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Hi, I have a question. In my calculus book, I always see the fundamental theorem for line integrals used for line integrals of vector fields, where f=M(x,y)i + N(x,y)j is a vector field.

The fundamental theorem tells me that if a vector field f is a gradient field for some function F, then f is path independent and then given a curve paramaterized by r(t) from a <= t <=b, ∫f°dr = F(r(a)) - F(r(b))

However, what is the analogue of this for line integrals of scalar fields over the arc length of a curve? You know, the ol' ∫f(x,y)ds type of line integrals where f(x,y) is just a regular function of two variables? I know the gradient is a vector field, so clearly f here isn't a gradient, but it could still be exact (meaning there's a function F such that ∂F/∂x = ∂F/∂y). If that's the case, is F a potential function for f, and can I apply the fundamental theorem?

Another issue I have is visualizing the problem. It's simple to visualize the line integral of a scalar field using arc length: it's just the "Area" of the "curtain" of f above the curve. But I can't for the life of me visualize the vector field version ∫Mdx +Ndy. But I know that for the vector field version, ∫f°dr = ∫f°Tds where T is the unit tangent vector so I know they're related by the tangent vector. How can I use the idea of the unit tangent vector to help visualize/understand the line integral of a vector field? The problem is my book just gives examples of physics and work, and to be honest, I just don't know anything about physics. How can I visualize it from a purely geometric perspective?

The fundamental theorem tells me that if a vector field f is a gradient field for some function F, then f is path independent and then given a curve paramaterized by r(t) from a <= t <=b, ∫f°dr = F(r(a)) - F(r(b))

However, what is the analogue of this for line integrals of scalar fields over the arc length of a curve? You know, the ol' ∫f(x,y)ds type of line integrals where f(x,y) is just a regular function of two variables? I know the gradient is a vector field, so clearly f here isn't a gradient, but it could still be exact (meaning there's a function F such that ∂F/∂x = ∂F/∂y). If that's the case, is F a potential function for f, and can I apply the fundamental theorem?

Another issue I have is visualizing the problem. It's simple to visualize the line integral of a scalar field using arc length: it's just the "Area" of the "curtain" of f above the curve. But I can't for the life of me visualize the vector field version ∫Mdx +Ndy. But I know that for the vector field version, ∫f°dr = ∫f°Tds where T is the unit tangent vector so I know they're related by the tangent vector. How can I use the idea of the unit tangent vector to help visualize/understand the line integral of a vector field? The problem is my book just gives examples of physics and work, and to be honest, I just don't know anything about physics. How can I visualize it from a purely geometric perspective?

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