Matrixdarstellung des totalen Differentials
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:
Sei Usubseteq mathbbR^n offen und sei f:Urightarrow mathbbR^m bei x_in U differenzierbar. Dann existiert für jedes vin mathbbR^n die Ableitung von f entlang v und es gilt partial_v fx_textD_x_fv. Insbesondere ist die totale Ableitung textD_x_fv eindeutig durch die partiellen Ableitungen bestimmt und es gilt textD_x_fpartial_fx_...partial_nfx_in textMat_mnmathbbR wobei letzteres auch die bf Jacobi-Matrix von f bei x_ genannt wird.
Solution:
Beweis. Nach Annahme existiert die totale Ableitung textD_x_f und es gilt fx_+hfx_+textD_x_fh+o||h|| für hrightarrow . Man setzt hsv für srightarrow und vin mathbbR^n womit gilt partial_vfx_limlimits_srightarrow fracfx_+sv-fx_slimlimits_srightarrow textD_x_fv+otextD_x_fv. Also existiert die Ableitungen von f entlang dem beliebigen Vektor v ist x_. Insbesonderen existieren alle partiellen Ableitung von f bei x_ und die partielle Ableitung in der j-ten Richtung für jin ...n stellt die j-te Spalte der Matrix textD_x_f dar wie behauptet.
Sei Usubseteq mathbbR^n offen und sei f:Urightarrow mathbbR^m bei x_in U differenzierbar. Dann existiert für jedes vin mathbbR^n die Ableitung von f entlang v und es gilt partial_v fx_textD_x_fv. Insbesondere ist die totale Ableitung textD_x_fv eindeutig durch die partiellen Ableitungen bestimmt und es gilt textD_x_fpartial_fx_...partial_nfx_in textMat_mnmathbbR wobei letzteres auch die bf Jacobi-Matrix von f bei x_ genannt wird.
Solution:
Beweis. Nach Annahme existiert die totale Ableitung textD_x_f und es gilt fx_+hfx_+textD_x_fh+o||h|| für hrightarrow . Man setzt hsv für srightarrow und vin mathbbR^n womit gilt partial_vfx_limlimits_srightarrow fracfx_+sv-fx_slimlimits_srightarrow textD_x_fv+otextD_x_fv. Also existiert die Ableitungen von f entlang dem beliebigen Vektor v ist x_. Insbesonderen existieren alle partiellen Ableitung von f bei x_ und die partielle Ableitung in der j-ten Richtung für jin ...n stellt die j-te Spalte der Matrix textD_x_f dar wie behauptet.
Meta Information
Exercise:
Sei Usubseteq mathbbR^n offen und sei f:Urightarrow mathbbR^m bei x_in U differenzierbar. Dann existiert für jedes vin mathbbR^n die Ableitung von f entlang v und es gilt partial_v fx_textD_x_fv. Insbesondere ist die totale Ableitung textD_x_fv eindeutig durch die partiellen Ableitungen bestimmt und es gilt textD_x_fpartial_fx_...partial_nfx_in textMat_mnmathbbR wobei letzteres auch die bf Jacobi-Matrix von f bei x_ genannt wird.
Solution:
Beweis. Nach Annahme existiert die totale Ableitung textD_x_f und es gilt fx_+hfx_+textD_x_fh+o||h|| für hrightarrow . Man setzt hsv für srightarrow und vin mathbbR^n womit gilt partial_vfx_limlimits_srightarrow fracfx_+sv-fx_slimlimits_srightarrow textD_x_fv+otextD_x_fv. Also existiert die Ableitungen von f entlang dem beliebigen Vektor v ist x_. Insbesonderen existieren alle partiellen Ableitung von f bei x_ und die partielle Ableitung in der j-ten Richtung für jin ...n stellt die j-te Spalte der Matrix textD_x_f dar wie behauptet.
Sei Usubseteq mathbbR^n offen und sei f:Urightarrow mathbbR^m bei x_in U differenzierbar. Dann existiert für jedes vin mathbbR^n die Ableitung von f entlang v und es gilt partial_v fx_textD_x_fv. Insbesondere ist die totale Ableitung textD_x_fv eindeutig durch die partiellen Ableitungen bestimmt und es gilt textD_x_fpartial_fx_...partial_nfx_in textMat_mnmathbbR wobei letzteres auch die bf Jacobi-Matrix von f bei x_ genannt wird.
Solution:
Beweis. Nach Annahme existiert die totale Ableitung textD_x_f und es gilt fx_+hfx_+textD_x_fh+o||h|| für hrightarrow . Man setzt hsv für srightarrow und vin mathbbR^n womit gilt partial_vfx_limlimits_srightarrow fracfx_+sv-fx_slimlimits_srightarrow textD_x_fv+otextD_x_fv. Also existiert die Ableitungen von f entlang dem beliebigen Vektor v ist x_. Insbesonderen existieren alle partiellen Ableitung von f bei x_ und die partielle Ableitung in der j-ten Richtung für jin ...n stellt die j-te Spalte der Matrix textD_x_f dar wie behauptet.
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