# What does it mean that the gradient is perpendicular/paralell to a vector?

1. Oct 20, 2011

### crocomut

For a solenoidal velocity field [ tex ] \nabla \cdot \mathbf{u} [ /tex ] which means that [ tex ] \nabla [/tex ] is perpendicular to [ tex ] \mathbf{u} [ /tex ].

Similarly, for an irrotational velocity field [ tex ] \nabla \times \mathbf{u} [ /tex ] which means that [ tex ] \nabla [/tex ] is parallel to [ tex ] \mathbf{u} [ /tex ].

So what exactly does it mean physically to have a gradient (of nothing) parallel/perpendicular to a vector?

PS - whats up with latex not working?

Last edited: Oct 20, 2011
2. Oct 20, 2011

### MisterX

Don't put spaces in square brackets for tags. I fixed this issue in the quote above.

What you've posted doesn't make much sense. There seems to be many problems with it. For a solenoidal velocity field $\mathbf{u}$, $\nabla \cdot \mathbf{u}$ is the divergence of u.

$\nabla$ is known as http://mathworld.wolfram.com/Del.html" [Broken], and not "a gradient". It may be thought of like a vector of differential operators.

Last edited by a moderator: May 5, 2017
3. Oct 20, 2011

### crocomut

Re: What does it mean that del is perpendicular/paralell to a vector?

Sorry, let me correct and ask again:

For a solenoidal velocity field $$\nabla \cdot \mathbf{u} = 0$$ which means that $$\nabla$$ is perpendicular to $$\mathbf{u}$$.

Similarly, for an irrotational velocity field $$\nabla \times \mathbf{u} = \mathbf{0}$$ which means that $$\nabla$$ is parallel to $$\mathbf{u}$$.

So what exactly does it mean physically to have $$\nabla$$ parallel/perpendicular to the velocity vector?

4. Oct 20, 2011

### MisterX

Del may be thought of as a vector of operators. Claiming that del is perpendicular to a vector in ℝ3 makes no sense, like assigning a real number value to the plus sign.

5. Oct 20, 2011

### crocomut

Your answer is exactly what I was thinking but, as you can see from http://i.imgur.com/VmbKS.jpg"in my hydrodynamics lecture, it is not what my professor claims. Hence the confusion.

Last edited by a moderator: Apr 26, 2017