<s>[INST] How do you define a symbol to be a constant? [/INST] To define a symbol to be a constant, you can use the `DefineConstant` function. For example: In[1]:= DefineConstant[c] This command will treat `c` as a constant.</s>


<s>[INST] How do you define a symbol to be a complex constant? [/INST] To define a symbol to be a complex constant, use the `DefineConstant` function with the `Type -> Complex` option. For example: In[2]:= DefineConstant[d, Type -> Complex] This command will treat `d` as a complex constant.</s>


<s>[INST] How do you change the Type of a constant after it has been defined? [/INST] To change the Type of a constant after it has been defined, use the `Declare` function. For example: In[3]:= Declare[d, Type -> {NonNegative, Real}] This command will update the type of `d`.</s>


<s>[INST] How do you create a Tensor? [/INST] To create a tensor, use the `DefineTensor` function. For example: In[4]:= DefineTensor[alpha, 1] This command defines `alpha` as a rank-1 tensor.</s>


<s>[INST] How do you define a contravariant 1-tensor? [/INST] To define a contravariant 1-tensor, use the `DefineTensor` function with the `Variance -> Contravariant` option. For example: In[5]:= DefineTensor[v, 1, Variance -> Contravariant] This defines `v` as a contravariant 1-tensor.</s>


<s>[INST] How do you create a scalar function? [/INST] To create a scalar function, define a rank-0 tensor using the `DefineTensor` function with the appropriate type. For example: In[6]:= DefineTensor[u, 0, Type -> {NonNegative, Real}] This command defines `u` as a scalar function with the specified type.</s>


<s>[INST] How do you specify tensor symmetries for higher-rank tensors? [/INST] To specify symmetries for higher-rank tensors, use the `Symmetries` option in the `DefineTensor` function. For example: In[7]:= DefineTensor[h, 2, Symmetries -> Symmetric] This defines `h` as a symmetric rank-2 tensor.</s>


<s>[INST] How do you associate a tensor with a bundle other than the default tangent bundle? [/INST] To associate a tensor with a different bundle, use the `Bundle` option in the `DefineTensor` function. For example: In[8]:= DefineTensor[omega, 2, Bundle -> fiber, Type -> Complex, Variance -> {Contravariant, Covariant}] This associates `omega` with the `fiber` bundle as a mixed variance complex tensor.</s>


<s>[INST] How do you change certain options, such as Type and Bundle, after a tensor has been defined? [/INST] To change options for a tensor after it has been defined, use the `Declare` function. For example: In[9]:= Declare[u, Type -> Complex] This updates the type of `u` to complex.</s>


<s>[INST] How do you expand any tensor expression in terms of the default basis? [/INST] To expand a tensor expression in terms of the default basis, use the `BasisExpand` function. For example: In[10]:= BasisExpand[alpha] This expands `alpha` into its component expressions multiplied by the default basis vectors.</s>


<s>[INST] How do you allow Ricci to correctly interpret an expression such as f[u]? [/INST] To allow Ricci to correctly interpret a function like `f[u]`, use the `DefineMathFunction` function. For example: In[11]:= DefineMathFunction[f, Type -> Complex] This command defines `f` as a mathematical function with complex type.</s>


<s>[INST] How do you get a tensor product? [/INST] To compute a tensor product, use the `TensorProduct` function. For example: In[12]:= TensorProduct[alpha, beta] This command will compute the tensor product of `alpha` and `beta`.</s>


<s>[INST] How do you output a wedge product? [/INST] To output a wedge product, use the `Wedge` function. For example: In[13]:= Wedge[alpha, beta] This command computes the wedge product of `alpha` and `beta`.</s>


<s>[INST] How do you get a symmetric product? [/INST] To compute a symmetric product, use the `SymmetricProduct` function. For example: In[14]:= SymmetricProduct[alpha, beta] This command will compute the symmetric product of `alpha` and `beta`.</s>


<s>[INST] How do you get a dot product? [/INST] To compute a dot product, use the standard vector notation. For example: In[15]:= DotProduct[a, e] This command computes the dot product of `a` and `e`.</s>


<s>[INST] How do you get an inner product? [/INST] To compute an inner product, use the `Inner` function. For example: In[16]:= Inner[a, b] This command computes the inner product of `a` and `b`.</s>


<s>[INST] How do you apply a constant factor in each term of x? [/INST] To apply a constant factor to each term in an expression, use the `FactorConstants` function. For example: In[17]:= FactorConstants[x] This command applies a constant factor in each term of `x`.</s>


<s>[INST] How do you expand all tensors in x into component expressions multiplied by the default basis vectors and covectors? [/INST] To expand all tensors in an expression into their component forms, use the `BasisExpand` function. For example: In[18]:= BasisExpand[2 * a] This expands the tensors in `2 * a` into their component expressions.</s>


<s>[INST] How do you convert a component expression for a tensor back to pure tensor form? [/INST] To convert a component expression back to pure tensor form, use the `BasisGather` function. For example: In[19]:= BasisGather[%18, a] This command converts the components of `a` back into its tensor form.</s>


<s>[INST] How do you cause one tensor to always be replaced by some other tensor expression, or to be replaced only when certain indices are inserted? [/INST] To define a substitution for a tensor, use the `DefineRelation` function. For example: In[20]:= DefineRelation[a, 2 * b] This command will substitute `a` with `2 * b` whenever `a` appears.</s>