Can you find the gradient of a vector?

[anban]

Homework Statement

I know you can find the gradient of a scalar using partial derivatives. Does it make sense to find the gradient of a vector, however?

A homework problem of mine asks to find the gradient of a vector. I'm starting to think it's a trick question...

Homework Equations

∇ dot V = the divergence of V
∇ cross V = the curl of V

The Attempt at a Solution

The equations above lead me believe that it doesn't make sense to take the gradient of a vector , but the gradient operator can be used in combination with a dot product or cross product to give similar information about the way a function behaves (divergence and curl). So, perhaps divergence and curl are like the vector version of a gradient?

 

[FeynmanIsCool]

V∇ !
Its not a trick question!

 

[FeynmanIsCool]

Sorry, my above comment makes no sense, that was the gradient of a scalar which you mentioned, I read too fast! I'm curious too...maybe someone more math minded can help?...

 

[Dick]

You can make the gradient of vector make some sense if you know what a tensor is. If not, then you may have misinterpreted the question. What is it?

 

[anban]

The specific question is "You are given some vector function V(x,y,z). Can the gradient operation be operated on V? If so, how would you interpret the result?" Very vague, I know.

I have not yet learned what a tensor is. My teacher is definitely one to give tricky questions, so I'm sort of just thinking this is a way to teach us that the gradient operation is used on scalars, and the divergence and curl operations are used on vectors. Is this correct? The way I make sense of this in my head is with an analogy like divergence & curl : vector as gradient: scalar. What do you think?

 

[Dick]

The specific question is "You are given some vector function V(x,y,z). Can the gradient operation be operated on V? If so, how would you interpret the result?" Very vague, I know.

I have not yet learned what a tensor is. My teacher is definitely one to give tricky questions, so I'm sort of just thinking this is a way to teach us that the gradient operation is used on scalars, and the divergence and curl operations are used on vectors. Is this correct? The way I make sense of this in my head is with an analogy like divergence & curl : vector as gradient: scalar. What do you think?

Ok, so it's just a thinking question. Gradient gives you a vector from a scalar. If you want to take the gradient of a vector you might think about taking the gradient of each component giving you a matrix.

 

[Ray Vickson]

Homework Statement

I know you can find the gradient of a scalar using partial derivatives. Does it make sense to find the gradient of a vector, however?

A homework problem of mine asks to find the gradient of a vector. I'm starting to think it's a trick question...

Homework Equations

∇ dot V = the divergence of V
∇ cross V = the curl of V

The Attempt at a Solution

The equations above lead me believe that it doesn't make sense to take the gradient of a vector , but the gradient operator can be used in combination with a dot product or cross product to give similar information about the way a function behaves (divergence and curl). So, perhaps divergence and curl are like the vector version of a gradient?

One way to think about the gradient is as a "linearization factor", so if v(x,y,z) is a scalar function we have
$$v(x+h_x, y+h_y z+h_z) = v(x,y,z) + <A,h> + O(|h|^2),$$ where A is a vector, h = (h_x,h_y,h_z) is a vector and <.,.> denotes the inner product. If that holds for all h, we must have A = grad v. Can you think of a similar representation when v is a vector?