Verify Laplace equation in rectangular coordinates.

In summary, the conversation discusses the Laplace equation in rectangular coordinates and how it is applied to a vector. The equation states that the Laplacian of the vector is equal to the sum of the second partial derivatives of each component of the vector with respect to each coordinate. The conversation also mentions the possibility of splitting the Laplacian into parts for easier computation.
  • #1
yungman
5,726
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Just want to verify Laplace equation in rectangular coordinates that:

[tex] \nabla ^2 \vec E = 0 [/tex]

[tex]\Rightarrow\; \nabla^2 \vec E = \left ( \frac {\partial^2}{\partial x^2} +\frac {\partial^2}{\partial y^2} +\frac {\partial^2}{\partial z^2} \right ) ( \hat x E_x +\hat y E_y + \hat z E_z) = 0 [/tex]

[tex]\hbox {(1)}\;\Rightarrow \;\frac {\partial^2 \vec E}{\partial x^2} = 0,\;\frac {\partial^2 \vec E}{\partial y^2} = 0 \;\hbox { and } \frac {\partial^2 \vec E}{\partial z^2} = 0 [/tex]

And

[tex]\hbox {(2)}\; \nabla ^2 \vec E = \nabla^2_{xy}\vec E + \frac {\partial^2 \vec E}{\partial z^2} = 0 \;\hbox { where }\; \nabla^2_{xy}\vec E = \left ( \frac {\partial^2}{\partial x^2} +\frac {\partial^2}{\partial y^2} \right ) ( \hat x E_x +\hat y E_y + \hat z E_z) [/tex]
 
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  • #2
(1) is not right (which implies (2) is not what you're looking for). Are you sure you don't mean

[tex]\nabla^2 \phi = 0[/tex]

as this is the typical potential you work with in classical physics.

If you mean what you wrote, what that tells you is that each component of this vector satisfies Laplace's equation; that is,

[tex]{{\partial^2 E_x}\over{\partial x^2}} + {{\partial^2 E_x}\over{\partial y^2}} + {{\partial^2 E_x}\over{\partial z^2}} = 0[/tex]

and so on for each component
 
  • #3
Pengwuino said:
(1) is not right (which implies (2) is not what you're looking for). Are you sure you don't mean

[tex]\nabla^2 \phi = 0[/tex]

as this is the typical potential you work with in classical physics.

If you mean what you wrote, what that tells you is that each component of this vector satisfies Laplace's equation; that is,

[tex]{{\partial^2 E_x}\over{\partial x^2}} + {{\partial^2 E_x}\over{\partial y^2}} + {{\partial^2 E_x}\over{\partial z^2}} = 0[/tex]

and so on for each component

Thanks for the reply. I don't mean [itex]\nabla^2 \phi[/itex]. I was referring to Laplacian of a vector where:

[tex] \nabla^2 \vec E = \left ( \frac {\partial^2}{\partial x^2} +\frac {\partial^2}{\partial y^2} +\frac {\partial^2}{\partial z^2} \right ) ( \hat x E_x +\hat y E_y + \hat z E_z) = 0 [/tex]

So You say [itex] \nabla^2 \vec E = 0 \;\hbox { don't mean }\;\;\frac {\partial^2 \vec E}{\partial x^2} = 0,\;\frac {\partial^2 \vec E}{\partial y^2} = 0 \;\hbox { and } \frac {\partial^2 \vec E}{\partial z^2} = 0 [/itex]

The second question is totally independent to the first question, even I got the first one wrong, that has no bearing on the second question.

What I meant in the second question is I can split the [itex] \nabla^2_{xyz} \;\hbox { into Laplacian in x and y plus Laplacian in z } \; \Rightarrow\;\nabla ^2_{xyz} \vec E = \nabla^2_{xy}\vec E + \frac {\partial^2 \vec E}{\partial z^2}[/itex]

Thanks

Alan
 
  • #4
I see, so yah, you can split the Laplacian into parts like that. However, again, what i said in the first part still matters. You'll be doing the Laplacian on each individual component, but it can be split up like that.
 
  • #5
Thanks
 

Related to Verify Laplace equation in rectangular coordinates.

1. What is the Laplace equation in rectangular coordinates?

The Laplace equation in rectangular coordinates is a partial differential equation that describes the relationship between the second partial derivatives of a function in three-dimensional space. It is written as ∇²f = 0, where ∇² is the Laplace operator and f is the function.

2. How is the Laplace equation derived in rectangular coordinates?

The Laplace equation can be derived from the heat equation by setting the time derivative to zero. It can also be derived from the Poisson equation by setting the source term to zero.

3. What is the significance of the Laplace equation in physics and engineering?

The Laplace equation appears in many physical and engineering problems, including heat transfer, electrostatics, and fluid dynamics. It is a fundamental equation that helps us understand the behavior of these systems and solve practical problems.

4. How is the Laplace equation verified in rectangular coordinates?

To verify the Laplace equation in rectangular coordinates, we must take the second partial derivatives of a given function with respect to each coordinate (x, y, and z) and then sum them. If the resulting expression is equal to zero, then the function satisfies the Laplace equation.

5. Are there any other coordinate systems in which the Laplace equation can be verified?

Yes, the Laplace equation can also be verified in other coordinate systems, such as cylindrical and spherical coordinates. However, the form of the equation may differ depending on the coordinate system used.

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