Need Help with Mathematical Proof? - Introducing Myself & Seeking Assistance

[advphys]

Dear all,
I am new to this forum and this is my first post.
I hope this is the right place to open this topic.

I have a homework for my mathematical methods for physics course and i am kind of stuck into a question:

v=ψ∇β
show that ∇xv is perpendicular to v.

Here v is a vector
x is cross product
∇ is gradient vector operator
ψ and β are non-vectoral functions

I know it seems obvious, but i need mathematical proof which probably contains some identities of triple cross product with del operator.
Like i said, i couldn't figure it out.

Already now, i want to thank you for all your help.

 

[lurflurf]

hint
∇xv=(∇ψ)x(∇β)
which is obvious from
∇x∇β=0

 

[jfgobin]

Hello AdvPhys,a) Use the identity of the curl of a scalar function multiplying a vector function
b) Use the fact that the curl of the gradient of a scalar function is 0
c) How do you prove that two vectors are perpendicular?
d) Conclude.

[Edit: previous post was giving too much info]

 

[advphys]

Thanks for replies.
But still can't figure out how to conclude the last scalar triple product gives 0.

Edit: I got that it is obvious. I have the same vector in both parts.

Again thank you for your help.

 

[blue_raver22]

Hello,

Welcome to the forum! It is great to see students seeking assistance with their mathematical proofs. I understand that you are stuck on a question regarding the perpendicularity of ∇xv and v. This is a common topic in mathematical methods courses and can be quite challenging.

To prove that ∇xv is perpendicular to v, we need to use the properties of the cross product and the gradient operator. Let's start by writing out the definition of the cross product:

a x b = |a| |b| sin(θ) n

where a and b are vectors, θ is the angle between them, and n is the unit vector perpendicular to both a and b. Now, let's apply this to our problem:

∇xv = |∇| |v| sin(θ) n

We can also write v as:

v = |v| cos(θ) u

where u is the unit vector in the direction of v. Now, we can substitute this into the previous equation:

∇xv = |∇| |v| sin(θ) n = |∇| |v| sin(θ) cos(θ) n = |∇| |v| sin(θ) cos(θ) u

Since u is in the direction of v, we can write this as:

∇xv = |∇| |v| sin(θ) cos(θ) v

Now, we know that sin(θ) cos(θ) is equal to 1/2 sin(2θ). So, we can rewrite the equation as:

∇xv = 1/2 |∇| |v| sin(2θ) v

Finally, we can use the identity for the triple cross product:

a x (b x c) = (a · c) b - (a · b) c

Applying this to our equation, we get:

∇xv = 1/2 |∇| |v| [(v · v) ∇ - (v · ∇) v]

Since v · v is equal to |v|^2, we can rewrite the equation as:

∇xv = 1/2 |∇| |v|^3 [∇ - (v · ∇) v]

Now, we know that the dot product of two perpendicular vectors is equal to 0. So