Contour integral facts for kids
More specifically, given a complex-valued function and a contour , the contour integral of along is written as or .
Calculating contour integrals with the residue theorem
For a standard contour integral, we can evaluate it by using the residue theorem. This theorem states that
The following examples illustrate how contour integrals can be calculated using the residue theorem.
Multivariable contour integrals
To solve multivariable contour integrals (contour integrals on functions of several variables), such as surface integrals, complex volume integrals and higher order integrals, we must use the divergence theorem. For right now, let be interchangeable with . These will both serve as the divergence of the vector field written as . This theorem states that:
In addition, we also need to evaluate , where is an alternate notation of . The divergence of any dimension can be described as
The following examples illustrate the use of divergence theorem in the calculation of multivariate contour integrals.
Let the vector field be bounded by the following conditions
The corresponding double contour integral would be set up as such:
We now evaluate by setting up the corresponding triple integral:
From this, we can now evaluate the integral as follows:
To evaluate this, we use the divergence theorem as stated before, and evaluate afterwards. Let , then:
From this, we now can evaluate the integral:
Thus, we can evaluate a contour integral of the fourth dimension.
- Cauchy's integral formula
- Line integral
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Contour integral Facts for Kids. Kiddle Encyclopedia.