Intuitive / self-apparent derivation of gradient in curvilinear coords

In summary, the conversation mainly revolves around finding a clear and intuitive explanation for obtaining the gradient in polar, cylindrical, and curvilinear coordinates. The person is looking for a method that can help students remember, derive, and understand the concept of gradient in different coordinate systems. They mention trying the directional derivative approach, but it has not been successful in conveying the concept to students. The focus should be on understanding the difference between what is happening and how it is described using coordinates, as the choice of coordinate system does not affect the gradient. The person also mentions having a vague recollection of an intuitive method but is unable to recall it.
  • #1
raxAdaam
32
0
Hi there -

I'm looking for a clear and intuitive explanation of how one obtains the gradient in polar / cylindrical / curvilinear coords.

I do a lot of tutoring, but am finding that the method I've been using (basically chain rule + nature of directional derivative) just doesn't roll with students: the directional derivative approach (i.e. defining the unit tangent vector to the curve) seems really unintuitive to students - either because they've not really seen it or because they've not internalized it.

I don't need a rigorous method that applies to any imaginable coord. system, just something to help them a) remember b) derive and c) understand the gradient (and eventually curl, div etc.) in different coordinate systems. I have some vague recollection of seeing something pretty intuitive a long time ago, but it has slipped my memory and I just use the above described method personally. Everything I've found online so far has been a little too vague about details and unclear with its notation, so now I'm here!

Thanks in advance for the help!



Rax
 
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  • #2
I would stress the difference between what is happening and its description via coordinates. The choice of the coordinate system is only a choice of description, not a choice which affects the gradient. Try to make an image.
 

FAQ: Intuitive / self-apparent derivation of gradient in curvilinear coords

What is a gradient in curvilinear coordinates?

A gradient in curvilinear coordinates is a vector that represents the rate of change of a function in a specific direction in a curved coordinate system.

What is the significance of the gradient in curvilinear coordinates?

The gradient in curvilinear coordinates is important because it helps us understand the change of a function in a curved coordinate system, which is often used to describe physical systems in science and engineering.

How is the gradient calculated in curvilinear coordinates?

The gradient in curvilinear coordinates is calculated using the partial derivative of the function with respect to each coordinate, multiplied by the corresponding unit vector in that direction.

Why is there a need for a different method of calculating the gradient in curvilinear coordinates?

In a curved coordinate system, the traditional method of calculating the gradient using the simple derivative does not work, as the direction of change is constantly changing due to the curvature of the coordinates. Therefore, a new method using partial derivatives is needed.

Is the intuitive / self-apparent derivation of gradient in curvilinear coordinates widely used in scientific research?

Yes, the intuitive / self-apparent derivation of gradient in curvilinear coordinates is widely used in scientific research, as it provides a clear understanding of the change of a function in a curved coordinate system and is essential in many fields of science and engineering.

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