- #1
Swapnil
- 459
- 6
What does the subscript [tex](\theta, \phi)[/tex] mean on the laplace operator? i.e.
[tex]{\nabla}^2 V_{(\theta, \phi)}[/tex]
[tex]{\nabla}^2 V_{(\theta, \phi)}[/tex]
Ooops.. I meant to put the subscript on the operator not on the function. Sorry about that.masudr said:I think the variables should be on the operator, and not on the potential. I've seen it most commonly like this:
[tex]\nabla^2_{r'} V(r-r')[/tex]
where the subscript is to remind us that, as CPL.Luke says, that we are differentiating with respect to the dashed variables (or undashed, as it is in your example)
quasar987 said:I've seen [tex]\nabla^2_{xy}[/tex] to mean
[tex]\nabla^2_{xy}=\frac{\partial^2}{\partial x^2}+\frac{\partial^2}{\partial y^2}[/tex]
In Another Notation Thing, "Del^2 V" represents the second derivative of the scalar field V with respect to the position vector.
"Del^2 V" is calculated by taking the Laplacian operator (∇^2) and applying it to the scalar field V. This can also be written as (∇^2)V.
The physical significance of Del^2 V is that it represents the rate of change or curvature of the scalar field V at a specific point in space. It is often used in physics and engineering to calculate forces and potential energy.
Yes, "Del^2 V" is equivalent to the sum of the second partial derivatives of V with respect to the x, y, and z coordinates. This is because the Laplacian operator (∇^2) is defined as the sum of the second partial derivatives.
Yes, "Del^2 V" can be negative. This indicates that the scalar field V is concave downward at that point. In physics, this may represent a region of lower potential energy or a force pulling in the opposite direction of the scalar field's gradient.