Divergence and rotational equal to zero - solutions?

In summary, the conversation discusses finding solutions to equations, specifically the vectorial Laplace equation, and using the curl and divergence of a vector field F to solve it. The conversation also mentions the involvement of a scalar field phi in the equation.
  • #1
Tosh5457
134
28
Hi, I'd like to know the solutions for these equations, and how to arrive at them. Is it possible to derive the general form of F(x,y,z) analytically? I'm still studying linear differential equations so I have no clue on what to do with partial differential equations...

div F = 0
curl F = 0

Thanks :)
 
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  • #3
dextercioby said:
Doesn't this lead to Laplace equation ?

Yes, it leads to the vectorial Laplace equation (each component's laplacian is 0).
 
  • #4
Vectorial ? No, scalar, take curl F=0. Then F = grad phi. Phi is a scalar. Phi will be involved in a scalar equation.
 
  • #5


Hi there,

Thank you for your question. The equations you have mentioned - divergence and rotational equal to zero - are known as the continuity and the solenoidal equations, respectively. These equations are commonly used in the study of fluid mechanics and electromagnetism.

To answer your question, the solutions for these equations depend on the specific form of the function F(x,y,z). In general, it is possible to derive the general form of F(x,y,z) analytically, but it may require advanced mathematical techniques such as Fourier transforms or Green's functions.

For the divergence equation, the general solution is given by the potential function, which can be derived by solving the Laplace equation. This potential function satisfies the condition that the divergence of F is equal to zero.

For the rotational equation, the general solution is given by the vector potential, which can be derived by solving the Helmholtz equation. This vector potential satisfies the condition that the curl of F is equal to zero.

If you are studying linear differential equations, I would recommend starting with simpler equations and gradually building up to partial differential equations. This will help you develop the necessary mathematical skills and techniques to solve more complex equations.

I hope this helps answer your question. Keep studying and exploring, and good luck with your studies!
 

FAQ: Divergence and rotational equal to zero - solutions?

What does it mean for divergence and rotational to be equal to zero?

When divergence and rotational are equal to zero, it means that a vector field is both incompressible (divergence=0) and irrotational (rotational=0). In other words, the vector field has no sources or sinks and no swirling motion.

What are the solutions to divergence and rotational equal to zero?

The solutions to divergence and rotational equal to zero depend on the specific vector field being considered. Generally, these solutions involve finding a potential function, which describes the vector field in terms of a scalar function.

How is divergence and rotational equal to zero used in physics?

In physics, divergence and rotational equal to zero are used to describe the behavior of fluids. In particular, they are used in fluid dynamics to understand the flow of incompressible and irrotational fluids, such as water or air.

Can a vector field have both divergence and rotational equal to zero?

Yes, a vector field can have both divergence and rotational equal to zero. This means that the vector field is both incompressible and irrotational, and is often referred to as a "solenoidal" vector field.

How is divergence and rotational equal to zero related to conservative forces?

Divergence and rotational equal to zero are related to conservative forces through the gradient of a potential function. When a vector field has both divergence and rotational equal to zero, it can be described by a potential function, and the gradient of this function gives the vector field's conservative force.

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