Review, things you should know about/know how to do for the final…Chapter 13-Vector Calculus
I)
Section 13.1: Vector fields
A. Be able to draw and/or match (equation with graph) simple vector fields
B. Know what conservative/potential/incompressible fields are
C. Know that gradients of scalar functions are vector fields in their own right
II)
Section 13.2: Line integrals
A. Know how to evaluate which may include
1.Parametrizing given curve (sometimes given)
2.Using formula
a. Line integrals over scalar fields:

C
tend




f (r )ds  f (r (t )) | r (t ) | dt where r  x(t ), y(t ) or
tstart

r  x(t ), y(t ), z(t ) . (Note you can also use (which are the same formulas as above, basically)
formula 3 or 9, as appropriate, from section 13.2.)
b. Line integrals over vector fields:

C
 


 tend  
F (r )  dr  F (r (t ))  r (t )dt where r  x(t ), y(t ) or
tstart

r  x(t ), y(t ), z(t ) . (Note you can also use (same formulas as above, basically) formula on pg 740.
3.Evaluating the resulting integral
B. Find work done by a force field
C. Understand the notation for a line integral over a vector field:
version:

C

C
 

F (r )  dr  Pdx  Qdy  Rdz or the 2-d
C
 

F (r )  dr  Pdx  Qdy and how to proceed when using this notation.
C
D. Obtain sign of a line integral over a vector field just by seeing a picture of the vector field and the
corresponding line (practice prob 16 in this section)
E. Be able to find proper orientation of curves when required (e.g. line integrals over vector field)
III)
Section 13.3: Fundamental Theorem for line integrals
A. Be able to evaluate line integral using FTC for line integrals (over vector fields). May include
  
1.Verifying a field is indeed conservative (sometimes given (requires   F  0 ))

2.Find a corresponding potential function for the vector field, find an f such that F  f
3.Finding corresponding locations of the endpoints of the curve being integrated over (sometimes given)
 



and use the FTC formula:  F (r )  dr  f (rend )  f (rstart )
C
B. Be able to use formula graphically as well (as in practice problems) and determine whether or not a field is
conservative graphically
IV)
Section 13.4: Green’s Theorem
A. Know formula and how to use it:
1.Formula:  Pdx  Qdy   (Qx  Py )dA
C
B.
V)
A.
B.
C.
D.
E.
D
2.Use to convert a integral over a closed curve into a double integral.
3.Use to convert a double integral of Qx-Py into an integral over a closed curve
Use orientation of the curve to determine the correct sign to use in Green’s formula
Section 13.5: Curl and Divergence
Know formulas and be able to calculate the curl and divergence of a vector field
Be familiar with the terminology corresponding to these differential operator, e.g. know what
 
  
incompressible (   F  0 ), irrotational (   F  0 ), conservative, potential fields correspond to in terms
of these differential operators
It would be good to know the alternative forms of Green’s theorem on pg 762 and 763…note that these are
just, pretty much, the stokes and divergence theorem in 2-d
Know what divergence and curl mean graphically…you can use this to check your answer on the exam
Know properties of these operators
VI)
A.
B.
C.
D.
E.
VII)
A.
1.Divergence of a curl is zero…hence if a vector field’s divergence is nonzero, it is not a
conservative vector field
Section 13.6: Parametric surfaces and their areas
Be able to parametrize surfaces given using Cartesian coordinates including
1.Using arbitrary u,v
2.Using (x,y), (y,z), or (x,z)
3.Using cylindrical or spherical coords or something similar
4.Finding the right domain corresponding to your new parameters
Be able to visualize them which may include
1.Using grid curves
2.Figuring out what Cartesian equation the parametric surface satisfies
Know formula for a tangent plane to a parametrized surface
Be able to find area of a parametrized surface over a given region/find the area of a surface represented in
Cartesian coordinates by parametrizing it first
Note: You may use either the formula in definition 6 on mg 771 or formulas of the kind seen in 9 on pg 772
(where you don’t have to take a cross-product) on the final
Section 13.7: Surface integrals
Evaluate surface integrals which may involve:
1.Parametrizing surface (sometimes given to you) including obtaining a proper domain for the parameters
2.Using one of the acceptable formulas which include but are not limited to


 
a. Scalar fields:  f (r )dS   f (r (u, v)) | ru  rv | dA (Also, as said in class, can instead use formulas
S
D
like 4 on pg 777, when appropriate)

 
 
 
b. Vector fields:  F (r )  dS   F (r (u, v))  (ru  rv )dA . (Again, as said in class, formulas of the
S
D
form 10 on pg 782 can also be used, when appropriate)
3.Evaluate the integral
B. Be able to obtain appropriate sign of the integral corresponding to the orientation of the surface.
VIII)
Section 13.8: Stokes’ Theorem
A. Be able to use Stokes theorem which may involve

 
 
  

 
1.Memorize the formula:  F (r )  dr   (  F )  dS   (  F )( r (u, v))  (ru  rv )dA
C
S
D
2.Use to convert line integrals into surface integrals (Remember to check what the curl looks like…to see
what you’re up against… before parametrizing your surface)
3.Use to convert integral of curl of a vector field over a surface into a line integral
4.Of course…be able to find the curl of a vector field
5.Evaluate resulting integrals
IX)
Section 13.9: The Divergence Theorem
A. Be able to use divergence theorem which may involve:
 
 
1.Memorize and use the formula:  F  dS   (  F )dV
V
S
2.Use to convert surface integrals into volume integrals
3.Be able to find divergence of the vector field
4.Evaluate resulting integrals
Important Note: No formulas will be given to you on the exam! This means you have to memorize all the
formulas from dot and cross product formulas, formulas for unit tangents, normals, binormals, curvature, and
their corresponding planes (for curves), green’s theorem, stokes theorem, divergence theorem, line integral
and surface integral formulas, cylindrical and spherical conversion formulas, lagrange multiplier formulas,
directional derivative, curl, divergence, and gradient operator formulas, tangent plane formulas, velocity and
acceleration formulas