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Fall 2007 - Enright's Class - Quiz 1

Fall 2007 - Enright's Class - Quiz 1 - -3-3 2 x-1,y-1,z-1...

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Quiz 1 Solutions MTH 20E Note that there were two separate versions of the quiz; while the numbers may differ slightly, the techniques used in each problem remain unchanged. Question 1a. The area of the parallelogram generated by A and B is given by k A × B k = k ± ± ± ± ± ± i j k 1 2 3 1 0 3 ± ± ± ± ± ± k = k 6 i - 2 k k = 40 . Question 1b. The volume can be calculated by computing the triple product A · ( B × C ) = ± ± ± ± ± ± 1 2 3 1 0 3 1 3 0 ± ± ± ± ± ± = 6 . Question 2. For a given curve φ ( t ) ,a t b , we know that the length of φ is given by L ( φ ) = Z a b k φ 0 ( t ) k dt. In this case, φ 0 ( t ) = ( - sin t, cos t, 3), and so L ( φ ) = Z 0 π 2 p sin 2 t + cos 2 t + 9 dt = Z 0 π 2 10 dt = 10 π 2 . Question 3. It is helpful to remember that the gradient vector for a surface F will be normal to the tangent plane to the surface at any given point. Exploiting this fact, we calculate F = (2 x - 5 y, - 5 x + 2 y, 2 z ) So, at the given point (1 , 1 , 1), our normal vector will be given by n := ( - 3 , - 3 , 2). Recalling that the formula for a plane with normal vector n through the point ( x 0 ,y 0 ,z 0 ) is described by n · ( x - x 0 ,y - y 0 ,z - z 0 ) = 0 we find that the equation for the tangent plane in question is given as
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Unformatted text preview: (-3 ,-3 , 2) · ( x-1 ,y-1 ,z-1) = 0 , that is,-3 x-3 y + 2 z =-4 . 1 Question 4. By utilizing independence of path, one could compute this integral directly by using any curve from (1,2,3) to (2,2,2); an example of one such path would be φ ( t ) = (1 + t, 2 , 3-t ) , ≤ t ≤ 1 . However, it is worth nothing that the function f ( x,y,z ) = xyz is a potential function for F ; i.e., ∇ f = F . Thus, by the Fundamental Theorem of Line Integrals, Z φ F d s = f (2 , 2 , 2)-f (1 , 2 , 3) = 2 . Question 5. a ) Since the divergence of a vector field is an expression denoting the level of compression within that field, and div ( curl ( F )) = 0 for any smooth vector field F , this statement is TRUE . b ) The curl of a vector field is a measure of rotational force that a vector field applies; since the curl of a gradient field is 0, this statement is TRUE . 2...
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Fall 2007 - Enright's Class - Quiz 1 - -3-3 2 x-1,y-1,z-1...

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