Kovariante und kontravariante Vektoren
About points...
We associate a certain number of points with each exercise.
When you click an exercise into a collection, this number will be taken as points for the exercise, kind of "by default".
But once the exercise is on the collection, you can edit the number of points for the exercise in the collection independently, without any effect on "points by default" as represented by the number here.
That being said... How many "default points" should you associate with an exercise upon creation?
As with difficulty, there is no straight forward and generally accepted way.
But as a guideline, we tend to give as many points by default as there are mathematical steps to do in the exercise.
Again, very vague... But the number should kind of represent the "work" required.
When you click an exercise into a collection, this number will be taken as points for the exercise, kind of "by default".
But once the exercise is on the collection, you can edit the number of points for the exercise in the collection independently, without any effect on "points by default" as represented by the number here.
That being said... How many "default points" should you associate with an exercise upon creation?
As with difficulty, there is no straight forward and generally accepted way.
But as a guideline, we tend to give as many points by default as there are mathematical steps to do in the exercise.
Again, very vague... But the number should kind of represent the "work" required.
About difficulty...
We associate a certain difficulty with each exercise.
When you click an exercise into a collection, this number will be taken as difficulty for the exercise, kind of "by default".
But once the exercise is on the collection, you can edit its difficulty in the collection independently, without any effect on the "difficulty by default" here.
Why we use chess pieces? Well... we like chess, we like playing around with \(\LaTeX\)-fonts, we wanted symbols that need less space than six stars in a table-column... But in your layouts, you are of course free to indicate the difficulty of the exercise the way you want.
That being said... How "difficult" is an exercise? It depends on many factors, like what was being taught etc.
In physics exercises, we try to follow this pattern:
Level 1 - One formula (one you would find in a reference book) is enough to solve the exercise. Example exercise
Level 2 - Two formulas are needed, it's possible to compute an "in-between" solution, i.e. no algebraic equation needed. Example exercise
Level 3 - "Chain-computations" like on level 2, but 3+ calculations. Still, no equations, i.e. you are not forced to solve it in an algebraic manner. Example exercise
Level 4 - Exercise needs to be solved by algebraic equations, not possible to calculate numerical "in-between" results. Example exercise
Level 5 -
Level 6 -
When you click an exercise into a collection, this number will be taken as difficulty for the exercise, kind of "by default".
But once the exercise is on the collection, you can edit its difficulty in the collection independently, without any effect on the "difficulty by default" here.
Why we use chess pieces? Well... we like chess, we like playing around with \(\LaTeX\)-fonts, we wanted symbols that need less space than six stars in a table-column... But in your layouts, you are of course free to indicate the difficulty of the exercise the way you want.
That being said... How "difficult" is an exercise? It depends on many factors, like what was being taught etc.
In physics exercises, we try to follow this pattern:
Level 1 - One formula (one you would find in a reference book) is enough to solve the exercise. Example exercise
Level 2 - Two formulas are needed, it's possible to compute an "in-between" solution, i.e. no algebraic equation needed. Example exercise
Level 3 - "Chain-computations" like on level 2, but 3+ calculations. Still, no equations, i.e. you are not forced to solve it in an algebraic manner. Example exercise
Level 4 - Exercise needs to be solved by algebraic equations, not possible to calculate numerical "in-between" results. Example exercise
Level 5 -
Level 6 -
Question
Solution
Short
Video
\(\LaTeX\)
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Exercise:
Gegeben seien die folgen beiden Paare von orthonormalen Basisvektoren. Finde die Komponenten des roten Vektors in beiden Basen und die entsprechen Transformationsmatrizen! center tikzpicturevect/.stylstealth' scope tkzInitxmax ymax tkzGridsub subxstep subystep tkzDefPoA tkzDefPoB tkzDefPoC tkzDrawSegmentsvect colorgreen!!black thickAB tkzDrawSegmentsvect colorgreen!!black thickAC tkzLabelSegmentbelow colorgreen!!blackAB e_x tkzLabelSegmentleft colorgreen!!blackAC e_y scope scopedashed yshiftcm tkzInitxmax ymax tkzGridsub subxstep subystep tkzDefPoA tkzDefPoB tkzDefPoC tkzDrawSegmentsvect colorgreen!!black thickAB tkzDrawSegmentsvect colorgreen!!black thickAC tkzLabelSegmentbelow colorgreen!!blackAB tilde e_x tkzLabelSegmentleft colorgreen!!blackAC tilde e_y scope scopexshiftcm yshiftcm tkzInitxmin- xmax ymin- ymax tkzGridsub subxstep subystep tkzDefPoA tkzDefPoB tkzDrawSegmentsvect colorred ultra thickAB scope tikzpicture center
Solution:
Der rote Vektor ist rotfinvariant. Aber die Komponenten des Vektors hängen von der gewählten Basis ab und sind damit verschieden: v e_x + e_y e_x e_y mqty e_x e_y mqtydmat mqty e_x e_y mqtydmat mqtydmatfracfrac mqty e_x e_y mqtydmatfracfrac mqty v frac tilde e_x + frac tilde e_y tilde e_x tilde e_y mqtyfrac frac Die Gleichungen zeigen dass man die Änderung der blaufkovarianten Basisvektoren wie folgt schreiben kann: tilde e_x tilde e_y e_x e_y e_x e_y mqtydmat Die blaufkontravarianten Komponenten des Vektors transformieren konträr mit der inversen Matrix: mqtyfrac frac mqtydmatfracfrac mqty
Gegeben seien die folgen beiden Paare von orthonormalen Basisvektoren. Finde die Komponenten des roten Vektors in beiden Basen und die entsprechen Transformationsmatrizen! center tikzpicturevect/.stylstealth' scope tkzInitxmax ymax tkzGridsub subxstep subystep tkzDefPoA tkzDefPoB tkzDefPoC tkzDrawSegmentsvect colorgreen!!black thickAB tkzDrawSegmentsvect colorgreen!!black thickAC tkzLabelSegmentbelow colorgreen!!blackAB e_x tkzLabelSegmentleft colorgreen!!blackAC e_y scope scopedashed yshiftcm tkzInitxmax ymax tkzGridsub subxstep subystep tkzDefPoA tkzDefPoB tkzDefPoC tkzDrawSegmentsvect colorgreen!!black thickAB tkzDrawSegmentsvect colorgreen!!black thickAC tkzLabelSegmentbelow colorgreen!!blackAB tilde e_x tkzLabelSegmentleft colorgreen!!blackAC tilde e_y scope scopexshiftcm yshiftcm tkzInitxmin- xmax ymin- ymax tkzGridsub subxstep subystep tkzDefPoA tkzDefPoB tkzDrawSegmentsvect colorred ultra thickAB scope tikzpicture center
Solution:
Der rote Vektor ist rotfinvariant. Aber die Komponenten des Vektors hängen von der gewählten Basis ab und sind damit verschieden: v e_x + e_y e_x e_y mqty e_x e_y mqtydmat mqty e_x e_y mqtydmat mqtydmatfracfrac mqty e_x e_y mqtydmatfracfrac mqty v frac tilde e_x + frac tilde e_y tilde e_x tilde e_y mqtyfrac frac Die Gleichungen zeigen dass man die Änderung der blaufkovarianten Basisvektoren wie folgt schreiben kann: tilde e_x tilde e_y e_x e_y e_x e_y mqtydmat Die blaufkontravarianten Komponenten des Vektors transformieren konträr mit der inversen Matrix: mqtyfrac frac mqtydmatfracfrac mqty
Meta Information
Exercise:
Gegeben seien die folgen beiden Paare von orthonormalen Basisvektoren. Finde die Komponenten des roten Vektors in beiden Basen und die entsprechen Transformationsmatrizen! center tikzpicturevect/.stylstealth' scope tkzInitxmax ymax tkzGridsub subxstep subystep tkzDefPoA tkzDefPoB tkzDefPoC tkzDrawSegmentsvect colorgreen!!black thickAB tkzDrawSegmentsvect colorgreen!!black thickAC tkzLabelSegmentbelow colorgreen!!blackAB e_x tkzLabelSegmentleft colorgreen!!blackAC e_y scope scopedashed yshiftcm tkzInitxmax ymax tkzGridsub subxstep subystep tkzDefPoA tkzDefPoB tkzDefPoC tkzDrawSegmentsvect colorgreen!!black thickAB tkzDrawSegmentsvect colorgreen!!black thickAC tkzLabelSegmentbelow colorgreen!!blackAB tilde e_x tkzLabelSegmentleft colorgreen!!blackAC tilde e_y scope scopexshiftcm yshiftcm tkzInitxmin- xmax ymin- ymax tkzGridsub subxstep subystep tkzDefPoA tkzDefPoB tkzDrawSegmentsvect colorred ultra thickAB scope tikzpicture center
Solution:
Der rote Vektor ist rotfinvariant. Aber die Komponenten des Vektors hängen von der gewählten Basis ab und sind damit verschieden: v e_x + e_y e_x e_y mqty e_x e_y mqtydmat mqty e_x e_y mqtydmat mqtydmatfracfrac mqty e_x e_y mqtydmatfracfrac mqty v frac tilde e_x + frac tilde e_y tilde e_x tilde e_y mqtyfrac frac Die Gleichungen zeigen dass man die Änderung der blaufkovarianten Basisvektoren wie folgt schreiben kann: tilde e_x tilde e_y e_x e_y e_x e_y mqtydmat Die blaufkontravarianten Komponenten des Vektors transformieren konträr mit der inversen Matrix: mqtyfrac frac mqtydmatfracfrac mqty
Gegeben seien die folgen beiden Paare von orthonormalen Basisvektoren. Finde die Komponenten des roten Vektors in beiden Basen und die entsprechen Transformationsmatrizen! center tikzpicturevect/.stylstealth' scope tkzInitxmax ymax tkzGridsub subxstep subystep tkzDefPoA tkzDefPoB tkzDefPoC tkzDrawSegmentsvect colorgreen!!black thickAB tkzDrawSegmentsvect colorgreen!!black thickAC tkzLabelSegmentbelow colorgreen!!blackAB e_x tkzLabelSegmentleft colorgreen!!blackAC e_y scope scopedashed yshiftcm tkzInitxmax ymax tkzGridsub subxstep subystep tkzDefPoA tkzDefPoB tkzDefPoC tkzDrawSegmentsvect colorgreen!!black thickAB tkzDrawSegmentsvect colorgreen!!black thickAC tkzLabelSegmentbelow colorgreen!!blackAB tilde e_x tkzLabelSegmentleft colorgreen!!blackAC tilde e_y scope scopexshiftcm yshiftcm tkzInitxmin- xmax ymin- ymax tkzGridsub subxstep subystep tkzDefPoA tkzDefPoB tkzDrawSegmentsvect colorred ultra thickAB scope tikzpicture center
Solution:
Der rote Vektor ist rotfinvariant. Aber die Komponenten des Vektors hängen von der gewählten Basis ab und sind damit verschieden: v e_x + e_y e_x e_y mqty e_x e_y mqtydmat mqty e_x e_y mqtydmat mqtydmatfracfrac mqty e_x e_y mqtydmatfracfrac mqty v frac tilde e_x + frac tilde e_y tilde e_x tilde e_y mqtyfrac frac Die Gleichungen zeigen dass man die Änderung der blaufkovarianten Basisvektoren wie folgt schreiben kann: tilde e_x tilde e_y e_x e_y e_x e_y mqtydmat Die blaufkontravarianten Komponenten des Vektors transformieren konträr mit der inversen Matrix: mqtyfrac frac mqtydmatfracfrac mqty
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