Does the gradiant change when stretching coordinates?

In summary, the question is whether a gradient in Cartesian coordinate system remains the same when the coordinates are stretched by a factor of 2. The response suggests that the gradient should remain the same as it is the rate of change of a function. It also suggests testing this by substituting 2x for x in a known problem. Additionally, the topic of converting units from m/s to mph is brought up.
  • #1
physicsfreak88
8
0

Homework Statement



Well, if you have given a gradiant in cartisian coordinate system? what happened to the gradiant if we stretched the coordinates by factor of 2?

Homework Equations

The Attempt at a Solution


I think gradiant should be the same as it's the rate of change of some function.
 
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  • #2
You could test this yourself by taking a known problem where you have the gradient and subbing in 2x for x and see how it flows through the differentiations to get the gradient.
 
  • #3
physicsfreak88 said:

Homework Statement



Well, if you have given a gradiant in cartisian coordinate system? what happened to the gradiant if we stretched the coordinates by factor of 2?

Homework Equations

The Attempt at a Solution


I think gradiant should be the same as it's the rate of change of some function.

If a car is traveling at x m/s, is it traveling at x mph?
 

1. How does stretching coordinates affect the gradient of a function?

Stretching coordinates can change the gradient of a function by altering the scale and shape of the coordinate system. This can result in a steeper or shallower gradient, depending on the direction and amount of stretching.

2. Does stretching coordinates always change the gradient of a function?

No, stretching coordinates does not always change the gradient of a function. If the stretching is done equally in all directions, the gradient will remain the same. Only if the stretching is done asymmetrically or in a specific direction will the gradient be affected.

3. Can stretching coordinates make the gradient of a function disappear?

No, stretching coordinates cannot make the gradient of a function disappear. The gradient is a fundamental property of a function and will always exist, regardless of any stretching or transformation of coordinates.

4. How can I calculate the new gradient after stretching coordinates?

The new gradient after stretching coordinates can be calculated by using the chain rule of differentiation. This involves taking the derivative of the original function with respect to the stretched coordinates and then multiplying it by the stretching factor.

5. Is there a limit to how much stretching can affect the gradient of a function?

Yes, there is a limit to how much stretching can affect the gradient of a function. If the stretching is too extreme, it can result in a highly distorted coordinate system, making it difficult to accurately calculate the gradient. In such cases, it may be necessary to use alternative methods, such as numerical differentiation, to determine the gradient.

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