Understanding Nabla and its Derivatives in 3D Systems

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The Nabla operator, denoted as ∇, generalizes the derivative to multiple dimensions, representing a vector of partial derivatives in 3D systems. It is used to compute curl, divergence, and gradient in electromagnetic theory, indicating how a scalar or vector field changes in space. In one dimension, the Nabla operator simplifies to the standard derivative, but in multiple dimensions, it encompasses derivatives along each axis. The discussion also highlights that partial derivatives represent the slope or rate of change of a function with respect to specific variables. Understanding these concepts is crucial for analyzing vector fields and their behaviors in three-dimensional space.
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Could some one explain what does Nabla operator actually signify ? I understand that the various products with nabla are used to find curl,divergence,gradient in EM, but what does Nabla represent in itself ? A more basic question would be, what does del operator(partial derivative) represent , in a 3d system ?
 
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It's a generalization of the derivative operator to multiple dimensions. That's all.

In one dimension, say along the x-axis, the derivative operator looks like this:

<br /> \frac{d}<br /> {{dx}} = \frac{\partial }<br /> {{\partial x}} = \vec i \frac{\partial }<br /> {{\partial x}}<br />

Since there's only one dimension, the "normal" derivative and partial derivative are the same. Also, there's only way way to take a derivative in one dimension -- along that dimension. Thus, the \vec i[/tex] is implied.<br /> <br /> In multiple dimensions, say x, y and z, it looks like:<br /> <br /> &lt;br /&gt; \nabla = &lt;br /&gt; \vec i \frac{\partial }&lt;br /&gt; {{\partial x}} + \vec j \frac{\partial }&lt;br /&gt; {{\partial y}} + \vec k \frac{\partial }&lt;br /&gt; {{\partial z}}&lt;br /&gt;<br /> <br /> Same thing, just with more dimensions.<br /> <br /> - Warren
 
I would disagree there, chroot.
The one-dimensional analogue of the partial derivative at a point, is the derivative with respect to the elements of some particular sequence converging to that point.

Remember that existence of all partial derivatives does not guarantee differentiability at that point; some similar restriction ought to be provable for "sequential" derivatives in the one-dimensional case.
 
arildno said:
I would disagree there, chroot.
The one-dimensional analogue of the partial derivative at a point, is the derivative with respect to the elements of some particular sequence converging to that point.

So you're saying that del doesn't reduce to the standard one-dimensional derivative d/dx? Can you explain a bit more?

- Warren
 
What I was referring to was, what does a partial derivative represent (in multiple dimensions) like it represents slope in one dimension co-ordinate system ?
 
a gradient vector. the direction in which the function is changing most rapidly, and the magnitude is the amount its changing
 
FunkyDwarf said:
a gradient vector. the direction in which the function is changing most rapidly, and the magnitude is the amount its changing
If it acts on a scalar field.
 
chroot said:
So you're saying that del doesn't reduce to the standard one-dimensional derivative d/dx? Can you explain a bit more?

- Warren
Hmm..what I meant is that a partial derivative is the derivative with respect to some proper subset of arguments in the vicinity of the point.
For example along the x-axis (or some line parallell to that) in 2-D, or along the rationals in the 1-D case.
 
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chroot said:
So you're saying that del doesn't reduce to the standard one-dimensional derivative d/dx? Can you explain a bit more?

- Warren

It's still a vector operator...so I would say that in one dimension:

\nabla = \hat{x}\frac{d}{dx}


I'm sure Arildno's answer was better, but I didn't follow what he was saying.
 
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