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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.

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

  1. The Fundamental Theorem of calculus;

  2. The Fundamental Theorem of line integrals;

  3. Green's theorem;

  4. Stokes' theorem;

  5. The divergence theorem.

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.

Strategy 6.5.1. Integral theorems: when to use what.

  • You want to evaluate a line integral along a curve in \(\mathbb{R}^n\) 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 \(\mathbb{R}^2\text{:}\) Green's theorem (Section 5.7).

  • You want to evaluate a line integral along a closed curve in \(\mathbb{R}^3\text{:}\) Stokes' theorem (Section 5.8).

  • You want to evaluate a surface integral along a surface in \(\mathbb{R}^3\) 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 \(\mathbb{R}^3\text{:}\) Divergence theorem in \(\mathbb{R}^3\) (Section 6.2).