Magnitude of the gradient of a constant scalar field

In summary: The dot product is the projection of one vector onto another, so it can change depending on the direction of the two vectors.
  • #1
Gamdschiee
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Hey!

Short definition: A gradient always shows to the highest value of the scalar field. That's why a gradient field is a vector field.

But let's assume a constant scalar field [tex]f(\vec r)[/tex] The gradient of f is perpendicular to this given scalar field f.

My Questions:
1. Why does the gradient points away? I mean yes, it is clear that there isn't any other higher value, so it just points away?

2. Does the magnitude of the gradient represent the alteration of the scalar field f, although the field itself is constant?
 
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  • #2
I don't understand your question. If the scalar field f is constant, the gradient is zero everywhere.
 
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  • #4
Gamdschiee said:
I wondered, because I read this thread: https://www.physicsforums.com/threads/constant-scalar-field.811084/
Isn't that the case that I've described?

That thread talks about a scalar field that is constant on a surface S, meaning that the field has the same value everywhere on the surface S, but varies elsewhere. I understood your OP to say the scalar field was constant everywhere.
 
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  • #5
phyzguy said:
. I understood your OP to say the scalar field was constant everywhere.
That is also how I understood it.
@Gamdschiee can you clarify, do you intend to ask about a scalar function which is constant everywhere or constant on some surface?
 
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  • #6
Ahh! Thank you, I see what I did wrong there. Yes, I meant that the scalar field is only constant on certain surface/line etc. - Sorry!

Let's view page 2 and fig. 1 of this link please: http://www.phys.ufl.edu/~pjh/teaching/phz3113/notes/week5.pdf

So you can say that those closed lines in this figure are the whole scalar field. And it makes sense when the gradient points away, because there is the higher value. Is that right?
 
  • #7
Gamdschiee said:
Ahh! Thank you, I see what I did wrong there. Yes, I meant that the scalar field is only constant on certain surface/line etc. - Sorry!

Let's view page 2 and fig. 1 of this link please: http://www.phys.ufl.edu/~pjh/teaching/phz3113/notes/week5.pdf

So you can say that those closed lines in this figure are the whole scalar field. And it makes sense when the gradient points away, because there is the higher value. Is that right?
Yes, that is correct. You can think of it as a topographical map.
 
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  • #8
I see thanks.
Now it's more clear to me. But how can you in general describe the magnitude of such graditude?
 
  • #9
The magnitude is equal to the rate of change of f in the steepest direction.
 
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  • #10
So basically you can say that a gradient from any location in the gradient field always points to the maximum value of the scalar field. And the magnitude from a certain picked gradient is just the slope at this certain location? The higher the slope the nearer you come to a inflection point.

Is that right?

And what do you exactly mean by "steepest direction"? Hasn't any direction (i.e. gradient) a different slope?
 
  • #11
Gamdschiee said:
So basically you can say that a gradient from any location in the gradient field always points to the maximum value of the scalar field.
Unfortunately, no. It points in the locally steepest direction, but the maximum is a global feature not a local one. If you follow the gradient around it will eventually bring you to the maximum, but not usually in a straight line.
 
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  • #12
Thank you, I think I got that now.

Here an example: http://d2vlcm61l7u1fs.cloudfront.net/media/dd4/dd4ee8d1-733b-4f93-a05f-b6178210dac1/phpxPlrNF.png

1. The maximum should be at C. It looks like a local maximum around the C-spot. Because you don't know how high the value is outside F and A e.g.
2. The longer the vectors the steeper is f at that location, so the f is steepest at F.
3. "The gradients are pointing in the locally steepest direction" - What does that exactly mean to point in the locally steepest direction? Does that mean, that the gradient always points to the nearest higher "slope-value" than its current "slope-value"?
 
  • #13
Gamdschiee said:
"The gradients are pointing in the locally steepest direction" - What does that exactly mean to point in the locally steepest direction?
Sorry, this is difficult to express in words. Imagine that f is a 2D scalar function f(x,y). One way that you could plot f is to draw a surface which is curved in 3D in such a way that the height is equal to the potential, ie z=f(x,y).

If you were a mountain climber on such a surface then at any point you could go uphill, downhill, or along the hill. The gradient tells you which direction is uphill and how steep the hill is at that point. If you walk in any other direction you won’t be climbing as steeply, thus the direction of the gradient is the “locally steepest direction”
 
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  • #14
Dale said:
Unfortunately, no. It points in the locally steepest direction, but the maximum is a global feature not a local one. If you follow the gradient around it will eventually bring you to the maximum, but not usually in a straight line.
There is no guarantee that a local maximum found in this manner will be a global maximum.
 
  • #15
jbriggs444 said:
There is no guarantee that a local maximum found in this manner will be a global maximum.
Yes, that is correct
 
  • #16
phyzguy said:
I don't understand your question. If the scalar field f is constant, the gradient is zero everywhere.
then why isn't the gradient of the dot product of two vectors always zero? Del (A.B) =0??
 
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  • #17
ChuckH said:
then why isn't the gradient of the dot product of two vectors always zero? Del (A.B) =0??
Because the dot product is not always constant. Indeed, why would it be?
 
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Related to Magnitude of the gradient of a constant scalar field

1. What is the definition of magnitude of the gradient of a constant scalar field?

The magnitude of the gradient of a constant scalar field is the measure of the rate of change of the scalar field in all directions. In other words, it represents how quickly the scalar field changes in value as you move in any direction.

2. How is the magnitude of the gradient of a constant scalar field calculated?

The magnitude of the gradient of a constant scalar field is calculated by taking the square root of the sum of the squares of the partial derivatives of the scalar field with respect to each variable. This can be represented as: √(∂f/∂x)^2 + (∂f/∂y)^2 + (∂f/∂z)^2

3. What is the significance of the magnitude of the gradient of a constant scalar field?

The magnitude of the gradient of a constant scalar field is significant because it provides information about the steepness and direction of the scalar field. A higher magnitude indicates a steeper change in the scalar field, while a lower magnitude indicates a more gradual change.

4. How does the magnitude of the gradient of a constant scalar field relate to the concept of a gradient?

The magnitude of the gradient of a constant scalar field is essentially the length of the gradient vector, which represents the direction and rate of change of the scalar field. The larger the magnitude, the larger the gradient vector and the steeper the change in the scalar field.

5. Can the magnitude of the gradient of a constant scalar field be negative?

No, the magnitude of the gradient of a constant scalar field is always a positive value. This is because it represents the absolute value of the gradient vector, which cannot be negative. However, the gradient vector itself can have a negative direction if the scalar field is decreasing in that direction.

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