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a1 - Phys 263 Assignment 1 Due Friday 1 For vectors A B and...

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Phys. 263 Assignment 1 Due: Friday, May 16, 2008 1. For vectors A , B and C : A = ˆ i + 2 ˆ j + 3 ˆ k, B = 3 ˆ i - 2 ˆ j + ˆ k, C = ˆ i + ˆ j - ˆ k calculate the following quantities: (a) A + B + C (b) A · ( B × C ) (c) C · ( A × B ) (d) A × B × C (e) ∇ · A (f) ∇ × B (g) ( A · B ) (h) ∇ × ( B × C ) 2. If A = A ( t ) and B is a constant vector, prove that d dt [ A · ( ˙ A × B )] = A · ( ¨ A × B ) . 3. Consider the function f ( r ), where r = x 2 + y 2 + z 2 is the magnitude of the radial vector r = ( x, y, z ). (a) Calculate the gradient of f ( r ). Express the result in terms of df dr and the unit vector (ˆ r = r /r ) in the radial direction. Hint: remember ∂f ( r ) ∂x = df ( r ) dr ∂r ∂x (b) Calculate ∇ · r . (c) Calculate ∇ ·
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