# Solution to Peculiar Problem: Can Closed Integral be Evaluated?

• Kolahal Bhattacharya
In summary, The conversation is about evaluating the closed integral of a vector F and a scalar dS using the divergence theorem or its corollaries. The person asking the question is unsure if it can be evaluated without any special relationship between F and the normal vector n. They also mention that they previously misread the question and solved for int{F.dS} instead of FdS. They speculate that the key to solving this may lie in using the triple product rule and considering dS as a separate vector from n.
Kolahal Bhattacharya

## Homework Statement

Can anyone say if this can at all be evaluated?
closed integral{F dS} using divrgence theorem/any of its corollary?
here F is a vector and dS is a scalar and there is no dot sign between them.

2. Homework Equations

## The Attempt at a Solution

I do not want to evaluate this.I just want to know if it is done.And how?

I don't think so, unless F has some special relation with the normal vector n. Is it a cross product with n, as in a problem you posted before?

OK,there is no cross product.This actually appeared in the exam I appeared yesterday.

I misread the question and worked as int{F.dS}...it's easy.
This observation may yield something-
Note that FdS=F(n dot dS).Using triple product rules, we have nx(FxdS)=F(n dot dS)-dS(n dot F) => FdS=F(n dot dS)=nx(FxdS)+dS(n dot F).

Hmm. Ok. I'll keep thinking about it. But I'm having trouble thinking about dS as a vector separate from n.

## 1. What is a closed integral?

A closed integral, also known as a line integral, is a type of mathematical integral that involves integrating a function along a specific path between two points. It is typically denoted by a curved integral symbol and is used in various fields of mathematics and science to solve problems involving physical quantities such as force, work, and displacement.

## 2. Why is evaluating a closed integral a peculiar problem?

Evaluating a closed integral can be a peculiar problem because it involves integrating a function along a specific path, rather than over a specific range of values as in a regular integral. This requires a different approach and set of techniques, making it a more challenging problem to solve.

## 3. Can all closed integrals be evaluated?

No, not all closed integrals can be evaluated. Some integrals may have no closed-form solution and require numerical methods to approximate their value. Additionally, the complexity of the function being integrated and the path of integration can also affect the evaluability of a closed integral.

## 4. What are some techniques for evaluating closed integrals?

Some common techniques for evaluating closed integrals include using Green's theorem, the fundamental theorem of line integrals, and Cauchy's integral theorem. Other methods such as using trigonometric substitutions or partial fraction decomposition can also be helpful in certain cases.

## 5. How is the solution to a closed integral problem typically presented?

The solution to a closed integral problem is typically presented as a numerical value or an expression involving variables, depending on the method used for evaluation. In some cases, the solution may also be presented graphically as a curve or surface in three-dimensional space.

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