Integral theorems: when to use what

Section 6.5 Integral theorems: when to use what

This is not really an independent section (or an independent lecture), but just a brief summary of the typical usages of the various integral theorems that we have seen so far.

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Objectives

You should be able to:

Determine which integral theorem may be useful to evaluate certain type of line and surface integrals.

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We have seen five integral theorems so far, all particular cases of the generalized Stokes' theorem:

The Fundamental Theorem of calculus;
The Fundamental Theorem of line integrals;
Green's theorem;
Stokes' theorem;
The divergence theorem.

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One of the main difficulties with the integral theorems of calculus is to determine which theorem may be helpful in a given situation. In this section I list a few typical situations for which integral theorems may be useful, highlighting the main applications of the integral theorems. You can use this list as a rule of thumb.

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Strategy 6.5.1. Integral theorems: when to use what.

You want to evaluate a line integral along a curve in $R^{n}$ for an exact one-form (or the gradient of a function): Fundamental Theorem of line integrals (Section 3.4).
You want to evaluate a line integral along a closed curve in $R^{2} :$ Green's theorem (Section 5.7).
You want to evaluate a line integral along a closed curve in $R^{3} :$ Stokes' theorem (Section 5.8).
You want to evaluate a surface integral along a surface in $R^{3}$ for an exact two-form (or the curl of a vector field): Stokes' theorem (Section 5.8).
You want to evaluate a surface integral along a closed surface in $R^{3} :$ Divergence theorem in $R^{3}$ (Section 6.2).