s1 - f r x 2 r df dr y 2 r df dr z 2 r df dr(8 = 3 f r r df...

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Phys. 263 Assignment 1 SOLUTIONS 1. (a) A + B + C = (5 , 1 , 3) (b) B × C = (1 , 4 , 5) so that A · ( B × C ) = 1 + 8 + 15 = 24. (c) A × B = (8 , 8 , - 8) so that C · ( A × B ) = 8 + 8 + 8 = 24. (d) A × B × C = A × ( B × C ) = ( - 2 , - 2 , 2) (e) to (h) are =0, since these are constant vectors 2. d dt [ A · ( ˙ A × B )] = ˙ A · ( ˙ A × B ) + A · d dt ( ˙ A × B ) We know that the direction of the cross product is orthogonal to ˙ A in the first term: = 0 + A · ( ¨ A × B + ˙ A × ˙ B ) = A · ( ¨ A × B ) . since ˙ B = 0. 3. (a) The gradient of f ( r ) is f ( r ) = ± ∂f ( r ) ∂x , ∂f ( r ) ∂y , ∂f ( r ) ∂z ² . (1) Since f ( r ) depends on x through the dependence of r on x , we write ∂f ( r ) ∂x i = df ( r ) dr ∂r ∂x i (2) For example, ∂r ∂x = x ( x 2 + y 2 + z 2 ) - 1 / 2 = x r (3) Therefore, we see f ( r ) = ( x, y, z ) 1 r df ( r ) dr (4) But the vector ( x, y, z ) is just the radial vector r , so that f ( r ) = r r df ( r ) dr (5) Finally, by definition, the unit vector in the radial direction ˆ r is just r /r , so we obtain f ( r ) = ˆ r df ( r ) dr (6) 1
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(b) ∇ · r = ± ∂x , ∂y , ∂z ² · ( x, y, z ) = 1 + 1 + 1 = 3 (7) (c) ∇ · r f ( r ) = ∂x [ xf ( r )] + ∂y [ yf ( r )] + ∂z [ zf ( r )] = 3
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Unformatted text preview: f ( r ) + x 2 r df dr + y 2 r df dr + z 2 r df dr (8) = 3 f ( r ) + r df dr since r 2 = x 2 + y 2 + z 2 . (d) From part (c), ∇ · r r n-1 = 3 r n-1 + r ( n-1) r n-2 = ( n + 2) r n-1 (9) 4. (a) The work integral is W = Z F · d r = Z (-ydx + xdy ) (10) including the path W =-Z 1 ydx + Z 1 xdy (11) In the first integral, we have to specify the value of y as x ranges from 0 to 1 (in this case y = 0). For the second integral, we specify the value of x = 1, so that W =-Z 1 dx + Z 1 1 dy = 1 (12) (b) In this case we specify y = 1 in the first integral and x = 0 in the second, giving W =-1 (13) Clearly, for this force, the work done depends on the choice of path. 2...
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This note was uploaded on 09/20/2010 for the course AMATH 261 taught by Professor Rogermelko during the Spring '10 term at Waterloo.

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s1 - f r x 2 r df dr y 2 r df dr z 2 r df dr(8 = 3 f r r df...

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