Proof: Nabla X (Nabla X a) = Nabla (Nabla · a) - Nabla^2 a

In summary: I think I'm just going to have to commit this to memory.In summary, the conversation discusses how to prove the equation \nabla\times(\nabla\times\vec{a})= \nabla(\nabla\cdot\vec{a})- \nabla^2\vec{a}, where a is a vector point function. The conversation also covers the use of curl, grad, and div in this proof, as well as the application of the Laplacian to vector functions. The main difficulty in the proof lies in expanding the brackets and dealing with different denominators.
  • #1
madmike159
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Homework Statement



Prove that:

[tex]\nabla[/tex]X([tex]\nabla[/tex]Xa) = [tex]\nabla[/tex]([tex]\nabla[/tex][tex]\cdot[/tex]a) - [tex]\nabla^{2}[/tex]a

where a is a vector point function.

(X is the cross product and that dot is a dot product.)

Homework Equations



curl, grad, div

The Attempt at a Solution



I have just done another question of the form curl curl A = grad div A - [tex]\nabla[/tex]^2A

I'm stuck on this one as I don't know what a vector point function is. I tried it with a unit vector, but that just gave me 0 = 0.
 
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  • #2
A "vector point function" is just a "vector field"- a function that defines a vector at each point. It is, in fact, exactly the kind of functions you have been applying [itex]\nabla\times[/itex] or [itex]\nabla\cdot[/itex] to all along.

(By the way, there is no "[text]" command in LaTex (there is a \text but I don't know why you would want that here).)
You want to show that
[tex]\nabla\times(\nabla\times\vec{a})= \nabla(\nabla\cdot\vec{a})- \nabla^2\vec{a}[/tex]

Okay, "just do it"- it's really just manipulation. Let [itex]\vec{a}= f(x,y,z)\vec{i}+ g(x,y,z)\vec{j}+ h(x,y,z)\vec{k}[/itex] and calculate both sides.

Of course,
[tex]\nabla\times\vec{a}= \left|\begin{array}{ccc}\vec{i} & \vec{j} & \vec{k} \\ \frac{\partial}{\partial x} & \vec{\partial x} & \frac{\partial}{\partial y} & \frac{\partial}{\partial z} \\ f(x,y,z) & g(x,y,z) & h(x,y,z)\end{array}\right|[/tex]
[tex]= \left(\frac{\partial h}{\partial y}- \frac{\partial g}{\partial z}\right)\vec{i}+ \left(\frac{\partial f}{\partial z}- \frac{\partial h}{\partial x}\right)\vec{j}+ \left(\frac{\partial g}{\partial x}- \frac{\partial f}{\partial y}\right)\vec{k}[/tex]

The [itex]\nabla(\nabla\cdot \vec{a})[/itex] should be straight forward. [itex]\nabla\cdot\vec{a}[/itex] is a scalar function and then you take grad of that.

But be careful about [itex]\nabla^2\vec{a}[/tex]. Strictly speaking,
[tex]\nabla^2= \frac{\partial^2}{\partial x^2}+ \frac{\partial^2}{\partial y^2}+ \frac{\partial^2}{\partial z^2}[/tex]
the "Laplacian" applies to scalar valued functions. To apply it to a vector function, apply it to each component separately:
[tex]\nabla^2\vec{a}= \left(\frac{\partial^2 f}{\partial x^2}+ \frac{\partial^2 f}{\partial y^2}+ \frac{\partial^2 f}{\partial z}^2\right)\vec{i}[/tex][tex]+ \left(\frac{\partial^2 g}{\partial x^2}+ \frac{\partial^2 g}{\partial y^2}+ \frac{\partial^2 g}{\partial z}^2\right)\vec{j}[/tex][tex]+ \left(\frac{\partial^2 h}{\partial x^2}+ \frac{\partial^2 h}{\partial y^2}+ \frac{\partial^2 h}{\partial z}^2\right)\vec{k}[/tex]
 
  • #3
I'm still having trouble with this (and my lecturer gave a strong hint that this would be in our exam).

I'm also having trouble getting LaTex to work.

I'll use d here (for [tex]\partial[/tex]).

For curl curl a i get

(d/dy (dg/dx - df/dy) - d/dz (df/dz - dh/dx)) i and so on for j and k. I think I have these right. What I'm not sure on is how to expand the brackets.

I have the same problem for grad div a where i get

d/dx (df/dx + dg/dy + dh/dz) i and so on for j and k.

For d/dx x df/dx I think you would get d^2f/dx^2, but I have no idea what to do when the bottoms are different.
 

What is the equation "Proof: Nabla X (Nabla X a) = Nabla (Nabla · a) - Nabla^2 a" used for?

The equation "Proof: Nabla X (Nabla X a) = Nabla (Nabla · a) - Nabla^2 a" is used in vector calculus to describe the relationship between the curl of a vector and the divergence of its gradient. It is often used in electromagnetism and fluid dynamics to understand the behavior of vector fields.

What is Nabla?

Nabla, written as ∇, is a mathematical operator used in vector calculus to represent the gradient, divergence, and curl of a vector field. It is often used to describe the rate of change and direction of a vector field at a given point.

What does the "X" symbol represent in the equation?

The "X" symbol in the equation represents the cross product, also known as the vector product, of two vectors. It is a mathematical operation that results in a vector perpendicular to both of the original vectors and has a magnitude equal to the product of their magnitudes multiplied by the sine of the angle between them.

What is the relationship between the curl and divergence in this equation?

The equation shows that the curl of a vector is equal to the divergence of its gradient minus the Laplacian of the vector itself. This relationship demonstrates the interdependence of these two vector operations and how they are connected in vector calculus.

How is this equation derived?

This equation can be derived using the fundamental theorem of calculus and vector identities. It involves taking the cross product of two gradients, using the product rule, and simplifying the resulting expression. The full derivation can be found in most vector calculus textbooks.

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