Finding a vector from the curl of a vector

In summary, the equation ##\nabla\phi=\nabla\times \vec{A}## can be rewritten in component form, but solving for ##\vec{A}## is not straightforward due to the coupling of equations. However, an integral solution similar to the Biot-Savart equation may be used, along with potential boundary conditions, to solve for ##\vec{A}##.
  • #1
user1139
72
8
Consider the following

\begin{equation}
\nabla\phi=\nabla\times \vec{A}.
\end{equation}

Is it possible to find ##\vec{A}## from the above equation and if so, how does one go about doing so?

[Moderator's note: moved from a homework forum.]
 
Last edited by a moderator:
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  • #2
Thomas1 said:
Consider the following

\begin{equation}
\nabla\phi=\nabla\times \vec{A}.
\end{equation}

Is it possible to find ##\vec{A}## from the above equation and if so, how does one go about doing so?

[Moderator's note: moved from a homework forum.]
What do you get if you write this in components?
 
  • #3
Working with Cartesian coordinates, I will be able to equate the respective components on the LHS and RHS. The problem comes when I want to find ##A_x##, ##A_y## and ##A_z## since the equations are now coupled.
 
  • #4
Thomas1 said:
Working with Cartesian coordinates, I will be able to equate the respective components on the LHS and RHS. The problem comes when I want to find ##A_x##, ##A_y## and ##A_z## since the equations are now coupled.
Yes, but it is still a linear equation system.
 
  • #5
Do you know how to solve it? I ran out of ideas.
 
  • #6
You have ##c_1=z-y\, , \,c_2=x-z\, , \,c_3=y-x## which is ##\begin{bmatrix}
0&-1&1\\1&0&-1\\-1&1&0
\end{bmatrix}\cdot \begin{bmatrix}
x\\y\\z\end{bmatrix}=\begin{bmatrix}
c_1\\c_2\\c_3
\end{bmatrix}##
so invert the matrix and solve it. Of course you only get ##x\triangleq\dfrac{\partial A_1}{\partial x_1} ,\ldots##
 
  • #7
The matrix is singular and hence, cannot be inverted.
 
  • #8
Thomas1 said:
The matrix is singular and hence, cannot be inverted.
Then you cannot get ##A.##
 
  • #9
Thomas1 said:
Consider the following

\begin{equation}
\nabla\phi=\nabla\times \vec{A}.
\end{equation}

Is it possible to find ##\vec{A}## from the above equation and if so, how does one go about doing so?

[Moderator's note: moved from a homework forum.]

Let [itex]A = \nabla \times F + \nabla \psi[/itex]. Then [tex]
\nabla \times A = \nabla(\nabla \cdot F) - \nabla^2F = \nabla \phi.[/tex] Adding a gradient to [itex]F[/itex] does not change its curl, which is all we care about, but does change its divergence, so we can assume [itex]\nabla \cdot F = 0[/itex]. That results in [itex]F[/itex] satisfying Poisson's equation [tex]
\nabla^2 F = - \nabla \phi.[/tex] This is three decoupled equations in the cartesian components of [itex]F[/itex]. There is no way to determine [itex]\psi[/itex] since [itex]\nabla^2 \psi = \nabla \cdot A[/itex] is not specified.
 
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  • #10
If there is any hope, I think you would also need some boundary conditions.
 
  • #11
There is an integral solution to ## \nabla \times A =\nabla \phi ##. It is basically a Biot-Savart integral type solution to ## \nabla \times B=\mu_o J ##, to solve for ##B ##. There is also the possibility of a homogeneous solution, (## \nabla \times A=0 ##), so that the solution for ## A ## is not unique by this method.
 
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  • #12
To write out the above Biot-Savart type solution, ## A(x)=\int \frac{\nabla \phi \times (x-x')}{4 \pi |x-x'|^3} \, d^3x' ##, where ##x ## and ## x' ## are 3 dimensional coordinate vectors.
 

FAQ: Finding a vector from the curl of a vector

What is a vector?

A vector is a mathematical quantity that has both magnitude (size) and direction. It is commonly represented by an arrow pointing in the direction of the vector with its length representing the magnitude.

What is the curl of a vector?

The curl of a vector is a mathematical operation that describes the rotation or "circulation" of a vector field. It is a measure of how much the vector field is rotating at a given point.

Why is it important to find a vector from the curl of a vector?

Finding a vector from the curl of a vector allows us to understand the behavior and characteristics of a vector field. It is particularly useful in fields such as fluid dynamics, electromagnetism, and thermodynamics.

How do you find a vector from the curl of a vector?

The process of finding a vector from the curl of a vector involves using the curl operator, which is a mathematical operation that takes the partial derivatives of a vector field. By solving the resulting equations, we can determine the components of the vector at a given point.

Are there any real-world applications of finding a vector from the curl of a vector?

Yes, there are many real-world applications of finding a vector from the curl of a vector. For example, in fluid dynamics, the curl of a velocity field can be used to determine the vorticity, which is important in understanding the behavior of fluids. In electromagnetism, the curl of the magnetic field can be used to determine the strength and direction of the electric current. Additionally, the curl of a vector field is used in many engineering and scientific simulations to model and predict the behavior of complex systems.

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