# Vector Calculus Independent Study Path

## Unit 8: Fundamental Theorems of Vector Calculus

In single variable calculus, the fundamental theorem of calculus related the integral of the derivative of a function over an interval to the values of that function on the endpoints of the interval. In this unit, we will examine two theorems which do the same sort of thing. Gauss' theorem relates the integral of the divergence of a vector field over a solid region to the integral of the vector field over the boundary of the region, and Stokes' theorem relates the integral of the curl of a vector field over a surface to the integral of the vector field around the boundary of the surface. In this unit, you will learn:
• Gauss' Theorem, and how to simplify certain flux or triple integrals using it.
• The flux integral of a divergenceless vector field over a closed surface is 0.
• If the region between two closed surfaces is divergencelss, then the flux over the two surfaces is the same. (In physics, this means that two surfaces which contain the same charges have the same electromagnetic flux).
• Stokes' Theorem, and how to simplify certain flux or work integrals using it.
• The flux integral of a curl field over a closed surface is 0.
• Green's Theorem (aka, Stokes' Theorem in the plane).
• If the region between two closed loops is curl-free, then the work done going around either loop is the same.
For more detailed instructions, see the Xdvi or PDF pages.

For a collection of all of the Theorems on one HTML page, with gaudy color figures, see the Fundamental Theorems of Vector Calculus. The form of the theorems, and the notation, is that of Calculus with Analytic Geometry, Second Edition, by George F. Simmons, and page references are to this volume.

Similarly, an outline of a proof of Stokes' Theorem, using Simmons' notation, is available (PDF only).

## Suggested Procedure

• Read and do some problems from
• Rogers Chapters 21 - 26, or
• Marsden and Tromba chapter 8.
• Take the Sample Test, Xdvi or PDF.
• Take a unit test.

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watko@athena.mit.edu