What is the gradient in polar coordinates?

In summary, the Laplacian operator is a second order differential operator and cannot be represented as a first order operator. The three first order differential operators in vector analysis are the gradient, divergence, and curl, which can be written as ##\nabla##, ##\nabla \cdot##, and ##\nabla \times## respectively. However, in curvilinear coordinates, these expressions may differ and cannot be blindly applied as in Cartesian coordinates. The correct expressions for these operators in general coordinates can be found in introductory textbooks on vector analysis.
  • #1
SeM
Hi, on this page: https://en.wikipedia.org/wiki/Laplace_operator#Two_dimensions

the Laplacian is given for polar coordinates, however this is only for the second order derivative, also described as \delta f . Can someone point me to how to represent the first-order Laplacian operator in polar coordinates? I found this at https://math.stackexchange.com/questions/586848/how-to-obtain-the-gradient-in-polar-coordinates, however, I am not sure its correct:

\nabla = \boldsymbol{e_r} \frac{\partial}{\partial r}+ \boldsymbol{e_{\theta}} \frac 1r \frac{\partial}{\partial \theta}.

Thanks!
 
Science news on Phys.org
  • #2
SeM said:
the first-order Laplacian operator
The Laplacian by definition is a second order differential operator. It is defined as the divergence of the gradient of a scalar field. There is no such thing as a first order Laplacian.

There are three first order differential operators you will encounter in basic vector analysis, the gradient, the divergence and the curl. You will often see them written as ##\nabla##, ##\nabla \cdot##, and ##\nabla \times##. However, you should be very careful here. The reason for expressing them like this is that it becomes correct to schematically write ##\nabla = \vec e_1 \partial_1 + \vec e_2 \partial_2 + \vec e_3 \partial_3## and apply the ##\cdot## and ##\times## as scalar and cross products with the derivatives acting on what is to the right. This only works in Cartesian coordinates. In curvilinear coordinates you will have to find the appropriate expressions and they will not involve writing ##\nabla## as a vector and blindly applying the scalar and cross products.

See https://en.wikipedia.org/wiki/Curvilinear_coordinates#Differentiation for the correct expressions in general coordinates. This is also something you should be able to find in any introductory textbook on vector analysis.

Please also be aware that the "A" tag you put on this thread would imply that you have a graduate level understanding of the subject and expect an answer at that level. Based on the content of the question, the thread level is at most an "I" and I have changed it accordingly.
 
  • #3
SeM said:
\nabla = \boldsymbol{e_r} \frac{\partial}{\partial r}+ \boldsymbol{e_{\theta}} \frac 1r \frac{\partial}{\partial \theta}.
By the way, this forum supports ##\LaTeX## but you need to put double-$ signs before and after the code:$$
\nabla = \boldsymbol{e_r} \frac{\partial}{\partial r}+ \boldsymbol{e_{\theta}} \frac 1r \frac{\partial}{\partial \theta}.
$$
 
  • #4
Thanks!
 

Related to What is the gradient in polar coordinates?

1. What is the gradient in polar coordinates?

The gradient in polar coordinates is a mathematical concept used to measure the rate of change of a function in a given direction at a specific point in polar coordinates. It is represented by a vector that points in the direction of the steepest increase of the function and its magnitude represents the rate of change.

2. How is the gradient calculated in polar coordinates?

The gradient in polar coordinates is calculated using the partial derivatives of the function with respect to the radial and angular coordinates. It can be expressed as ∇f = (∂f/∂r)er + (1/r)(∂f/∂θ)eθ, where er and eθ are unit vectors in the radial and angular directions, respectively.

3. What is the significance of the gradient in polar coordinates?

The gradient in polar coordinates is important in many areas of science and engineering, as it helps to determine the direction and magnitude of the maximum change of a function. It is used in fields such as fluid mechanics, electromagnetics, and heat transfer to analyze the behavior of physical systems and optimize their performance.

4. How does the gradient in polar coordinates relate to the gradient in Cartesian coordinates?

The gradient in polar coordinates and Cartesian coordinates are related through a transformation called the Jacobian. This transformation allows us to convert between the two coordinate systems and express the gradient in terms of either coordinate system. However, the gradient vector will have different components and magnitudes in each coordinate system.

5. Can the gradient in polar coordinates be negative?

Yes, the gradient in polar coordinates can be negative. This means that the function is decreasing in the direction of the gradient vector. However, the gradient vector itself is always positive, as it represents the magnitude of the maximum change of the function at a given point in polar coordinates.

Similar threads

  • Advanced Physics Homework Help
Replies
9
Views
2K
Replies
6
Views
6K
Replies
1
Views
2K
Replies
7
Views
4K
  • Special and General Relativity
Replies
14
Views
866
Replies
2
Views
966
  • Classical Physics
Replies
22
Views
2K
  • Advanced Physics Homework Help
Replies
2
Views
280
  • Linear and Abstract Algebra
2
Replies
41
Views
3K
Replies
3
Views
1K
Back
Top