric calculation of instantaneous operations between vectors </small></center>

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	</center>		</center>

	[[File:V-3-~~cat-color~~.png\|thumb\|center\|450px\|link=]]		[[File:V-3-eng.png\|thumb\|center\|450px\|link=]]
	<center><small>'''Figure V.3''' The time derivative of a vector with constant direction and variable value is parallel to the vector, and it is independent from the reference frame: '''(a)''' sign(<math>\dot\us</math>) = sign(<math>\us</math>); '''(b)''' sign(<math>\dot\us</math>) = -sign(<math>\us</math>)</small></center>		<center><small>'''Figure V.3''' The time derivative of a vector with constant direction and variable value is parallel to the vector, and it is independent from the reference frame: '''(a)''' sign(<math>\dot\us</math>) = sign(<math>\us</math>); '''(b)''' sign(<math>\dot\us</math>) = -sign(<math>\us</math>)</small></center>

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	[[File:V-4-~~cat-color~~.png\|thumb\|center\|570px\|link=]]		[[File:V-4-eng.png\|thumb\|center\|570px\|link=]]
	<center><small>'''Figure V.4''' The time derivative of a vector with constant value and variable direction is orthogonal to the vector, and depends on the reference frame		<center><small>'''Figure V.4''' The time derivative of a vector with constant value and variable direction is orthogonal to the vector, and depends on the reference frame
	</small></center>		</small></center>
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	The concept '''rate of change of the orientation''' (<math>\dot{\theta}</math>) calls for the previous introduction of the <span style="text-decoration: underline;">[[C1. Configuració d'un sistema mecànic#C1.3 Orientació d'un sòlid rígid amb moviment pla\|'''angle of orientation''']]</span> (<math>\theta</math>), defined on plane <math>\Pi</math> from a fixed direction in <math>\Pi</math> and vector <math>\uvec</math>. The orientation of that plane in R and the rate of change of orientation<math>\dot{\theta}</math> of <math>\uvec</math> may be combined in just one mathematical object: the <span style="text-decoration: underline;">[[C2. Moviment d'un sistema mecànic#C2.4 Velocitat angular d’un sòlid rígid\|'''angular velocity''']]</span> of the vector with respect to R. It is a vector orthogonal to the plane and with value <math>\dot{\theta}</math>. The direction of the vector is given by the screw rule ('''Figure V.5'''). The generic notation that will be used in this course for the angular velocity of an object relative to a frame R is <math>\Omegavec^{\textup{object}}_\Rs</math>.		The concept '''rate of change of the orientation''' (<math>\dot{\theta}</math>) calls for the previous introduction of the <span style="text-decoration: underline;">[[C1. Configuració d'un sistema mecànic#C1.3 Orientació d'un sòlid rígid amb moviment pla\|'''angle of orientation''']]</span> (<math>\theta</math>), defined on plane <math>\Pi</math> from a fixed direction in <math>\Pi</math> and vector <math>\uvec</math>. The orientation of that plane in R and the rate of change of orientation<math>\dot{\theta}</math> of <math>\uvec</math> may be combined in just one mathematical object: the <span style="text-decoration: underline;">[[C2. Moviment d'un sistema mecànic#C2.4 Velocitat angular d’un sòlid rígid\|'''angular velocity''']]</span> of the vector with respect to R. It is a vector orthogonal to the plane and with value <math>\dot{\theta}</math>. The direction of the vector is given by the screw rule ('''Figure V.5'''). The generic notation that will be used in this course for the angular velocity of an object relative to a frame R is <math>\Omegavec^{\textup{object}}_\Rs</math>.

	[[File:V-5-~~cat-color~~.png\|thumb\|center\|450px\|link=]]		[[File:V-5-eng.png\|thumb\|center\|450px\|link=]]
	<center><small>'''Figure V.5''' Angular velocity of a vector with constant value relative to a reference frame</small></center>		<center><small>'''Figure V.5''' Angular velocity of a vector with constant value relative to a reference frame</small></center>

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	'''Particular case: Time derivative of a vector with constant vaue but with an orientation evolving in a general way relative to the reference frame'''		'''Particular case: Time derivative of a vector with constant vaue but with an orientation evolving in a general way relative to the reference frame'''

	It can be proved that, when the vector <math>\uvec</math> to be differentiated does not evolve on a plane but has a 3D evolution, the result of the time derivative can be obtained in the same way through its angular velocity <math>\Omegavec^{\uvec}_\Rs</math> ['''Batlle, J.A., Barjau, A. (2020) chapter 1 in Rigid body kinematics. Cambridge University Press''']. The determination of that angular velocity (units <span style="text-decoration: underline;">[[C1. ~~Configuració d'un sistema mecànic~~\|'''C1''']]</span> i <span style="text-decoration: underline;">[[C2. ~~Moviment d'un sistema mecànic~~\|'''C2''']]</span>), however, is more complicated.		It can be proved that, when the vector <math>\uvec</math> to be differentiated does not evolve on a plane but has a 3D evolution, the result of the time derivative can be obtained in the same way through its angular velocity <math>\Omegavec^{\uvec}_\Rs</math> ['''Batlle, J.A., Barjau, A. (2020) chapter 1 in Rigid body kinematics. Cambridge University Press''']. The determination of that angular velocity (units <span style="text-decoration: underline;">[[C1. Configuration of a mechanical system\|'''C1''']]</span> i <span style="text-decoration: underline;">[[C2. Movement of a mechanical system\|'''C2''']]</span>), however, is more complicated.


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	[[File:V6-~~cat~~-~~180ppp~~.png\|thumb\|center\|650px\|link=]]		[[File:V-6-eng.png\|thumb\|center\|650px\|link=]]
	<center><small>'''Figure V.6''' Time derivative of a vector relative to a reference frame: general case</small></center>		<center><small>'''Figure V.6''' Time derivative of a vector relative to a reference frame: general case</small></center>

Bros: /* V.3 Analytical representation of a vector */

2023-03-16T18:25:54Z

V.3 Analytical representation of a vector

← Older revision		Revision as of 18:25, 16 March 2023
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	In '''Figure V.6''', the vector basis B = (1,2,3) does not change its orientation relative to R (it is a fixed basis in R), whereas the vector basis B = (1',2',3') does change its orientation relative to R but not relative to R’ (it is a fixed basis in R’ but a moving one in R): <math>\velang{B}{R} = \vec 0</math>, <math>\velang{B'}{R} \not= \vec{0}</math>, <math>\velang{B}{R}\neq\vec{0}</math>, <math>\velang{B'}{R'}~~\neq~~\vec{0}</math>.		In '''Figure V.6''', the vector basis B = (1,2,3) does not change its orientation relative to R (it is a fixed basis in R), whereas the vector basis B = (1',2',3') does change its orientation relative to R but not relative to R’ (it is a fixed basis in R’ but a moving one in R): <math>\velang{B}{R} = \vec 0</math>, <math>\velang{B'}{R} \not= \vec{0}</math>, <math>\velang{B}{R'}\neq\vec{0}</math>, <math>\velang{B'}{R'}=\vec{0}</math>.

Bros: /* V.3 Analytical representation of a vector */

2023-03-16T18:25:27Z

V.3 Analytical representation of a vector

← Older revision		Revision as of 18:25, 16 March 2023
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	In '''Figure V.6''', the vector basis B = (1,2,3) does not change its orientation relative to R (it is a fixed basis in R), whereas the vector basis B = (1',2',3') does change its orientation relative to R but not relative to R’ (it is a fixed basis in R’ but a moving one in R): <math>\velang{~~\Bs~~}{~~\Rs~~} = \vec 0</math>, <math>\velang{~~\Bs~~'}{~~\Rs~~} \not= \vec{0}</math>, <math>\velang{~~\Bs~~}{~~\Rs~~}\neq\vec{0}</math>, <math>\velang{~~\Bs~~'}{~~\Rs~~'}\neq\vec{0}</math>.		In '''Figure V.6''', the vector basis B = (1,2,3) does not change its orientation relative to R (it is a fixed basis in R), whereas the vector basis B = (1',2',3') does change its orientation relative to R but not relative to R’ (it is a fixed basis in R’ but a moving one in R): <math>\velang{B}{R} = \vec 0</math>, <math>\velang{B'}{R} \not= \vec{0}</math>, <math>\velang{B}{R}\neq\vec{0}</math>, <math>\velang{B'}{R'}\neq\vec{0}</math>.

Bros at 18:23, 16 March 2023

2023-03-16T18:23:55Z

← Older revision		Revision as of 18:23, 16 March 2023
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	\newcommand{\evec}{\overline{\textbf{e}}}		\newcommand{\evec}{\overline{\textbf{e}}}
	\newcommand{\Omegavec}{\overline{\mathbf{\Omega}}}		\newcommand{\Omegavec}{\overline{\mathbf{\Omega}}}
			\newcommand{\velang}[2]{\Omegavec^{\textrm{#1}}_{\textrm{#2}}}
	\newcommand{\ds}{\textrm{d}}		\newcommand{\ds}{\textrm{d}}
	\newcommand{\ts}{\textrm{t}}		\newcommand{\ts}{\textrm{t}}

Bros: /* V.3 Analytical representation of a vector */

2023-03-16T18:23:01Z

V.3 Analytical representation of a vector

← Older revision		Revision as of 18:23, 16 March 2023
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	In '''Figure V.6''', the vector basis B = (1,2,3) does not change its orientation relative to R (it is a fixed basis in R), whereas the vector basis B = (1',2',3') does change its orientation relative to R but not relative to R’ (it is a fixed basis in R’ but a moving one in R): <math>\~~Omegavec^~~\~~Bs_~~\Rs = \vec 0</math>, <math>\~~Omegavec^~~{\Bs'}_{\Rs} \not= \vec{0}</math>, <math>\~~Omegavec~~{\Bs}{\Rs}\neq\vec{0}</math>, <math>\~~Omegavec~~{\Bs'}{\Rs'}\neq\vec{0}</math>.		In '''Figure V.6''', the vector basis B = (1,2,3) does not change its orientation relative to R (it is a fixed basis in R), whereas the vector basis B = (1',2',3') does change its orientation relative to R but not relative to R’ (it is a fixed basis in R’ but a moving one in R): <math>\velang{\Bs}{\Rs} = \vec 0</math>, <math>\velang{\Bs'}{\Rs} \not= \vec{0}</math>, <math>\velang{\Bs}{\Rs}\neq\vec{0}</math>, <math>\velang{\Bs'}{\Rs'}\neq\vec{0}</math>.

Bros: /* Operations along time: time derivative */

2023-03-16T18:21:41Z

Operations along time: time derivative

← Older revision		Revision as of 18:21, 16 March 2023
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	===Operations along time: time derivative===		===Operations along time: time derivative===

	Projecting a vector on a vector basis <math>(\evec_1, \evec_2, \evec_3)</math> means expressing it as an addition of three orthogonal ~~vector s~~<math>\uvec=\sum_{\is}^{}\us_\is\evec_\is</math>.		Projecting a vector on a vector basis <math>(\evec_1, \evec_2, \evec_3)</math> means expressing it as an addition of three orthogonal vectors <math>\uvec=\sum_{\is}^{}\us_\is\evec_\is</math>.
	If it is a fixed basis in R (the reference frame where the operation is performed), those vectors have constant direction. Hence:		If it is a fixed basis in R (the reference frame where the operation is performed), those vectors have constant direction. Hence:

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	</center>		</center>

	~~De manera que la derivació temporal d'un vector es pot escriure analíticament de la següent manera:~~

	<center>		<center>

Bros: /* Operations along time: time derivative */

2023-03-16T18:20:52Z

Operations along time: time derivative

← Older revision		Revision as of 18:20, 16 March 2023
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	<center>		<center>
	<math>\frac{\ds(\us_\is\evec_\is)}{\ds\ts} =\dot{\us}_\is\evec_\is</math> , <math>\left\{\left.\frac{\ds\uvec}{\ds\ts}		<math>\frac{\ds(\us_\is\evec_\is)}{\ds\ts} =\dot{\us}_\is\evec_\is</math> , <math>\left\{\left.\frac{\ds\uvec}{\ds\ts}
	\right]_{\Rs}\right\}_\Bs=\frac{\ds}{\ds\ts}\left\{\uvec\right\}_\Bs=		\right]_{\Rs}\right\}_\Bs\equiv\frac{\ds}{\ds\ts}\left\{\uvec\right\}_\Bs=
	\begin{Bmatrix}\dot{\us}_1		\begin{Bmatrix}\dot{\us}_1
	\\\dot{\us}_2		\\\dot{\us}_2

Bros at 18:20, 16 March 2023

2023-03-16T18:20:01Z

← Older revision		Revision as of 18:20, 16 March 2023
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	\newcommand{\Ps}{\textbf{P}}		\newcommand{\Ps}{\textbf{P}}
	\newcommand{\Ss}{\textbf{S}}		\newcommand{\Ss}{\textbf{S}}
	\newcommand{\deg}{^\textsf{o}}</math>		\newcommand{\deg}{^\textsf{o}}
			\newcommand{\vec}[1]{\overline{#1}}</math>
	==V.1 Geometric representation of a vector==		==V.1 Geometric representation of a vector==
	Vectors may be represented geometrically through an image, showing their '''direction''' with an arrow and their '''value''' with a scalar (positive or negative) ('''Figure V.1'''). Both direction and value may change with time.		Vectors may be represented geometrically through an image, showing their '''direction''' with an arrow and their '''value''' with a scalar (positive or negative) ('''Figure V.1'''). Both direction and value may change with time.
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	In '''Figure V.6''', the vector basis B = (1,2,3) does not change its orientation relative to R (it is a fixed basis in R), whereas the vector basis B = (1',2',3') does change its orientation relative to R but not relative to R’ (it is a fixed basis in R’ but a moving one in R): <math>\Omegavec^\Bs_\Rs = \vec 0</math>, <math>\Omegavec^\~~Bs_~~{\Rs'} \not= \vec{0}</math>.		In '''Figure V.6''', the vector basis B = (1,2,3) does not change its orientation relative to R (it is a fixed basis in R), whereas the vector basis B = (1',2',3') does change its orientation relative to R but not relative to R’ (it is a fixed basis in R’ but a moving one in R): <math>\Omegavec^\Bs_\Rs = \vec 0</math>, <math>\Omegavec^{\Bs'}_{\Rs} \not= \vec{0}</math>, <math>\Omegavec{\Bs}{\Rs}\neq\vec{0}</math>, <math>\Omegavec{\Bs'}{\Rs'}\neq\vec{0}</math>.


	~~En canvi, la base <math>B'=(1',2',3')</math> canvia d’orientació respecte de R però no respecte de R' (és una base fixa a R' però mòbil a R):~~ , <math>\Omegavec^{\Bs'}_\Rs \~~not=~~ 0</math>, <math>\Omegavec^{\Bs'}_{\Rs'} = 0</math>.


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	-------------		-------------

	==V.4 ~~Operacions entre~~ vectors ~~amb representació analítica~~==		==V.4 Operations between vectors with analytical representation==


	===~~Operacions instantànies~~: ~~suma~~, ~~producte escalar~~, ~~producte vectorial.~~===		===Instantaneous operations: addition, scalar product, vector product===

	~~Les operacions instantànies entre~~ vectors ~~es poden fer a través de les~~ bases ~~vectorials~~. ~~En ser instantànies~~, ~~el caràcter fix o mòbil de la base és irrellevant~~. ~~El que és fonamental és que tots dos~~ vectors ~~estiguin projectats a la mateixa base~~.		Instantaneous operations between vectors may be carried out through vector bases. As they are instantaneous, being fixed or moving bases is not relevant. What is absolutely necessary is that both vectors are projected on the same basis.

	<center>		<center>
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	===~~Operacions al llarg del temps~~: ~~derivació temporal (mètode analític)~~===		===Operations along time: time derivative===
	~~Aquest mètode també es coneix com '''derivació en base'''.~~

	~~Projectar un vector en una base vectorial <math>(\evec_1, \evec_2, \evec_3)</math> és expressar-lo com a suma de tres vectors ortogonals:~~

	~~<center><math>\uvec=\sum_{\is}^{}\us_\is\evec_\is</math></center>~~

	~~Si la base és fixa respecte de la referència~~ R ~~on es calcula la derivada~~, ~~aquests~~ vectors ~~no canvien d’orientació, i per tant~~:		Projecting a vector on a vector basis <math>(\evec_1, \evec_2, \evec_3)</math> means expressing it as an addition of three orthogonal vector s<math>\uvec=\sum_{\is}^{}\us_\is\evec_\is</math>.
			If it is a fixed basis in R (the reference frame where the operation is performed), those vectors have constant direction. Hence:

	<center>		<center>
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	</center>		</center>

	~~Si la base és mòbil respecte de~~ R, ~~aquests~~ vectors ~~canvien d’orientació amb velocitat~~ angular <math>~~\left\{~~\Omegavec^\Bs_\Rs~~\right\}_\Bs~~</math>:		If it is a moving basis in R, those vectors change its orientation according to an angular velocity <math>\Omegavec^\Bs_\Rs</math>:

	<center>		<center>