Consider the scalar field, (F), and the vector fields ü(r) and ü(F). Using index notation, prove the following identities: (a) V. (ø T) = ū . Vø + ¢ỹ ·õ. (b) V× (ø i) = pỹ ×ũ – ở × Vø. (c) V·(ũ x ū) = · ỹ × ũ – - Ÿ × ū. (d) V × (ũ x ĩ) = . ỹ ū – ūỹ · ū +ūỷ •ỡ – ủ · ỹ ū. Introduce parentheses on the right-hand side of these equations to indicate the order of operations and improve legibility by identifying scalar and vector quantities.
Consider the scalar field, (F), and the vector fields ü(r) and ü(F). Using index notation, prove the following identities: (a) V. (ø T) = ū . Vø + ¢ỹ ·õ. (b) V× (ø i) = pỹ ×ũ – ở × Vø. (c) V·(ũ x ū) = · ỹ × ũ – - Ÿ × ū. (d) V × (ũ x ĩ) = . ỹ ū – ūỹ · ū +ūỷ •ỡ – ủ · ỹ ū. Introduce parentheses on the right-hand side of these equations to indicate the order of operations and improve legibility by identifying scalar and vector quantities.
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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Transcribed Image Text:Consider the scalar field, ø(r), and the vector fields ü(r) and ū(r). Using index
notation, prove the following identities:
(a) V. (ø T) = ū . Vo + ¢ỹ · õ.
(b) V× (ø t) = ¢ỹ ×ð – i x Vø.
(c) V·(ū× í) = · ỹ × ū – ū · ỹ x ū.
(d) V× (ũ x ī) = ở . ỹũ – ū ỹ ·ũ +ūÿ·¯ – ủ · ỹ ở.
V •
Introduce parentheses on the right-hand side of these equations to indicate the order
of operations and improve legibility by identifying scalar and vector quantities.
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