Gradient in general co-ordinate system

In summary, the discussion revolves around the definition of the gradient in a general co-ordinate system. While the Wikipedia page defines it as a differential operator, the book by Gonzalez and Stuart presents a different definition using arbitrary frames. The question arises on how to apply this notation to show the presence of a 1/r term in the gradient in cylindrical coordinates. It is clarified that the ei in the book's definition are not necessarily unit vectors and can be considered as the Graßmann product of differential forms.
  • #1
sebb1e
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I know that for a general co-ordinate system, the gradient can be expressed as it is at the bottom of this page:

http://en.wikipedia.org/wiki/Orthogonal_coordinates#Differential_operators_in_three_dimensions

However, the book I am working from (A First Course in Continuum Mechanics by Gonzalez and Stuart) defines it as:

"Let {ei} be an arbitrary frame. Then grad phi(x)= (partial d phi by d xi)(x) ei where (x1,x2,x3) are the coordinates of x in ei"

I don't understand how this relates to the actual definition, for example given the definition in my book, how do I show that the gradient in cylindricals contains a 1/r in front of the unit theta term?

I presume that these ei are completely general so they are not unit vectors?

I need to understand this notation properly so I can apply it to deformation gradients etc
 
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  • #2
Yes, non Cartesian coordinates are not necessarily unit vectors, especially if one denotes the distance to the origin. The trick is to consider them as the Graßmann product of differential forms: ##dx\wedge dy \wedge dz##. From there you can substitute ##x,y,z## and recalculate the product.
 

1. What is gradient in general co-ordinate system?

The gradient in general co-ordinate system is a mathematical concept that describes the rate of change of a function with respect to its variables. It is represented by a vector that points in the direction of the steepest increase of the function.

2. How is gradient calculated in general co-ordinate system?

The gradient in general co-ordinate system is calculated using partial derivatives. The partial derivative of a function with respect to a particular variable is found by treating all other variables as constants and differentiating the function with respect to the chosen variable.

3. What is the physical interpretation of gradient in general co-ordinate system?

The gradient in general co-ordinate system has a physical interpretation as a measure of the direction and magnitude of the flow of a physical quantity. For example, in a temperature gradient, the direction of the gradient points towards the direction of increasing temperature and its magnitude indicates the rate of change of temperature.

4. How is gradient used in real-world applications?

The gradient in general co-ordinate system is used in various real-world applications, such as in physics, engineering, and economics. It is used to model and analyze physical systems, optimize processes, and make predictions based on the rate of change of a particular quantity.

5. What is the relationship between gradient and slope?

The gradient in general co-ordinate system is closely related to the concept of slope in one-dimensional space. In one-dimensional space, the gradient is equivalent to the slope of a tangent line to the graph of a function. However, in higher dimensions, the gradient is a vector that points in the direction of steepest increase of the function, while the slope is a scalar quantity that describes the steepness of the function in a particular direction.

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