# Line and surface integral pdf

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Published: 23.03.2021  ## Line, Surface and Volume Integrals

In mathematics , a line integral is an integral where the function to be integrated is evaluated along a curve. The function to be integrated may be a scalar field or a vector field. The value of the line integral is the sum of values of the field at all points on the curve, weighted by some scalar function on the curve commonly arc length or, for a vector field, the scalar product of the vector field with a differential vector in the curve. This weighting distinguishes the line integral from simpler integrals defined on intervals. In qualitative terms, a line integral in vector calculus can be thought of as a measure of the total effect of a given tensor field along a given curve. ## PHYS. 321 - ELECTROMAGNETIC THEORY - PART 2

In mathematics , particularly multivariable calculus , a surface integral is a generalization of multiple integrals to integration over surfaces. It can be thought of as the double integral analogue of the line integral. Given a surface, one may integrate a scalar field that is, a function of position which returns a scalar as a value over the surface, or a vector field that is, a function which returns a vector as value. If a region R is not flat, then it is called a surface as shown in the illustration. Surface integrals have applications in physics , particularly with the theories of classical electromagnetism. To find an explicit formula for the surface integral over a surface S , we need to parameterize S by defining a system of curvilinear coordinates on S , like the latitude and longitude on a sphere. Let such a parameterization be x s , t , where s , t varies in some region T in the plane.

The amount of the fluid flowing through the surface per unit time is also called the flux of fluid through the surface. For this reason, we often call the surface integral of a vector field a flux integral. If water is flowing perpendicular to the surface, a lot of water will flow through the surface and the flux will be large. On the other hand, if water is flowing parallel to the surface, water will not flow through the surface, and the flux will be zero. The choice of normal vector orients the surface and determines the sign of the fluid flux. As shown in the following figure, we chose the upward point normal vector. In multivariable calculus, we have double integrals, triple integrals, line integrals, surface integrals — where does it end? As it turns out, any kind of integral can be​.

## PHYS. 321 - ELECTROMAGNETIC THEORY - PART 2

Soletf : R3! The terms path integral, curve integral, and curvilinear integral are also used. The surface integral is defined as, where dS is a "little bit of surface area. Created by Christopher Grattoni. Some examples are discussed at the end of this section. 