Is the summation notation for three equivalent expressions?

In summary, summation notation, also known as sigma notation, is a mathematical shorthand notation used to represent the sum of a series of numbers or terms. It is read as "the sum of," followed by the expression or equation to be summed up, and the values or indices of the variable that will be substituted into the expression or equation. The limits of summation indicate the starting and ending points of the summation, and it is commonly used in science to represent repeated calculations or processes. Some common properties of summation notation include the distributive and associative properties, as well as following the same rules as regular algebraic notation.
  • #1
schwarzschild
15
1
Are the following three equivalent?

[tex] P_{\alpha}A^{\beta}\tilde{\omega}^{\beta}(\vec{e_{\beta}} ) [/tex] [tex]= \sum_{\alpha = 0}^{3}{P_{\alpha}\tilde{\omega}^{\alpha}(\sum_{\beta = 0}^{3}{A^{\beta}\vec{e}_{\beta}) = \sum_{\alpha = 0}^{3}{P_{\alpha}A^{\alpha} [/tex]
 
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  • #2
The second equality is correct, assuming that the omegas are dual basis vectors. The first expression doesn't make sense, but it looks like a typo. Change the second beta to an alpha, and the first equality is good too.
 
  • #3
Don't use two sets of tex /tex tags; enclose the whole equality in one set of tex /tex tags.
 

1. What is summation notation?

Summation notation, also known as sigma notation, is a mathematical shorthand notation used to represent the sum of a series of numbers or terms. It is denoted by the Greek letter sigma (Σ) followed by the expression or equation to be summed up, and the limits of the summation.

2. How do you read summation notation?

Summation notation is read as "the sum of," followed by the expression or equation to be summed up, and the values or indices of the variable that will be substituted into the expression or equation. For example, "the sum of x squared from 1 to n" would be read as "the sum of x squared from 1 to n."

3. What are the limits of summation?

The limits of summation, also known as the upper and lower bounds, indicate the starting and ending points of the summation. The lower bound is the value at which the summation starts, while the upper bound is the value at which the summation ends. These values are typically represented by numbers or variables.

4. How is summation notation used in science?

Summation notation is commonly used in science to represent repeated calculations or processes, such as in physics equations, statistical formulas, and chemistry reactions. It allows for a more concise and efficient representation of complex mathematical expressions and can be used to solve problems involving series, sequences, and patterns.

5. What are some common properties of summation notation?

Some common properties of summation notation include the distributive property, which allows you to distribute a constant or coefficient to each term in the expression being summed, and the associative property, which allows you to change the grouping of terms without changing the result of the summation. Additionally, summation notation follows the same rules as regular algebraic notation, such as the commutative and identity properties.

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