Support Eq, tensor_cross, nabla symbol and component() method #587
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Hey @WenyinWei , let's discuss more about this. |
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I would collect more info and record it here, about how other symbol engines handle such stuff. |
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Let’s support$\nabla$ symbol in
Eq,tensor_cross,Eqandcomponent()method. Furthermore, allow the user to express the equation in compact (tensor symbol) form or component form, e.g.Compact form:$\mathrm{A} \cdot \mathrm{B}$
Component form:$A^i_j B^j_k \hat{e}_i \hat{e}^j\hat{e}_j \hat{e}^k = A^i_j B^j_k \hat{e}_i \hat{e}^k$
🎯 Goal
Let Einsteinpy be a standard package in the domain of CAS-tensor. I found einsteinpy even more powerful and elegant than the
sympy.tensormodule and anything else. Let’s build it into a perfect package.Introducing the
Eqclass in einsteinpy would magnificently enhance the expression capability of einsteinpy and allow we express the equality relation of tensor concisely and easy-to-understand.The users may choose to express their tensor relation in compact and symbolic form. Let$\nabla\times$ in the compact equation form, while $\epsilon_{ijk}$ in the component equation form. Similar is the
tensor_crossto be liketensor_prod.Users of einsteinpy would benefit a lot from these features because they don’t even need to write formulas by hand in
.ipynbMarkdown blocks.💡 Possible solutions
Inherit a class
TEqfromsympy.Eq, therefore in einsteinpy we can flexibily reuse and override the pre-existing methods of sympy and utilize its functionality.It’s easy to deduce the component form from the compact one, however, not vice versa. So I suggest to preserve the compact form info in the
Eqclass instead of the info of the components.Let the
tensor_prodallow multiple indexes contraction. It turns out to be cumbersome and easy to make mistakes if the user have to calculate the indexes by hand for multiple contraction. They know the indexes exactly but they would make mistakes if they have to compute indexes one contraction by one contraction, because the indexes shift; the same astensor_cross.The$\nabla$ symbol in $\nabla$ , $\nabla \times$ and $\nabla \cdot$ . I do not suggest
TEqcan be a lazy symbol, which doesn’t transform to the components automatically but exists as a pure symbol. To make it smart enough to be able to derive the component form, the combination withChristoffel tensorsare necessary to implement thedoit()method to compute the lazy symbol.component()orcomponents()might be better for tensors.The$\nabla$ ’s laziness might be helpful to make the vector and tensor identities possible. It would spend a lot of work on pattern matching to make $\nabla$ intelligent.
Wish someday it knows:$\nabla \times (\nabla \times \vec{A}) = \nabla (\nabla \cdot \vec{A}) - \nabla^2 \vec{A}$
📋 Steps to solve the problem
2021.5.11 Submitted the request and communicate with the community.
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