Understanding Scalar Fields: Tools for Studying Vector Field Behavior

[Jhenrique]

After watch this video , I understood that for study the behavior of the vector field, just use 2 tools, the line integral and the surface integral, and actually too, the divergence and the curl. In accordance with this, the maxwell's equations are justly the line integral, the surface integral, the divergence and the curl of the eletric and magnetic field.

Ok, let's assume that the eletric and magnetic field would be scalar fields, so, how would be the maxwell's equations? In other words, which are the standard tools for study scalar field?

 

[Matterwave]

Expressed in terms of the potentials: $\vec{E}=-\vec{\nabla}\phi-\frac{\partial \vec{A}}{\partial t}$ and $\vec{B}=\vec{\nabla}\times \vec{A}$

We have the two inhomogenous Maxwell equations:

$$\nabla^2\phi+\frac{\partial}{\partial t}\left(\vec{\nabla}\cdot \vec{A}\right)=\frac{\rho}{\varepsilon_0}$$
$$\Box \vec{A}+\vec{\nabla}\left(\vec{\nabla} \cdot \vec{A}-\frac{1}{c^2}\frac{\partial \phi}{\partial t}\right)=\mu_0 \vec{J}$$

Is this what you wanted? The scalar field $\phi$ is in there.

 

[Jhenrique]

It isn't. Given an any function, we have 2 standard tools for study it: differentiation and integration. Given a vector field, we have 4 standard tools for analyze it: curl, divergence, circulation and flux. So, given a scalar field, which standard tools we have for investigate it?

 

[Matterwave]

The analogous operations (sort of) to what you're saying would be the gradient, and the regular volume integral. But I don't know why you call this "standard tools to investigate". The curl and divergence are just two operations we can take, in 3-dimensional space only for the curl, of a vector field.

 

[Jhenrique]

Yeah, I thought in the gradient and I was waiting that you say me this...

realize that if the curl = ν, divergence = ξ, circulation = Γ and flux = Φ, so the following relanships are true:
$$\nu = \frac{dΓ}{dA}$$ $$\xi = \frac{d\Phi}{dV}$$
In english: the curl in a point is a local/infinitesimal circulation and the divergence in a point is a local/infinitesimal flux. So the circulation global of a region A limited by a curve s, is the integral of each infinitesimal/local circulation inside this curve s $(\Gamma = \int_{A} \nu dA)$; and the global flux of a region V limited by a surface S, is the integral of each infinitesimal/local flux inside this surface S $(\Phi = \int_{V} \xi dV)$.

I think that you know of what I'm talking about...

Happens that I can't make this analogy with the gradient, if the gradient is the analogous of curl/divergence, so which operation is the analogous of circulation/flux for scalar field so that exist a connection between the differential operation (gradient) and the integral operation (like this above)?

 

[Matterwave]

The only one that comes to mind is the Fundamental theorem of calculus:

$$\int_a^b \frac{d f}{dx} dx=f(b)-f(a)$$

The left hand side is a (like a) gradient in 1 dimension.

In higher dimensions this equation has the form:

$$\int_a^b \vec{\nabla}\phi\cdot d\vec{r}=\phi(b)-\phi(a)$$

This is known as the gradient theorem.

 

[Jhenrique]

What I want to say is: exist the divergence (local) and the global divergence (flux), like exist the curl (local) and the global curl (circulation), so, exist some kind of "global gradient" too?

 

[Matterwave]

Yes...that would be the value of the scalar field at the two end points. Right side of my equation.

If the scalar field is potential energy field $U(x,y,z)$, for example, by the analogies you are drawing the "global gradient" would be the change in potential energy between two points $\Delta U$. I gave you the answer in post #6 through the gradient theorem.

I would definitely not CALL this a "global gradient". Just as "global divergence" and "global curl" have no real meaning, "global gradient" has has no real meaning either. Differential operators always act locally, not globally.