Practice Orthogonality and Eigenfunction Expansion - 18.8 | 18. Separation of Variables, Use of Fourier Series | Mathematics (Civil Engineering -1)
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18.8 - Orthogonality and Eigenfunction Expansion

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Learning

Practice Questions

Test your understanding with targeted questions related to the topic.

Question 1

Easy

What is the definition of orthogonality in the context of functions?

💡 Hint: Think of angles and how they relate to functions.

Question 2

Easy

Explain the importance of orthogonality in computing Fourier coefficients.

💡 Hint: Remember how each function overlaps with another.

Practice 1 more question and get performance evaluation

Interactive Quizzes

Engage in quick quizzes to reinforce what you've learned and check your comprehension.

Question 1

What is the significance of orthogonality in Fourier series?

  • It ensures functions add up correctly.
  • It allows for overlapping contributions.
  • It prevents distortion in the series.
  • It guarantees maximum amplitude.

💡 Hint: Think about how functions interact with each other.

Question 2

True or False: The inner product of two orthogonal functions over a defined interval equals zero.

  • True
  • False

💡 Hint: Recall the definition of orthogonality.

Solve and get performance evaluation

Challenge Problems

Push your limits with challenges.

Question 1

Consider the eigenfunctions sin(nπx/L) and sin(mπx/L). Prove their orthogonality by evaluating their inner product over the interval [0, L].

💡 Hint: Use integration by parts or trigonometric identities.

Question 2

Derive the formula for the nth Fourier coefficient of a function using inner products. Explain how this relates to the concept of orthogonality.

💡 Hint: Recall the definition of inner products and apply to your function.

Challenge and get performance evaluation