Connection Between Vector Fields & Potential Functions

In summary, the question is about the existence of a potential function for a vector field when the curl is equal to 0. This is not always true, as the field must also be defined on a simply-connected open subset of three-dimensional space. Mathematically, this can be shown through Helmholtz decomposition. Intuitively, a nonzero curl implies a non-conservative field while a zero curl corresponds to a conservative field, which can be represented by a scalar potential function. This is because a conservative field is path-independent, allowing for the conservation of energy. In a simply-connected region, this can be proved using Stokes' theorem. However, this is not always the case in every physical circumstance.
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Hi, upon studying vector calculus and more precisely about the curl I stumbled upon a question : why is it that there is always a potential function of a vector field when the curl of this vector field is equal to 0?
 
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  • #2
Is this a homework question? If so, what have you already tried? You have to show your work. If you need a hint, then consider Stokes' theorem ##\oint F\cdot dr = \int (\nabla \times V)\cdot dA##. Using this you can very easily show that there exists a ##\varphi## such that ##V = \nabla \varphi##; try to show it.

Just for the sake of being complete, I should warn you that this is not always true; ##\nabla \times V = 0## only implies ##V = \nabla \varphi## if ##V## is defined on an open subset of ##\mathbb{R}^{3}## that is simply-connected i.e. every path in the open subset can be continuously deformed to a point. If this doesn't hold then ##\nabla \times V = 0## will not imply that ##V = \nabla \varphi##.
 
  • #3
There are a number of ways to answer this question. Mathematically, you can [pretty much] always take a vector field and separate it into a part with zero curl and a part with zero divergence--this is called Helmholtz decomposition. In other words, given a vector field [itex]\mathbf{F}[/itex], we can write it as: [tex]\mathbf{F}=-\nabla\phi+\nabla\times\mathbf{A}[/tex]
Where the first term [itex]-\nabla\phi[/itex] has zero curl by to the identity Curl(Grad(f))=0 for any scalar field f, and the second term has zero divergence by the identity Div(Curl(v))=0 for any vector field v. The second term can be chosen to represent only the curl of the field, so if we set it to zero, then we have [itex]\mathbf{F}=-\nabla\phi[/itex]. This is a very mathematical way to answer your question and it is not that intuitive.

A more intuitive way might be to think about what a curl means physically. Let's assume there is a nonzero curl in a vector field, let's call it an electric field. This implies that if you sit a charge down at rest in a region with nonzero curl, the field could force it to start going in a big loop and return to its original spot, arriving back at the spot with a nonzero velocity (and it could continue looping, picking up more and more velocity each loop.). In other words, the field returned the particle to its original spot but added some kinetic energy. In this sense the field does not conserve kinetic energy, so we call it a "non-conservative field".

On the other hand, if we assume a particle's gain in kinetic energy between any two points is independent of the path it were to take between those points, then a particle could never traverse a closed loop and pick up kinetic energy (since that loop is equivalent to having stood still the whole time, since the particular path doesn't matter--just the endpoints.) This is the case that corresponds to zero curl. Because the kinetic energy gained between any two points is independent of path, we could totally describe this kind of field by picking a reference point called "0 energy" and then labeling all other points with the value of the kinetic energy that the particle would have gained or lost going from the reference point to each other point. Thus we can think of such a path-independent field as one which does conserve energy, or a "conservative field". This labeling of all points by a scalar energy is basically exactly what the potential field [itex]\phi(\mathbf{x})[/itex] is, so we have constructed our representation of a conservative field with a scalar field.

Wikipedia has a nice discussion about your question: http://en.wikipedia.org/wiki/Conservative_field
 
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  • #4
(Actually, it's not strictly true: Every conservative field is irrotational (curl vanishes), since the curl of the gradient of an arbitrary vector field is zero, but every irrotational field is conservative only in a simply connected domain. It does, however, hold in most physical circumstances, for example.)

In simply connected region, it can be proved like this: A potential function exists iff the line integral of the field over an arbitrary closed curve vanishes (use fundamental theorem of line integrals), ie, the field is path-independent (also used as the definition of a conservative field). You can then write the said integral, consider a surface that has the curve as its boundary and use Stokes' theorem to show that the integral vanishes if the curl is zero to prove this.

I seriously should stop leaving tabs open for like 30 minutes, coming back to them and posting a reply without refreshing the page.
 
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1. What is the relationship between vector fields and potential functions?

Vector fields and potential functions are closely related in that a vector field can be derived from a potential function and vice versa. A potential function is a scalar function that describes the potential energy of a particle at any given point in space. A vector field, on the other hand, is a function that assigns a vector to every point in space. The gradient of a potential function is equal to the vector field, and the potential function can be obtained by integrating the vector field.

2. How are vector fields and potential functions used in physics?

Vector fields and potential functions are commonly used in physics to describe the forces acting on a particle or object in a given space. In classical mechanics, the vector field represents the force acting on a particle, while the potential function represents the potential energy of the particle. These concepts are also used in electromagnetism, where the vector field represents the electric or magnetic field, and the potential function represents the electric or magnetic potential.

3. Can all vector fields be derived from a potential function?

No, not all vector fields can be derived from a potential function. A vector field is said to be conservative if it can be derived from a potential function. However, some vector fields, such as those with closed loops, are non-conservative and cannot be derived from a potential function. These non-conservative vector fields do not have a unique potential function, and their line integrals depend on the path taken.

4. What is the significance of a vector field being conservative?

A vector field being conservative means that the work done by the field on a particle moving through it is path-independent. This means that the path taken by the particle does not affect the work done by the field. In other words, the value of the line integral of a conservative vector field only depends on the endpoints of the path. This property is essential in physics as it simplifies calculations and allows for easier analysis of systems.

5. How are vector fields and potential functions visualized?

Vector fields and potential functions can be visualized using vector field plots and contour plots, respectively. Vector field plots use arrows to represent the direction and magnitude of the vector at each point in space, while contour plots use curves to represent the potential function's values at different points. These visualizations can help in understanding the behavior of the vector field and potential function in a given space and aid in solving problems in physics.

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