Explanation of the Alternator

In summary, the Alternator $$Alt(T)$$ is a multilinear function defined as $$Alt(T):= \frac{1}{k!} \sum_{\sigma \in S_n} sgn(\sigma) T (v_{\sigma(1)},...,v_{\sigma(k)}))$$ where $$S_n$$ is the group of permutations and sgn is the signum of the permutation. The alternator is called alternating because it produces a minus sign when two argument vectors are changed, and it counts the number of mismatches in a tensor. This generalizes to tensors of different ranks by replacing $$\dfrac{1}{k!}$$ with $$\binom{k+l}{k}$$.
  • #1
let us define the Alternator $$Alt(T)$$ where T is a multilinear function
$$Alt(T):= \frac{1}{k!} \sum_{\sigma \in S_n} sgn(\sigma) T (v_{\sigma(1)},...,v_{\sigma(k)}))$$.

Further recognize that


is the group of permutations and sgn the signum of the permutation.
May someone explain me why the alternator is alternating, thus to say if I change two argument vectors in Alt, then a minus will appear, as an example

$$w(v_1,...,v_n)=- w(v_n,...,v_1)$$
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  • #2
It should be formulated a bit more precise. Let's assume we have a tensor ##T=v_1\otimes \ldots \otimes v_k##.
Then the alternator makes an alternating tensor out of it, that is it is a mapping from non-alternating to alternating tensors, because it counts the number of mismatches (##\tau = (12)##):
\operatorname{Alt}_2(T)=\operatorname{Alt}_2(v_1\otimes v_2)= \dfrac{1}{2} \left(\operatorname{sgn}(\operatorname{id}) v_{\operatorname{id}(1)}\otimes v_{\operatorname{id}(2)} + \operatorname{sgn}(\tau)v_{\tau(1)}\otimes v_{\tau(2)} \right)=\dfrac{1}{2}\left(v_1\otimes v_2 - v_2\otimes v_1\right)
If you now look on what ##\operatorname{Alt}_2## did with ##T=v_1\otimes v_2##, you will find ##\operatorname{Alt}_2(v_1\otimes v_2)=-\operatorname{Alt}_2(v_2\otimes v_1)## which is why it is called alternator. This generalizes to ##k-##homogenous tensors and by replacing ##\dfrac{1}{k!}## by ##\binom{k+l}{k}## to pairs of tensors of different ranks.
  • #3
Thank you very much for your answer, I got it.

What is an alternator?

An alternator is an electrical device that converts mechanical energy into electrical energy. It is commonly used in vehicles to recharge the battery and power the electrical systems while the engine is running.

How does an alternator work?

An alternator works by using a magnetic field and a rotating loop of wire to generate electricity. As the engine runs, a belt turns the alternator pulley, causing the magnetic field to rotate and induce an electrical current in the wire loop. This current is then converted into usable electricity.

What are the main components of an alternator?

The main components of an alternator include the rotor, stator, diode rectifier, voltage regulator, and bearings. The rotor is the rotating part that creates the magnetic field, while the stator is the stationary part that contains the wire loop. The diode rectifier converts the alternating current produced by the alternator into direct current, and the voltage regulator controls the output voltage. Bearings help to support and stabilize the rotating parts of the alternator.

What are the signs of a failing alternator?

Some common signs of a failing alternator include dimming headlights, a dead battery, strange noises coming from the alternator, and warning lights on the dashboard. If you notice any of these symptoms, it is important to have your alternator checked and potentially replaced to avoid further damage to your vehicle.

How can I maintain my alternator?

To maintain your alternator, it is important to regularly check the belt that drives it for wear and tear, and replace it if necessary. Keeping the alternator and surrounding area clean can also help prevent debris from interfering with its operation. It is also important to have your alternator checked during routine vehicle maintenance and replace it when needed to ensure optimal performance.

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