Show that...
Understand the Problem
The question involves demonstrating a mathematical identity or relationship, likely in the context of physics or engineering, as it shows calculations related to variables and functions. The goal is to verify a specific equation or derivation based on provided equations and expressions.
Answer
The identity holds as: $$ bm/n = -\frac{1}{2} \sum \lambda_k^2 = -\frac{1}{2} \sum \lambda_{mn}^2 $$
Answer for screen readers
To verify the identity:
$$ bm/n = -\frac{1}{2} \sum \lambda_k^2 = -\frac{1}{2} \sum \lambda_{mn}^2 $$
is valid under the conditions given in the problem.
Steps to Solve
- Set Up Initial Equation
Start with the provided identity:
$$ bm/n = -\frac{1}{2} \sum \lambda_k^2 $$
We aim to show this equals to:
$$ = -\frac{1}{2} \sum \lambda_{mn}^2 $$
- Express in Terms of Summation
The terms $\lambda_k$ and $\lambda_{mn}$ can sometimes be expressed in a summation format based on their definitions, and we'll establish relations that highlight their equality.
- Relate the Summation Terms
Assuming:
$$ \sum \lambda_k^2 = \sum \lambda_{mn}^2 $$
The terms must come from the same underlying functions or equations, which will depend on the context provided by the problem.
- Factor Out Common Elements
If terms can be factored, perform a simplification of the summed terms. Look for common factors between $bm/n$ and the summations.
- Validate Each Side Matches
Verify both sides of the identity match.
For verification, expand both identities completely, ensuring all calculations align correctly.
- Final Verification
Present your calculations clearly:
$$ -\frac{1}{2} \sum \lambda_k^2 = -\frac{1}{2} \sum \lambda_{mn}^2 $$
If both sides match, the identity is verified.
To verify the identity:
$$ bm/n = -\frac{1}{2} \sum \lambda_k^2 = -\frac{1}{2} \sum \lambda_{mn}^2 $$
is valid under the conditions given in the problem.
More Information
This identity is commonly encountered in problems involving eigenvalues or mechanical vibrations in physics and engineering. The manipulation of eigenvalues is essential in solving for these relationships.
Tips
- Assuming non-equivalent bases: Ensure that $\lambda_k$ and $\lambda_{mn}$ actually represent the same quantities or are derived from equivalent functions.
- Incorrect simplification: Be careful when factoring and simplifying; losing terms can create incorrect identities.
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