Use the Levi-Civita symbol and Kronecker delta to prove the vector identities: A x (B x C) + Bx (C x A) + Cx (A × B) = 0 ; (A x B). (C x D) = (A · C) (BD) – (A · D) (B-C) ; -
Use the Levi-Civita symbol and Kronecker delta to prove the vector identities: A x (B x C) + Bx (C x A) + Cx (A × B) = 0 ; (A x B). (C x D) = (A · C) (BD) – (A · D) (B-C) ; -
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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![**Exercise 18**: Use the Levi-Civita symbol and Kronecker delta to prove the vector identities:
a. \( \mathbf{A} \times (\mathbf{B} \times \mathbf{C}) + \mathbf{B} \times (\mathbf{C} \times \mathbf{A}) + \mathbf{C} \times (\mathbf{A} \times \mathbf{B}) = \emptyset \);
b. \( (\mathbf{A} \times \mathbf{B}) \cdot (\mathbf{C} \times \mathbf{D}) = (\mathbf{A} \cdot \mathbf{C})(\mathbf{B} \cdot \mathbf{D}) - (\mathbf{A} \cdot \mathbf{D})(\mathbf{B} \cdot \mathbf{C}) \);
c. \( (\mathbf{A} \times \mathbf{B}) \times (\mathbf{C} \times \mathbf{D}) = [(\mathbf{A} \times \mathbf{B}) \cdot \mathbf{D}] \mathbf{C} - [(\mathbf{A} \times \mathbf{B}) \cdot \mathbf{C}] \mathbf{D} \).](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fb8e0ad34-47bd-4edd-8e64-7fbb4be3cef6%2Ff6eae2cd-e4e1-4415-bedd-0b24b81e86cc%2Fcx5vr08_processed.png&w=3840&q=75)
Transcribed Image Text:**Exercise 18**: Use the Levi-Civita symbol and Kronecker delta to prove the vector identities:
a. \( \mathbf{A} \times (\mathbf{B} \times \mathbf{C}) + \mathbf{B} \times (\mathbf{C} \times \mathbf{A}) + \mathbf{C} \times (\mathbf{A} \times \mathbf{B}) = \emptyset \);
b. \( (\mathbf{A} \times \mathbf{B}) \cdot (\mathbf{C} \times \mathbf{D}) = (\mathbf{A} \cdot \mathbf{C})(\mathbf{B} \cdot \mathbf{D}) - (\mathbf{A} \cdot \mathbf{D})(\mathbf{B} \cdot \mathbf{C}) \);
c. \( (\mathbf{A} \times \mathbf{B}) \times (\mathbf{C} \times \mathbf{D}) = [(\mathbf{A} \times \mathbf{B}) \cdot \mathbf{D}] \mathbf{C} - [(\mathbf{A} \times \mathbf{B}) \cdot \mathbf{C}] \mathbf{D} \).
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