Final Exam
CURVE
The MEDIAN score on the final was 152. This was fairly good, between those of the
first and second midterms. The curve is:
- A: 186 and above
- A-: 170 and above
- B+: 159 and above
- B: 145 and above
- B-: 132 and above
- C+: 123 and above
- C: 113 and above
- C-: 100 and above
- D+: 96 and above
- D: 92 and above
- D-: 84 and above
- F: below 84
A Little General Information:
The final exam will cover material from the entire course. However, roughly
40% to
50% of the exam (rather than one quarter) will focus on Chapter 14.
Assignment of partial credit will depend on the problem. Some problems (e.g.
true/false) may be "all or nothing" i.e. with no possible partial credit).
The problems will be mostly independent of one another, so that if you can't solve
one problem, it should not affect your ability to get full credit on
other problems.
I will not use a fixed grading scale (such as "90 = A, 80 = B etc."). I
will set a curve based on the overall difficulty of the exam and the performance
of the class (in particular, if everyone gets a nearly-perfect score on
every exam, I would be very happy to give all As!). This will be announced after
the exam on the course web page.
Summary of concepts
Section 14.1: Vector Fields
- Geometric meaning of a vector field
- Plots of vector fields in the plane
- Gradient vector field of a function
- Divergence of a vector field
- Curl of a vector field
- Notation for gradient, divergence, and curl in terms of Nabla (upside-down Delta)
Sections 14.2 and 14.3: Line integrals, the Fundamental Theorem, and Independence of Path
- Definition and evaluation of a line integral with respect to arclength (ds)
- Definition and evaluation of a line integral with respect to coordinate
variables (dx, dy, dz)
- Relationship between the two kinds of line integrals (F.T ds versus
Pdx + Qdy + Rdz), Equation 15 on p. 1029
- Work and how to calculate it with a line integral
- Independence of path (Definition on p. 1033)
- Independence of path is equivalent to the integrand being a gradient (Theorem 2, p. 1034)
- Conservative vector fields
- How to tell when a vector field is conservative
- How to compute a potential of a conservative vector field
- Conservation of energy
Section 14.4: Green's Theorem
- Statement of Green's Theorem
- How to use Green's Theorem to compute line integrals
- How to use the Corollary to Green's Theorem (p. 1042) to compute area
- Flux of a vector field across a curve
Section 14.5: Surface integrals
Definition and evaluation of integral with respect to surface area (dS)
- Definition and evaluation of surface integral with respect to coordinate elements (dydz, etc.)
- Relation between integral of . n dS and integral of
P dy dz + Q dz dx + R dx dy (Equation 19, p. 1053).
- Flux of a vector field across a surface
Section 14.6: Divergence Theorem
- Statement of Divergence Theorem
- Analogy of Green's Theorem and Divergence Theorem to Fundamental Theorem of Calculus in one variable
- Physical/geometric meaning of the Divergence Theorem
- Using the Divergence Theorem to compute surface integrals
Section 14.7: Stokes's Theorem
- Statement of Stokes's Theorem
- Physical/geometric meaning of Stokes's Theorem
- Using Stokes's Theorem to evaluate line integrals or surface integrals
Typical computational tasks
Compute the gradient of a function, or the divergence or curl of a vector field
Evaluate a line integral (two types of them)
Use a line integral to compute mass or center of mass of a wire
Determine whether a vector field is conservative
Find a potential function for a conservative vector field (two methods)
Show that a given line integral is independent of path
Use Green's Theorem to compute a line integral
Use the Corollary to Green's Theorem to find the area of a region
Use Green's Theorem to calculate work
Calculate the flux of a vector field across a curve in the plane or a surface
in 3-dimensional space
Find a parametric equation for a surface in 3-dimensional space
Calculate the area of a surface
Calculate a surface integral with respect to dS or with respect to
coordinate elements
Find the centroid of a surface in 3-dimensional space
Use the Divergence Theorem to calculate a surface integral
Use Stokes's Theorem to evaluate a surface integral
Use Stokes's Theorem to evaluate a line integral
Things to remember:
- Any formulas you need for the kinds of calculations mentioned above. It's a
good idea to prepare a list of all the formulas you will need, and then
memorize those. Of course, you shouldn't bring that list to the exam.
Main course page.