# Calculus: Integral along a curve.

• LCSphysicist
In summary: The line integral is equal to ##2\pi##, whereas the double integral of the curl of ##\mathbf{F}## over the enclosed region is equal to ##0##. This shows that the implication does not hold in general.
LCSphysicist
Homework Statement
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Relevant Equations
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Let $F = (P(x,y),Q(x,y))$ a field of vector class 1 in the ring $A={(x,y): 4<x²+y²<9}$ and $x,y$ reals.

I am having trouble to understand why this alternative is wrong:

If $\partial P /\partial y = \partial Q /\partial x$ for every x,y inside A, so $\int_{C} Pdx + Qdy = 0$ for every circumference $\epsilon$ A.

I mean, the condition implies that $Curl F = 0$, and we have that the field of vector is C1, so we don't need to worry with anomalies or problems that could appear as $(...)/0$. In fact, $\int_{C} Pdx + Qdy = \int_{S} \nabla \times F \space ds = \int_{S} 0 \space ds = 0$

Where is my error? If i am wrong, could you give me an example of a vector field that does not satisfies the implication? Where is the error in my demonstration?

The annulus ##A## is not simply connected, therefore ##\mathbf{F}(x,y) = (P(x,y), Q(x,y))^T## is not a conservative vector field even if ##\dfrac{\partial P}{\partial y} = \dfrac{\partial Q}{\partial x}##. For example, consider the line integral of a vector field ##\mathbf{F}(x,y) = \dfrac{1}{x^2+y^2} (-y,x)^T## around a closed curve ##C: \mathbf{r}(t) = a(\cos{t}, \sin{t})^T## for ##t \in [0, 2\pi]## and with ##a \in (2,3)##.

LCSphysicist

## 1. What is the purpose of finding the integral along a curve in calculus?

The integral along a curve is used to find the total area under a curve. This can be useful in various real-world applications, such as calculating the distance traveled by an object with a changing velocity or finding the total volume of a three-dimensional object.

## 2. How is the integral along a curve different from a regular integral?

The integral along a curve is a type of line integral, which involves integrating a function along a specific path or curve. This is different from a regular integral, which involves finding the area under a curve between two points on the x-axis.

## 3. What is the process for finding the integral along a curve?

The process for finding the integral along a curve involves breaking the curve into small segments, finding the integral for each segment, and then adding them together to get the total area under the curve. This can be done using various methods, such as the Riemann sum or the Fundamental Theorem of Calculus.

## 4. Can the integral along a curve be negative?

Yes, the integral along a curve can be negative. This can happen if the curve lies below the x-axis, resulting in a negative area. It is important to pay attention to the orientation of the curve when finding the integral along a curve.

## 5. What are some real-life applications of the integral along a curve?

The integral along a curve has many real-life applications, such as calculating the work done by a varying force, finding the center of mass of an object, and determining the total charge of a moving particle. It is also used in fields such as physics, engineering, and economics.

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