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
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.
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 12/31/99 by Don Power