Gradient of a Vector Function in Other Co-ordinate Systems

In summary, the conversation is about trying to find the gradient of a vector function in polar and spherical coordinates. The speaker is familiar with the gradient in cartesian coordinates but is having trouble extending this concept to other coordinate systems. They are seeking help and any insight on how to find the gradient in these other coordinate systems.
  • #1
DylanB
52
0

Homework Statement


I am trying to figure out how to take the gradient of a vector function in polar and spherical co-ordinates.


Homework Equations





The Attempt at a Solution


I am aware of how the gradient of a vector function in cartesian co-ords looks, simply the second order tensor


[tex]
(\boldsymol{\nabla}\mathbf F)_{ij} = \frac{\partial F_i(\boldsymbol x)}{\partial x_j}[/tex]


I am having trouble extending this idea to polar and spherical co-ords. The del operator is easy enough to derive in different co-ordinates but finding the second order tensor I am having difficulties.
 
Last edited:
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  • #2
Any luck figuring out how to take a gradient of a vector field in spherical coordinates? I am also stumped on this and would appreciate any insight you have. Thanks!
 

1. What is a vector function?

A vector function is a mathematical function that maps a set of inputs to a set of outputs, where each input and output is a vector. It can also be described as a function that takes in a vector as an input and produces another vector as an output.

2. What is the gradient of a vector function?

The gradient of a vector function is a vector that describes the rate of change of the function with respect to each input variable. It is a multi-dimensional generalization of the concept of a derivative in one dimension.

3. How is the gradient of a vector function calculated in other coordinate systems?

In other coordinate systems, the gradient of a vector function is calculated by taking the partial derivatives of the function with respect to each input variable and then expressing them in terms of the new coordinate system. This can be done using transformation matrices or using the chain rule.

4. Why is the gradient of a vector function important?

The gradient of a vector function is important because it provides information about the direction and magnitude of the function's rate of change at any given point. It is useful in many fields such as physics, engineering, and economics for solving optimization problems and understanding the behavior of systems.

5. Can the gradient of a vector function be negative?

Yes, the gradient of a vector function can be negative. The magnitude of the gradient represents the rate of change, while the direction of the gradient indicates the direction in which the function is changing. Therefore, the gradient can be negative if the function is decreasing in a certain direction.

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