# Why Is the Line Integral of a Parallel Vector Field Zero on a Sphere?

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In summary: Your name]In summary, the conversation discusses the conditions under which the line integral \int\stackrel{C}{} F . dR would be equal to zero. The conversation defines terms such as vector field and position vector, and explains that if the vector field F is everywhere parallel to the position vector R, the line integral will always be zero. This is because the dot product between two perpendicular vectors is always zero.
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## Homework Statement

If the vector field F(x,y,z) is everywhere parallel to R and C is a curve drawn on a sphere with the center at the origin, then:
$$\int$$$$\stackrel{C}{}$$ F . dR = 0
why?

## The Attempt at a Solution

Im not exactly sure if I understand the problem, but this is the explanation i came up with: If R is the position vector of the curve (which lies on a sphere), and vector field F lies parallel to R we know that F will be perpendicular everywhere to C, thus satisfying this condition... is this right? is R the position vector in this case (i'm somewhat confused because the book I'm using is somewhat inconsistent with notation). If someone could please tell me if I'm doing this right that would be greatly appreciated.

Thank you for your question. I am a scientist and I would like to offer some clarification on this problem. First, let me define some terms. A vector field is a mathematical function that assigns a vector to every point in a given space. In this case, F(x,y,z) is a vector field that assigns a vector to every point in the three-dimensional space (x,y,z). The position vector, denoted as R, specifies the position of a point in the space relative to a chosen origin. So, in this case, R is the position vector of the curve C, which lies on a sphere with the center at the origin. Now, let's look at the integral \int\stackrel{C}{} F . dR. This is known as the line integral, and it represents the sum of the dot product between the vector field F and the position vector dR along the curve C.

Now, to answer your question, if the vector field F is everywhere parallel to R, this means that the dot product between F and dR will always be zero, since the dot product of two parallel vectors is equal to the product of their magnitudes. Therefore, the line integral will always be zero, regardless of the curve C or the sphere it lies on. This is because the dot product only takes into account the component of one vector in the direction of the other, and since F and dR are always perpendicular, their dot product will always be zero.

I hope this explanation helps clarify the problem for you. If you have any further questions, please don't hesitate to ask. Keep up the good work in your studies!

## 1. What is a line integral?

A line integral is a mathematical concept that describes the integration of a function along a given curve or line. It is used to calculate the total area under a curve or the total work done along a path.

## 2. How is a line integral different from a regular integral?

A line integral is different from a regular integral in that it involves integrating a function over a specific curve or path, rather than over a specific interval. This means that the limits of integration are determined by the curve or path, rather than by fixed values.

## 3. What are the applications of line integrals?

Line integrals have many applications in physics and engineering, such as calculating work done by a force along a path, finding the mass of a wire with varying density, and determining the flow of a vector field along a curve.

## 4. What is the relationship between line integrals and vector calculus?

Line integrals are closely related to vector calculus, as they involve the integration of vector fields along a path. In vector calculus, line integrals are often used to calculate the circulation and flux of a vector field.

## 5. How can I visualize a line integral?

A line integral can be visualized by imagining a small segment of the curve or path and calculating the contribution of that segment to the total area or work. By adding up all of these small contributions, the line integral can be approximated and visualized as the total area or work along the entire curve or path.

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