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, 

Work = U(B) - U(A)

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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