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

[EngageEngage]

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.

 

[mighty2000]

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!