The first part demonstrates that the zero tensor acts as the additive identity for tensor addition, just like the number zero in regular arithmetic. The second part establishes that this property-having all zero components-is invariant under coordinate transformations. This means that if a tensor is the zero tensor in one coordinate system, it will be the zero tensor in all others. This is proven by the tensor transformation law, which shows that multiplying the zero components by the rotation matrix results in zero components in the new coordinate system, confirming the zero tensor is a true tensor.
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$\complement\cdots$Counselor
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the zero tensor acts as the additive identity for tensor addition. + This is analogous to how the number zero functions in scalar arithmetic. The relation $T+0 \otimes 0=T$ holds true because the components of the zero tensor, $(0 \otimes 0){i j}$*, are all equal to zero. This ensures that when you add the zero tensor to any other tensor $T$, its components $T{i j}$* remain unchanged.
The property of a tensor having all zero components is invariant under coordinate transformations. This means that if the components of a tensor are all zero in one coordinate system, they will also be zero in any other valid coordinate system. The tensor transformation law, $A_{k l}^{\prime}=R_{k i} R_{l j} A_{i j}$, confirms this. Since the components of the zero tensor in the initial system are $A_{i j}=0$, applying the transformation results in $A_{k l}^{\prime}= R_{k i} R_{l j}(0)=0$. This proves that the zero tensor is a true tensor-a mathematical object whose properties are independent of the coordinate system used to describe it.
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