Condition of a vector field F being conservative is curl F = 0,

[Kolahal Bhattacharya]

Main Question or Discussion Point

When we say condition of a vector field F being conservative is curl F=0,does it mean that F=F(r)?.I know normally it does not look so.Please,then site an example where F is not a function of r,but still curl F=0.

 

[dextercioby]

By "curl of F=0" we mean

$$\nabla_{\vec{r}} \times \vec{F}=0$$

for an $\vec{F}=\vec{F}\left(\vec{r}\right)$

If $\vec{F}\neq \vec{F}\left(\vec{r}\right)$ then

$$\nabla_{\vec{r}} \times \vec{F}\equiv 0$$

 

[D H]

When we say condition of a vector field F being conservative is curl F=0,does it mean that F=F(r)?.I know normally it does not look so.Please,then site an example where F is not a function of r,but still curl F=0.

One example: A constant vector field $\vec F(x\hat x + y\hat y + z\hat z) = a\hat x + b\hat y + c\hat z$ has no curl.

The curl of a gradient is necessarily zero:
$$\vec F(\vec x) = \nabla \phi(\vec x)$$

So all you need to do is come up with a scalar function $\phi(\vec x)$ that cannot be expressed as a function of $||\vec x||$.

The constant vector field corresponds to $\phi(\vec x) = ax + by + cz$, where $\vec x = x\hat x + y\hat y + z\hat z$. Then $\nabla \phi(\vec x) = a\hat x + b\hat y + c\hat z$.

 

[Kolahal Bhattacharya]

I thank you both.And I was not interested about constant fields.
However,What about F=F(v) where v=dr/dt
And also if curl F=0 where F=F(t),or,F=F(v) does it mean the field is conservative?

 

[D H]

A picked a constant vector field as a simple counterexample. Any vector field that can be expressed as the gradient of a scalar function has zero curl.

For the latter, $\vec F = \vec F(\vec v), \vec v = d\vec r/dt$, the curl is zero since the partials of $\vec F$ with respect to components of $\vec r$ are zero. Drag in a constant density fluid satisfies these conditions, and is definitely not conservative.

 

[StatMechGuy]

A picked a constant vector field as a simple counterexample. Any vector field that can be expressed as the gradient of a scalar function has zero curl.

For the latter, $\vec F = \vec F(\vec v), \vec v = d\vec r/dt$, the curl is zero since the partials of $\vec F$ with respect to components of $\vec r$ are zero. Drag in a constant density fluid satisfies these conditions, and is definitely not conservative.

Unless you're talking about a viscous fluid and $\mathbf{v} = \mathbf{v}(\mathbf{r}, t)$ is the velocity field. But then things are still more complicated.

Generally velocity/time dependent forcing fields are not conservative. I really dislike it when classes take the perspective that if the curl is zero, then it has to be conservative.

 

[Kolahal Bhattacharya]

DH:If F=F(v) has curl F=0,then what do you mean by this?

Drag in a constant density fluid satisfies these conditions, and is definitely not conservative.

StatMechGuy:I really did not understand:

I really dislike it when classes take the perspective that if the curl is zero, then it has to be conservative

 

[D H]

Generally velocity/time dependent forcing fields are not conservative. I really dislike it when classes take the perspective that if the curl is zero, then it has to be conservative.

I agree. A zero curl simply means the field is irrotational, period.

 

[vanesch]

When we say condition of a vector field F being conservative is curl F=0,does it mean that F=F(r)?.I know normally it does not look so.Please,then site an example where F is not a function of r,but still curl F=0.

A vector field assigns a vector to each point of the "base space": it is a mapping v(p) from the base space M into a vector space V. In the settings where curl and so on make sense, it can be shown that, if curl_p v = 0 over M, AND IF M IS SIMPLY CONNECTED (no "holes"), that there exists a scalar function f(p) over M, such that v(p) = grad f.

Now, nothing stops you from adding extra parameters to this problem. That is, if you consider a "vector field" which is in fact a *family* of vector fields:
v(p,lambda), with p in M, but lambda any other (set of) parameters, such as time or whatever, well the same theorem holds, for each individual member (indicated by lambda) of the family: if curl_p v(p,lambda) = 0 then v(p,lambda) = grad_p f(p,lambda).