Why do we use line integrals?

A line integral allows for the calculation of the area of a surface in three dimensions. Or, in classical mechanics, they can be used to calculate the work done on a mass m moving in a gravitational field. Both of these problems can be solved via a generalized vector equation.

What is line integral in complex analysis?

In complex analysis, the line integral is defined in terms of multiplication and addition of complex numbers. Suppose U is an open subset of the complex plane C, f : U → C is a function, and is a curve of finite length, parametrized by γ: [a,b] → L, where γ(t) = x(t) + iy(t). The line integral.

What is the use of complex integration?

Complex integration is an intuitive extension of real integration. Since a complex number represents a point on a plane while a real number is a number on the real line, the analog of a single real integral in the complex domain is always a path integral.

What exactly is a line integral?

A line integral (sometimes called a path integral) is the integral of some function along a curve. One can integrate a scalar-valued function along a curve, obtaining for example, the mass of a wire from its density. One can also integrate a certain type of vector-valued functions along a curve.

What is a complex integral?

Definition of complex integration : the integration of a function of a complex variable along an open or closed curve in the plane of the complex variable.

Where is complex integration used in real life?

Complex analysis is used in analog electronic design. Filters are characterized by singularities of a complex transfer function. Impedance is modeled as a complex value in AC circuits such as audio amplifiers. The wave function of quantum mechanics and quantum field theory is complex-valued.

When to use complex line integral of a function?

Complex line integral. If the parametrization is continuously differentiable, the line integral can be evaluated as an integral of a function of a real variable: When is a closed curve, that is, its initial and final points coincide, the notation is often used for the line integral of f along .

How do you find the value of the line integral?

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).

What is the line integral of a scalar field?

Line integral of a scalar field. The line integral over a scalar field f can be thought of as the area under the curve C along a surface z = f(x,y), described by the field. The trajectory of a particle (in red) along a curve inside a vector field.

Why is the line integral of a vector field path independent?

For this reason, a line integral of a conservative vector field is called path independent . The line integral has many uses in physics. For example, the work done on a particle traveling on a curve C inside a force field represented as a vector field F is the line integral of F on C.