# Vector Fields and the Fundamental Theorem

### Summary and Review

In this chapter, we explored 2 and 3 dimensional vector fields,
where a *vector field* is an assignment of a vector to each
point in the plane (or in space, respectively). If the vector field
**F** is the gradient of a function *U*(*x,y,z*), then **F**
is said to be *conservative* and *U* is said to be a *potential*
for **F. **If the curl of **F** is zero, then **F** is
conservative and has a potential.

Line integrals are integrals
over oriented curves that are situated in vector fields. They are
calculated by **pulling back** into the parameter for a
parameterization of the curve, and in applications, they are often used
to determine the amount of work in displacing an object from point *A*
to point *B*. If **F** is conservative with a potential *U*,
then it can be shown that the total work is equal to the change in
potential between *A* and *B*. That is,

This also implies that work performed by a conservative field is **independent
of path. **
When a field is not conservative, then it can be explored using **Stoke's
theorem ** in 3-dimensions, which reduces to *Green's
theorem* in 2-dimensions. In particular, Stoke's theorem,
Green's theorem, and the closely-related **divergence theorem **
are all embodiments of the fundamental theorem of calculus for more than
one variable which says something about the *flux* of a
vector field. Indeed, using *differential forms*, it
can be shown that all of the integral theorems are special cases of **Stoke's
theorem for differential forms**.

There are other ideas and concepts introduced in this chapter, and you should
re-read the individual sections in addition to this summary. The
review materials are based on both the ideas above and some of those in
the chapter not mentioned here. Review questions and solutions are
in web page form on the left and in pdf form on the right. For
maximum benefit, you should attempt to answer the questions *before*
you look at the solutions.

You will need Acrobat reader in order to open and print the pdf files.

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