Merced College; Don Power

 

MULTIVARIATE CALCULUS, CH 16 - STUDY GUIDE

 

1.      On this graph, draw arrows in all 4 quadrants to identify the direction of the vectors in the vector field F(x,y)=[-x,y]:

    

 

 

2.  Evaluate a line integral.  Examples:

a.  , where F(x,y,z) = -4xyi+8yj+2k and C is the parabola y = x2, z=1 from A(0,0,1) to B(2,4,1).  That is, find the work done by the force F as the point of application moves along C from A to B.

b.   where C is the circle r(t) = 3 cos t i + 3 sin t j from 0 £ t £ 2p

 

3.  Special cases for line integrals:

    

     a.  Conservative function. 

 

              2D Example:   along any path from (0,p) to (1,2p/3)

 

3D Example: 

 

          -- Determine whether a vector field is conservative.

          -- Find the potential function for a conservative vector field.

          -- Evaluate a line integral by applying the fundamental theorem of line integrals.

 

 

     b.  Closed curve around a surface in the xy-plane (Green's Theorem)

          Ex:   where C is the region bounded by the x-axis, x=1, and y=x3.

    

     c.  Closed curve around a surface in space (Stokes' Theorem)

          -- Apply Green's theorem (where the surface is a plane parallel to any coordinate plane)

          -- Evaluate a line integral using Stokes' theorem (for a 3-dimensional surface or 3D F)

 

          Ex:  F(x,y,z) = z2i + x2j + y2k, S is surface z = x2, 0 £ x £ z, 0 £ y £ 4

              S contains no y, so the surface is a cylinder.  Visualize: graph z = x2 in xz-plane

              For boundaries of region, set x=0 and set x=z in z = x2

 

Ex:  F(x,y,z) = -ln(sqrt(x2+y2))i + arctan(x/y)j + k, S is surface z = 9-2x-3y over one petal of r = 2 sin (2q) in the first octant.

   Region:  r will be minimum when sin (2q) = 0, and maximum when sin (2q) = 1

          Solve for 2q, then solve for q.

 

3.  Find the surface area of the triangle with vertices (4,0,2), (0,3,1) and (0,0,5) by two methods:

     a. Compute the surface integral (Start by finding an equation of the plane  z=ax+by+c).

     b.  Use the cross product of vectors to find the area.

 

 

4.  Use the divergence theorem to evaluate  , whereF(x,y,z) = ______, n is the outer unit normal to the surface S, and S is the surface of the solid enclosed by ____ and _______

 

 

 

Return to:  Merced College; Don Power               Updated 12/16/09 by Don Power