2009+EECS145+1st+Midterm - 3 An electrical field is given...

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1 EECS145 Electrical Engineering Analysis Fall 2009 First Midterm Examination Chin C. Lee 1:00-1:50PM, Oct. 19, Monday Each problem has 20 points. 1. For f(x,y,z) = [x(y + z)] 1/2 , obtain grad f(x,y,z), calculate grad f(x,y,z) at point (1, 2, 3), and find the directional derivative at (1,2,3) along direction of the vector (1,-1,1). [Note: Directional derivative is the magnitude of the gradient projected on the direction specified]. 2. F (x, y, z) is a vector function. Prove that div ( curl F ) = 0. You are required to present your proof procedure step by step in details to receive full credit.
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Unformatted text preview: 3. An electrical field is given by E = (1/r 3 )( x , y , z ), where r = ( x , y , z ) is the position vector. Derive the electrostatic potential (voltage) between r = R 1 and r = R 2 . 4. Evaluate ∫∫ s G • d S , where G = (z, y, x) and S is the slant side of the tetrahedron defined by x + y + z ≤ 1, x ≥ 0, y ≥ 0, and z ≥ 0. [Note: Bottom side of the tetrahedron lies on the x-y plane]. 5. Evaluate ∫∫∫ v xyz dxdydz throughout the interior of a cylinder defined by x 2 + y 2 = 4, z = 2, and z = 10....
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