Line, Surface, and Volume Integrals

A line integral for a vector field is defined as the integral of the dot product of a vector function with an infinitesimal displacement vector along a prescribed path. If the prescribed path is closed, the integrals reduce to a closed-line integral. The closed-contour integral of the vector field is referred to in terms of the circulation of the vector field around the closed path. A vector with zero circulation around every closed path is called a conservative field, while one with non-zero circulation around a closed path is called a non-conservative field. For example, the line integral of a force along a prescribed path yields work. For a closed path, the net work done by the field or against the field is zero.

A surface integral is the integral of the dot product of a vector function with an infinitesimal area vector over a prescribed surface. The area vector is always directed perpendicular to its surface at its given location. The surface integral of a vector is the flux of this vector through the surface. If the prescribed path or surface is closed, the integrals reduce to a closed-surface integral. For a closed surface, the direction of the area vector points outward. For open surfaces, the direction depends on the contour enclosing the surface. The direction of the area vector is chosen so that it is always perpendicular to the surface and points outward from the region enclosed by the contour. According to the right-hand rule, if the fingers are along the direction of travel with the palm facing the interior of the surface, then the thumb points along the positive area vector. For example, consider a vector function that is the product of the density and velocity of a fluid. The surface integral of this vector can be used to calculate its flux through the surface.

Surface and line integrals of vector fields are more significant in physics than those of scalar fields because they have a direct physical interpretation related to the flow of energy or matter. Scalar fields are mathematical entities that represent physical quantities with only magnitude, such as temperature, pressure, or electric potential. A line integral of a scalar field is the integration of the scalar function along a path in the domain of the function, measuring the total contribution of the scalar function along the path. For example, the line integral of temperature along a path represents the total amount of heat gained or lost along the path. Similarly, a surface integral of a scalar field is the integration of the scalar function over a surface in the domain of the function, measuring the total contribution of the scalar function over the surface. For instance, the surface integral of fluid density over a surface represents the total amount of fluid passing through the surface. However, the direct physical interpretation of scalar field integrals is limited compared to that of vector field integrals, making the latter more important in physics.

The integral of the product of a scalar function and infinitesimal volume is the volume integral. For example, the volume integral of the density function gives the total mass. Similarly, the volume integral of the energy density represents the total energy stored in the volume. When the scalar function in the volume integral is replaced with a vector function, it reduces to a combination of integrals of scalar functions.