18.02 - ESG

Fundamental Theorems of Calculus

This page presents the Fundamental Theorems of Calculus, emphasizing the theorems of Vector Calculus, in one place with figures. The presentation is motivated by the geometrical interpretation of the regions of integration and their respective boundaries.

The text of the theorems (presented here in red) is from Calculus with Analytic Geometry, Second Edition, by George F. Simmons, and page references are to this volume. A similar page, with the notation from and references to Multivariable Calculus with Analytic Geometry, Fifth Edition, by Edwards & Penney, is linked from the figure on the right above, or from here

The theorems are

• The Fundamental Theorem of Calculus (Simmons Page 208)
• The Fundamental Theorem of Calculus for Line Integrals (Simmons, Page 758)
• Green's Theorem (Simmons Page 765)
• Stokes' Theorem (Simmons Page 781)
• Gauss's Theorem (Simmons Page 775)

The Fundamental Theorem of Calculus

Unless qualified, the term ``The Fundamental Theorem of Calculus" is conventionally taken to apply to a function of a single variable, integrated over an interval of the real line.

Here, the region of integration is bounded by the two points a and b, and the function F(x), an antiderivative of f(x), is evaluated on these boundary points.

The Fundamental Theorem of Calculus for Line Integrals

The region of integration has been generalized to any path in the plane (the extension to three or more independent variables is straightforward, but the figures are not).

Here, the region of integration is a curve with endpoints labeled ``A'' and ``B''. In the above figure, the blue arrows represent the vector field F and the magenta curves are level sets of f. In analogy with the previous theorem, for line integrals the scalar function f is an antiderivative in the sense that F = grad f.

Green's Theorem

If F is the gradient of a scalar field, and if in the Fundamental Theorem of Calculus for Line Integrals the points A and B coincide, then that theorem tells us that the line integral of F around a closed (with appropriate restrictions on the region R) vanishes. If F is not the gradient of a scalar, the line integral can be related to the variations of F in the enclosed region. Green's theorem is for curves in a plane.

Stokes' Theorem

For contours that do not lie in a plane, Green's Theorem is generalized as Stokes' Theorem.

Gauss's Theorem

If a surface to which Stokes' Theorem is applied were to be deformed in such a way as to create a closed surface, or if two surfaces which share a boundary are joined, the resulting surface has no boundary, and so Stokes' Theorem says that the integral of the normal component of the curl (circulation) of a vector function over a closed surface vanishes. If the integrand is not the curl of some vector function, then the surface integral is related to the variations of the integrand in the interior of the closed region.

(Following Simmons, integrals over closed surfaces will be written without a circle over the integral signs.)