finalreview

# finalreview - 1(20 points Find the total ux upward throught...

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1. (20 points) Find the total flux upward throught the upper hemisphere ( z 0) of the sphere x 2 + y 2 + z 2 = a 2 of the vector field T ( x, y, z ) = x 3 3 i + yz 2 + e zx j + ( zy 2 + y + 2 + sin( x 3 )) k . Hint Use the divergence theorem around some ”simple” region. you’ll also need to compute one more flux integral. Use symmetry. 2. (20 points) Suppose a vector field is defined by F ( x, y, z ) = ( y 2 z ) i +(2 xyz ) j + ( xy 2 + 4 z ) k . a) Determine whether there is a scalar function φ ( x, y, z ) defined everywhere in space such that φ = F . If there is such a φ , find one; if there are none, explain why not. b) Determine whether there is a vector field A ( x, y, z ) defined everywhere in space such that × A = F . If there is such an A , find one; if there are none, explain why not. c) Compute the integral W F · T ds where W is the circular helix whose position vector is given by R ( t ) = (cos( t )) i +(sin( t )) j + t k for t [0 , π ]. Use information from a), if you like. 3. ( points) What is your favorite vector field?
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