Math 6

Spring 2007

Final Review

Directions: all problems by hand unless indicated otherwise by the presence of an (M) at the beginning of the problem.

  1. What conditions have to apply to all the integral theorems for their conclusions to hold?
  2. State each of the vector integral theorems in appropriate notation
  3. Describe what each of the above theorems says in words.
  4. Show specifically how each of the theorems is related to the fundamental theorem of calculus.
  5. Write in cylindrical coordinates
  6. the equation of the cylinder
  7. The equation of the cone
  8. What specifically does each of the following refer to and in what kind of integrals is it used?
  9. ds
  10. dS
  11. dS
  12. dA

Write the area element dA in

  1. rectangular coordinates
  2. polar coordinates
  3. n
  4. T
  5. dV

Write the volume element dV in

  1. rectangular coordinates
  2. cylindrical coordinates
  3. spherical coordinates
  4. dt
  5. dz dy dx
  8. Evaluate by hand (sometimes, one of the integral theorems will help)
  9. , where C is the portion of y = x2 between x = 0 and x = 2, followed by the line segments between the origin and (2,0) and then the segment from (2,0) to (2,4)
  10. counterclockwise along the unit circle
  11. , where S is the portion of the plane 2x + y + z = 2 lying in the first octant
  12. Use spherical coordinates to find , where S is the sphere
  13. Find the flux of the vector field over the portion of the paraboloid below z = 4.
  14. Let Q be the part of the paraboloid above the xy-plane. Using cylindrical coordinates, find the flux of the vector field through the solid whose upper surface is Q and whose bottom is the intersection of Q in the xy-plane.
  16. Prove that the curl of a conservative vector field is zero.
  17. Prove that
  18. Write down a vector field R that has positive divergence and negative curl.
  19. Write down the triple integral that represents the volume of any solid.
  20. Write the double integral that represents the area of any closed, planar curve.
  21. (M) Find the surface area of the part of the paraboloidthat lies below the plane z = 5.
  22. Maximize y-x subject to the constraint that the solution lie on the unit circle. ()
  23. If the temperature on an elliptical plate is given by and the plate has a defining equation of , find the maximum and minimum temperatures on the plate as well as their locations. (min = -1 at (-1,0); max of at
  24. The plane x + y + z = 12 intersects the paraboloid in an ellipse. Find the point on the ellipse that’s closest to the origin. (2,2,8)
  25. Locate and classify all critical points for the surface (saddle at (0,0); min at
  26. If , find
  27. Find the direction of maximum change of f at the given point as well as the value of the change:
  28. at (2,1) <4, -3> value of 5
  29. at (2,0) <1,0>, value of 1
  30. For a real gas, van der Waal’s equation states that

P = pressure, V = volume, T = temperature, n = moles of gas, R is the universal gas constant, and a and b are constants. Compute and interpret and