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4: Line and Surface Integrals

• • Contributed by Michael Corral
• Professor (Mathematics) at Schoolcraft College

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A line integral is an integral where the function to be integrated is evaluated along a curve and a surface integral is a generalization of multiple integrals to integration over surfaces. It can be thought of as the double integral analog of the line integral. Given a surface, one may integrate over its scalar fields (that is, functions which return scalars as values), and vector fields (that is, functions which return vectors as values). Surface integrals have applications in physics, particularly with the theories of classical electromagnetism.

• 4.1: Line Integrals
In this section, we will see how to define the integral of a function (either real-valued or vector-valued) of two variables over a general path (i.e. a curve) in $$\mathbb{R}^2$$ . This definition will be motivated by the physical notion of work. We will begin with real-valued functions of two variables.
• 4.2: Properties of Line Integrals
We know from the previous section that for line integrals of real-valued functions (scalar fields), reversing the direction in which the integral is taken along a curve does not change the value of the line integral.
• 4.3: Green’s Theorem
We will now see a way of evaluating the line integral of a smooth vector field around a simple closed curve. A vector field $$\textbf{f}(x, y) = P(x, y)\textbf{i} + Q(x, y)\textbf{j}$$ is smooth if its component functions $$P(x, y)$$ and $$Q(x, y)$$ are smooth. We will use Green’s Theorem (sometimes called Green’s Theorem in the plane) to relate the line integral around a closed curve with a double integral over the region inside the curve:
• 4.4: Surface Integrals and the Divergence Theorem
We will now learn how to perform integration over a surface in $$\mathbb{R}^3$$ , such as a sphere or a paraboloid. Recall from Section 1.8 how we identified points $$(x, y, z)$$ on a curve $$C$$ in $$\mathbb{R}^3$$ , parametrized by $$x = x(t), y = y(t), z = z(t), a ≤ t ≤ b$$, with the terminal points of the position vector.
• 4.5: Stokes’ Theorem
So far the only types of line integrals which we have discussed are those along curves in $$\mathbb{R}^ 2$$ . But the definitions and properties which were covered in Sections 4.1 and 4.2 can easily be extended to include functions of three variables, so that we can now discuss line integrals along curves in $$\mathbb{R}^ 3$$ .
• 4.6: Gradient, Divergence, Curl, and Laplacian
In this final section we will establish some relationships between the gradient, divergence and curl, and we will also introduce a new quantity called the Laplacian. We will then show how to write these quantities in cylindrical and spherical coordinates.
• 4.E: Line and Surface Integrals (Exercises)
Problems and select solutions to the chapter.

Thumbnail: The definition of surface integral relies on splitting the surface into small surface elements. Each element is associated with a vector dS of magnitude equal to the area of the element and with direction normal to the element and pointing outward. Image used with permission (Public Domain; McMetrox)