hilbert2 said:At least in the tensor-related textbook I've been reading most recently, it is said that the same index should not be repeated more than twice in any expression. For instance, if ##\mathbf{a},\mathbf{b}## and ##\mathbf{c}## are three-component vectors, you shouldn't use the shorthand notation ##a_i b_i c_i = a_1 b_1 c_1 + a_2 b_2 c_2 + a_3 b_3 c_3##.
emq said:Is (Tii)2 equivalent to (∑i = 1nTii)2? That is, when you encounter parentheses with Einstein summation, you perform the summation first and then apply any mathematical operations indicated by the parentheses? The solutions manual gives a solution to a problem I've been working out seems to indicate this is the case, but I haven't seen it stated as a rule.
The summation convention is used in scientific notation to simplify and condense mathematical expressions that involve repeated summations. It allows for a more compact and efficient representation of these expressions.
Yes, parentheses can be used in summation convention expressions to indicate grouping or order of operations. However, it is important to follow the standard conventions for parentheses, such as using them in pairs and using them to enclose the correct terms.
The summation convention is typically used when there are repeated terms in a mathematical expression, such as in a series or sequence. It is also commonly used in vector and tensor calculations. If you are unsure, consult a reference or your instructor for guidance.
Yes, there are several rules for simplifying summation convention expressions. These include the distributive property, combining like terms, and using known formulas or identities. It is important to also pay attention to the index or variable used in the summation, as it may change throughout the expression.
To convert a summation convention expression into standard notation, you can expand the notation by writing out each term in the summation and then simplifying the resulting expression. It is also helpful to use known formulas or identities to simplify the expression further.