Understanding the del operator in electromagnetics

[yungman]

Gradient is the derivative of a scalar field. Divergence and Curl are both derivative of a vector field.

1) Are "scalar field" and "vector field" imply it is spatial dependent...ie they are function of any single point in space...ie the value of scalar and vector field is different at every single point in space if it is not a constant field.

2) Does this mean all three ( Gradient, Div, Curl ) are point form?...ie each point give different result.

3) Is "point form" means the function is different at every single individual point in space?

4) In electromagnetics, there are $\nabla'$ that represent operation respect to source point and $\nabla$ represent operation respect to field point of interest. I am confuse with this. Can anyone explain this?I just want to have a clearer understanding of the del operator.

Thanks

Alan

 

[mathman]

From a mathematical point of view, the answers to your first three questions are all yes. I'll leave your last question to a physicist.

 

[yungman]

From a mathematical point of view, the answers to your first three questions are all yes. I'll leave your last question to a physicist.

Thanks for clearing these up.

Alan

 

[HallsofIvy]

Be careful. When saying that something is a function of space means that it can have a different value at each point. It does not follow that it cannot have the same value at two distinct points nor does it mean the function cannot be a "constant" function- the same at all points.

 

[jasonRF]

4) In electromagnetics, there are $\nabla'$ that represent operation respect to source point and $\nabla$ represent operation respect to field point of interest. I am confuse with this. Can anyone explain this?

Essentially, in this case there is a function $$g$$ of both $$\mathbf{r} = (x,y,z)$$ and $$\mathbf{r'}=(x',y',z')$$. In other words

$$g(\mathbf{r,r'}) = g(x,y,z,x',y',z')$$.

Here is what the notation you asked about means:

$$\nabla g(\mathbf{r, r'}) = \hat{x} \frac{\partial g}{\partial x} + \hat{y} \frac{\partial g}{\partial y} + \hat{z} \frac{\partial g}{\partial z}$$

$$\nabla' g(\mathbf{r, r'}) = \hat{x} \frac{\partial g}{\partial x'} + \hat{y} \frac{\partial g}{\partial y'} + \hat{z} \frac{\partial g}{\partial z'}$$

Yes, the primed coordinates are very often used for source locations.

Similar kinds of notation can be used as well in other contexts. For example, in kinetic theory we see functions of position, velocity and time, $$f(\mathbf{r,v},t)$$. Here $$\mathbf{v} = (v_x, v_y, v_z)$$. In this case,

$$\nabla f(\mathbf{r, v},t) = \hat{x} \frac{\partial f}{\partial x} + \hat{y} \frac{\partial f}{\partial y} + \hat{z} \frac{\partial f}{\partial z}$$

$$\nabla_{\mathbf{v}} f(\mathbf{r, v},t) = \hat{x} \frac{\partial f}{\partial v_x} + \hat{y} \frac{\partial f}{\partial v_y} + \hat{z} \frac{\partial f}{\partial v_z}$$


Good luck,

Jason