May 8, 1996            Key to Final Exam                 Math 311

1. (20) Let be defined by Let A denote the matrix representation of L with respect to the standard bases in and . Remember, is the space of all polynomials of degree two or less, and is the space of all matrices with real entries.
   1. Find A

      , , . Thus, the matrix representation of this linear transformation is:

      

   2. What is the rank of A?

      The matrix A is row equivalent to the matrix:

      

      Thus, A has rank three.

   3. Is the matrix in the range of L.

   The equation has a solution, if and only if the equation has a solution. The augmented matrix of this system has rank 4. That is, its last row contains three zeros and a one. Hence the last row represents an equation without a solution. Thus, the matrix B is not in the range of L.

2. (20) Define and give examples of each of the following terms: (No examples, no credit.)

   (a) Eigenvalues and eigenvectors. (b) Linear transformation.

   See your text or notes for these definitions.

3. (20) A flea is hopping around the surface of the ellipsoid . Suppose the temperature at any point of a region containing the ellipsoid is given by T(x,y,z) = 3x-y+5z+7. To what point on the ellipsoid should the flea hop, if it desires to be coolest? What is the temperature at this point?

   The easiest way to find the point on the ellipsoid with minimal temperature is to use the method of Lagrange multipliers. The equation 3pt (T) + 3pt (f)= , where f is the constraining function, leads to the following system of equations:

   

   Place the expressions for x, y, and z in the constraining equation, then solve for . There are two solutions, . The negative value of gives that point where the temperature is largest, . The positive value of gives the point (-0.275, 1.061, -0.915). The minimum temperature, which is attained at this point, is .

4. (20) Let be the unit square in space, that is . Let T(u,v)=(1+u+3v,2+u-2v). Spatial units are in inches. Thus, has an area of one square inch.
   1. Sketch the region . Be sure to give the coordinates of the vertices of .

      

      The easiest way to sketch the image of a set is to determine where the boundaries of the set are mapped. , or ; , or . The other two parts of the boundary are plotted the same way.

   2. What is the area of .

      

   3. Suppose that the region has a electrical charge density of coulombs per square inch. What is the total charge on the region ? It will suffice to express your answer as an integral, but be sure that it is in a form that involves an integration over the region .

      

5. (20) Let be the solid region in the first octant of bounded above by the plane x+y+z=1. That is . Let .
   1. curl(F) =? div(F) =?

      .

   2. State Stoke's theorem and the Divergence theorem. Be sure to explain what your symbols represent.

      See text or class notes.

   3. Let denote that part of the boundary of that lies in the plane y=0. Use the unit normal to that points in the negative y direction, and compute the surface integral .

      

   4. Compute the flux of F. That is, compute the value of the integral .

      

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