Can you find the gradient of a vector?

In summary, the gradient operation can be performed on a vector, although it may not have the same interpretation as when applied to a scalar. The gradient operator can be combined with dot and cross products to provide information about the behavior of a vector function, similar to how it provides information about a scalar function. However, the exact meaning of the gradient of a vector is still up for debate and may involve concepts like tensors.
  • #1
anban
20
0

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?
 
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  • #2
V∇ !
Its not a trick question!
 
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  • #3
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?...
 
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  • #4
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?
 
  • #5
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?
 
  • #6
anban said:
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.
 
  • #7
anban said:

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
[tex] v(x+h_x, y+h_y z+h_z) = v(x,y,z) + <A,h> + O(|h|^2), [/tex] 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?
 

What is the gradient of a vector?

The gradient of a vector is a mathematical concept that describes the rate of change of a vector field in a given direction. It is a vector itself, with a magnitude and direction that indicates the direction of steepest ascent or descent of the vector field at a specific point.

How is the gradient of a vector calculated?

The gradient of a vector is calculated using partial derivatives. The partial derivative of each component of the vector is taken with respect to each variable in the vector field. These partial derivatives are then combined to form a vector that represents the gradient.

Why is the gradient of a vector important?

The gradient of a vector is important because it provides valuable information about the behavior of a vector field. It can be used to determine the direction of maximum change of the vector field, and can also be used to find the minimum or maximum points of a function.

Can you find the gradient of a vector in any direction?

Yes, the gradient of a vector can be found in any direction. It is a vector itself, so it has both magnitude and direction. The direction of the gradient vector indicates the direction of maximum change of the vector field at a specific point.

How is the gradient of a vector used in real-world applications?

The gradient of a vector is used in a variety of real-world applications, such as physics, engineering, and economics. It is used to analyze and model physical systems, optimize processes, and solve problems in various fields. Examples include predicting weather patterns, optimizing traffic flow, and determining the most efficient route for a delivery truck.

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