Update docs

README.md

@@ -6,11 +6,7 @@
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-Tensorial provides useful tensor operations (e.g., contraction; tensor product, `⊗`; `inv`; etc.) written in the [Julia programming language](https://julialang.org).
-The library supports arbitrary size of non-symmetric and symmetric tensors, where symmetries should be specified to avoid wasteful duplicate computations.
-The way to give a size of the tensor is similar to [StaticArrays.jl](https://github.com/JuliaArrays/StaticArrays.jl), and symmetries of tensors can be specified by using `@Symmetry`.
-For example, symmetric fourth-order tensor (symmetrizing tensor) is represented in this library as `Tensor{Tuple{@Symmetry{3,3}, @Symmetry{3,3}}}`.
-Einstein summation macro and automatic differentiation functions are also provided.
+Tensorial provides useful tensor operations, such as contraction, tensor product (`⊗`), and inversion (`inv`), implemented in the Julia programming language. The library supports tensors of arbitrary size, including both symmetric and non-symmetric tensors, where symmetries can be specified to avoid redundant computations. The approach for defining the size of a tensor is similar to that used in [StaticArrays.jl](https://github.com/JuliaArrays/StaticArrays.jl), and tensor symmetries can be specified using the `@Symmetry` macro. For instance, a symmetric fourth-order tensor (a symmetrized tensor) is represented in this library as `Tensor{Tuple{@Symmetry{3,3}, @Symmetry{3,3}}}`. The library also includes an Einstein summation macro `@einsum` and functions for automatic differentiation, such as `gradient` and `hessian`.
 
 ## Speed
 

docs/src/Tensor Type.md

@@ -2,7 +2,7 @@
 
 ## Type parameters
 
-All tensors are represented by a type `Tensor{S, T, N, L}` where each type parameter represents following:
+All tensors in Tensorial.jl are represented by the type `Tensor{S, T, N, L}`, where each type parameter represents the following:
 
 - `S`: The size of `Tensor`s which is specified by using `Tuple` (e.g., 3x2 tensor becomes `Tensor{Tuple{3,2}}`).
 - `T`: The type of element which must be `T <: Real`.
@@ -13,10 +13,7 @@ Basically, the type parameters `N` and `L` do not need to be specified for const
 
 ## `Symmetry`
 
-If possible, specifying the symmetry of the tensor is good for performance since Tensorial.jl provides the optimal computations.
-The symmetries can be applied using `Symmetry` in type parameter `S` (e.g., `Symmetry{Tuple{3,3}}`).
-`@Symmetry` macro can omit `Tuple` like `@Symmetry{2,2}`.
-The following are examples to specify symmetries:
+Specifying the symmetry of a tensor can improve performance, as Tensorial.jl provides optimized computations for symmetric tensors. Symmetries can be applied using `Symmetry` in the type parameter `S` (e.g., `Symmetry{Tuple{3,3}}`). The `@Symmetry` macro simplifies this process by allowing you to omit `Tuple`, as in `@Symmetry{2,2}`. Below are some examples of how to specify symmetries:
 
 - ``A_{(ij)}`` with 3x3: `Tensor{Tuple{@Symmetry{3,3}}}`
 - ``A_{(ij)k}`` with 3x3x2: `Tensor{Tuple{@Symmetry{3,3}, 2}}`

docs/src/index.md

@@ -2,11 +2,7 @@
 
 ## Introduction
 
-Tensorial provides useful tensor operations (e.g., contraction; tensor product, `⊗`; `inv`; etc.) written in the [Julia programming language](https://julialang.org).
-The library supports arbitrary size of non-symmetric and symmetric tensors, where symmetries should be specified to avoid wasteful duplicate computations.
-The way to give a size of the tensor is similar to [StaticArrays.jl](https://github.com/JuliaArrays/StaticArrays.jl), and symmetries of tensors can be specified by using `@Symmetry`.
-For example, symmetric fourth-order tensor (symmetrizing tensor) is represented in this library as `Tensor{Tuple{@Symmetry{3,3}, @Symmetry{3,3}}}`.
-Einstein summation macro and automatic differentiation functions are also provided.
+Tensorial provides useful tensor operations, such as contraction, tensor product (`⊗`), and inversion (`inv`), implemented in the Julia programming language. The library supports tensors of arbitrary size, including both symmetric and non-symmetric tensors, where symmetries can be specified to avoid redundant computations. The approach for defining the size of a tensor is similar to that used in [StaticArrays.jl](https://github.com/JuliaArrays/StaticArrays.jl), and tensor symmetries can be specified using the `@Symmetry` macro. For instance, a symmetric fourth-order tensor (a symmetrized tensor) is represented in this library as `Tensor{Tuple{@Symmetry{3,3}, @Symmetry{3,3}}}`. The library also includes an Einstein summation macro `@einsum` and functions for automatic differentiation, such as `gradient` and `hessian`.
 
 ## Installation