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Chapter 10 Problem Set

Page history last edited by sidjaggi 13 years, 3 months ago

 Problem set 10 Vector Integral Calculus

 

     1.   (Line/path integrals) Problem set 10.1q.1 3, 5, 6. Calculate Formula for the fo=[llowing data. If F is a force.

 

          (a)  Formula, C the parabola Formula from Formula to Formula

          (b)  Formula as in Prob.1 a. C from A straight to Formula, then vertically up to Formula

          (c)  FormulaFormulaFormula . Sketch C.
          (d)  Formula clockwise along the circle with center Formula from Formula to Formula

 

     2.  (Line/path integrals) Problem set 10.1 q17,18 Evaluate (8) with F or f and C as follows.

          (a)  Formula, C the helix. Formula

          (b)  Formula, C the hypocycloid. Formula

 

     3.  (Path-independent integrals) Problem set 10.2, problems 1, 3, 5, 7. Show that the form under the integral sign is exact in the plane Formula or in space Formula

          (a)  Formula

          (b)  Formula

          (c)  Formula

          (d)  Formula

 

     4.  (Path-independent integrals) Problem set 10.2, problems 11, 13, 15, 17,19.Check for path independence and if independent, integrate from (0,0,0) to (a,b,c).

          (a)  Formula

          (b)  Formula

          (c)  Formula

          (d)  Formula

          (e)  Formula

 

     5.  (Double Integrals) Problem set 10.3, problems 5, 6, 9. Describe the region of integration and evaluate. (Show details)

          (a)  Formula

          (b)  Formula

          (c)  Formula

 

     6.  (Double Integrals) (a) Problem set 10.3, problems 10. Integrate Formula over the triangular region with vertices (0,0), (1,1), (1,2).

           (b) Sketch the region R contained by the curve Formula. Compute its area (Hint: The area equals the volume of the function f(x,y)=1 in the region R. Use polar coordinates and the appropriate Jacobean to computes this volume.)


     7.  (Double Integrals) Problem set 10.3, 13. The first octant section cut from the region inside the cylinder Formula by the planes Formula.


     8.  (Centre of gravity) Problem set 10.3, 14, 15. Find the center of gravity Formula of a mass of density Formula in the given region R.
          (a)  R the semi-disk Formula.

          (b)  

 

 

     9.  (Triple Integrals) 

        (a)  A cylinder of mass 1Kg has height 2 m and radius 1m (it is bounded by the surfaces FormulaFormula.

             (i) Compute its moment of inertia about the z-axis

             (ii) Compute its moment of inertia about the x-axis.

        (b) A ball of radius 1 has a density function of 2-r Kg/m3. What is its mass? (Hint: Use spherical coordinates)

 

     10. (Surface integrals)

         (a) Find parametric representations for the following surfaces, and corresponding unit normal vectors. (Kreyszig 10.5, problems 12, 15)

              (i) The plane 5x+y-3z=30

              (ii) The sphere Formula

          (b) Find the following integrals (Kreyszig 10.6, problems 2, 7)

              (i) The flux integral of the function Formula over the surface Formula

              (ii) The flux integral of the function Formula over the surface of the sphere with radius 1 and centered at the origin. 

 

  11. (Divergence theorem) Verify the answer of 10(b)(ii) above using the divergence theorem.

 

  12. (Stokes's theorem) (Kreyszig 10.9, Problem 9) Compute the following surface integral directly, and also using Stokes's theorem. The function is Formula, and the surface is the square Formula.

 

 

 

 

 

 

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